Infectious Disease Modelling Research Progress

PUBLIC HEALTH IN THE 21ST CENTURY SERIES

INFECTIOUS DISEASE MODELLING RESEARCH PROGRESS

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PUBLIC HEALTH IN THE 21ST CENTURY SERIES Health-Related Quality of Life Erik C. Hoffmann (Editor) 2009. ISBN: 978-1-60741-723-1 Cross Infections: Types, Causes and Prevention Jin Dong and Xun Liang (Editors) 2009. ISBN: 978-1-60741-467-4 Swine Flu and Pig Borne Diseases Viroj Wiwanitkit 2009. ISBN: 978-1-60876-291-0 Family History of Osteoporosis Afrooz Afghani (Editor) 2009. ISBN: 978-1-60876-190-6 Biological Clocks: Effects on Behavior, Health and Outlook Oktav Salvenmoser and Brigitta Meklau (Editors) 2010. ISBN: 978-1-60741-251-9 Infectious Disease Modelling Research Progress Jean Michel Tchuenche and C. Chiyaka (Editors) 2010. ISBN: 978-1-60741-347-9

PUBLIC HEALTH IN THE 21ST CENTURY SERIES

INFECTIOUS DISEASE MODELLING RESEARCH PROGRESS

JEAN MICHEL TCHUENCHE AND

CHRISTINAH CHIYAKA EDITORS

Nova Science Publishers, Inc. New York

Copyright © 2009 by Nova Science Publishers, Inc.

All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Infectious disease modelling research progress / [edited by] Jean Michel Tchuenche ... [et al.]. p. ; cm. Includes bibilographical references. ISBN 978-1-61470-090-6 (eBook) 1. Communicable diseases--Epidemiology. I. Tchuenche, Jean Michel. [DNLM: 1. Communicable Diseases--epidemiology. 2. Epidemiologic Research Design. 3. Models, Theoretical. WA 110 I433 2009] RA651.I54 2009 362.196'9--dc22 2009016556

Published by Nova Science Publishers, Inc. New York

Dedication In memory of C.O.A. Sowunmi (1934-2007)

CONTENTS Preface

ix

Chapter 1

Epidemiology of Corruption and Disease Transmission as a Saturable Interaction: The SIS Case C.O.A. Sowunmi

1

Chapter 2

A Mathematical Analysis of Influenza with Treatment and Vaccination H. Rwezaura, E. Mtisi and J.M. Tchuenche

31

Chapter 3

A Theoretical Assessment of the Effects of Chemoprophylaxis, Treatment and Drug Resistance in TB Individuals Co-infected with HIV/AIDS C.P. Bhunu and W. Garira

85

Chapter 4

When Zombies Attack!: Mathematical Modelling of an Outbreak of Zombie Infection P. Munz, I. Hudea, J. Imad and R.J. Smith

133

Chapter 5

A Review of Mathematical Modelling of the Epidemiology of Malaria C. Chiyaka, Z. Mukandavire, S. Dube, G. Musuka and J.M. Tchuenche

151

Chapter 6

A Mathematical Model of the Within – Vector Dynamics of the Plasmodium Falciparum Protozoan Parasite M. Teboh-Ewungkem, Thomas Yuster and Nathaniel H. Newman

177

Chapter 7

Mycobacterium Tuberculosis Treatment and the Emergence of a Multi-drug Resistant Strain in the Lungs G. Magombedze, W. Garira, E. Mwenje and C. P. Bhunu

197

Chapter 8

Mathematical Modeling for Tumor Growth and Control Strategies Sanjeev Kumar, Deepak Kumar and Rashmi Sharma

229

viii Chapter 9 Index

Contents With-in Host Modelling: Their Complexities and Limitations G. Magombedze

253 261

PREFACE As someone once said, publications in mathematical biology are so numerous that they are becoming themselves an epidemic. Previous collections in this field contain important materials as a basis for modern ground breaking research in disease dynamics, but their focus is either too narrow, or is a collections of conference papers, written for the advanced and experienced research students. This book attempts to complement the gap and is therefore intended to encourage the growing demand for interdisciplinary research which is still at a low level in the developing world where most of the diseases considered are prevalent. Training at the interface of mathematics and biology is increasing today, and virtually any advance in diseases dynamics requires a sophisticated mathematical approach in order to map out the parameters that drive them for control and containment of epidemic outbreaks. The goal of this book is two-fold: To expose students to the usefulness and applicability of mathematical knowledge in designing public health policies, and to introduce them to interdisciplinary research at the frontiers of mathematics and biology. They are meant to provide a glimpse into the diverse world of epidemiological modeling and to invite interested readers to experience, through a selection of epidemic diseases of global concern, the fascinating mathematical techniques at the interface between biology and mathematics. The materials presented herein describe, thoroughly analyze and interpret the dynamics of infectious diseases. The chapters are independent and can be used as basic case studies for any existing text material for upper undergraduate and graduate courses with a variety of audiences. The focus of this volume is essentially on the epidemiology of corruption, tuberculosis, influenza, tumor growth and malaria. The book is organized as follows: The first Chapter analyses the epidemiology of corruption and disease transmission as a saturable interaction (SIS case). Chapter 2 is a robust and in-depth study of an influenza model with treatment and vaccination. In Chapter 3, a theoretical assessment of the effects of chemoprophylaxis, treatment and drug resistance in TB infected individuals co-infected with HIV/AIDS is studied. Modelling an outbreak of zombies is considered in Chapter 4. Zombies are fictional, but it uses movies and popular culture to treat a zombie outbreak like a regular disease outbreak, while Chapter 5 is a review of previous studies on the epidemiology of malaria. A model of the sporogony cycle of Plasmodium falciparum, one of the agents responsible for malaria is studied in Chapter 6. Chapter 7, which is parallel to Chapter 3; deals with the treatment of mycobacterium tuberculosis and the emergence of a multi-drug resistant strain in the lungs. A model for tumor growth and its control is investigated in

x

Jean Michel Tchuenche and C. Chiyaka

Chapter 8. The complexity and limitation of within host dynamics models is briefly commented in Chapter 9. We would like to express our sincere appreciation to the reviewers and the members of the editorial board of Nova Science Publishers, Inc., involved in the oversight of this book. Finally, despite all the support in the production, at the end, the responsibility for the final product is entirely ours. Jean M. Tchuenche Christinah Chiyaka

(Dar es Salaam, TZ) (Bulawayo, Zim) December, 2008.

In: Infectious Disease Modelling Research Progress ISBN 978-1-60741-347-9 c 2009 Nova Science Publishers, Inc. Editors: J.M. Tchuenche, C. Chiyaka, pp. 1-30

Chapter 1

E PIDEMIOLOGY OF C ORRUPTION AND D ISEASE T RANSMISSION AS A S ATURABLE I NTERACTION : T HE SIS C ASE∗ C.O.A. Sowunmi† 6 Are Close, New Bodija, Ibadan 200221, Nigeria

Abstract The first section of this chapter explores corruption, modelled as a non-fatal transmissible infection which divides a community into 4 compartments: Susceptible, Infective, Removed and Resistant. The Removed class has age structure. The corruptionfree steady state, though unstable, can be stabilized by feedback control. The efficiency of the measure depends on a number of parameters, especially the rates at which susceptibles become resistant, and the infective are removed or become resistant. The second part analyses disease transmission as a saturable interaction. The use of the concept of saturable interactions as a framework for modelling disease transmission is explored with two SIS cases. In both cases, it is found that the disease-free steady state is always present whereas the endemic steady state may not be possible. Also, the instability of the disease-free steady state is part of the sufficient conditions for the existence of the endemic steady state. Finally, sufficient conditions for the local asymptotic stability and instability of the steady states are obtained.

Keywords: Corruption, non-fatal communicable disease, equilibria, stability, feedback control, Lyapunov, disease transmission, saturability, Age Structure.

1.

Epidemiology of Corruption

1.1.

Introduction

Official corruption, popularly called corruption, is here defined simply as giving or receiving a bribe. It is a habit that can be acquired and also lost. It is also possible to resist both ∗

Compiled by J.M. Tchuenche and published posthumously with permission from author’s wife, Prof. Bisi Sowunmi. † E-mail address: [email protected], [email protected]

2

C.O.A. Sowunmi

the giving and receiving of a bribe. It is thus similar in some respects to the communicable diseases which have been modelled in numerous publications, yet it differs in other respects which make it interesting to model. Corruption has been as much in the news as HIV/AIDS and should be accorded as much mathematical attention by way of modelling. Corruption divides a community in which it is active into four compartments, namely: (i) Susceptibles i.e., those who have neither taken nor offered a bribe but will take or offer it if sufficiently tempted, (ii) Infectives who have once taken or offered a bribe and will do so again. In particular, they tempt the susceptibles, thereby turning them into infectives, (iii) Resistants, who will no longer succumb to temptation, no matter what, and lastly, (iv) Removed who were once infective but are now rendered incapable of receiving or offering a bribe for a while (they could be in prison, on suspension or whatever). When they cease to be removed, they either go to the Infectives or the Resistant. The removed class has age structure with infinite life span. Everyone is born susceptible. Neither the birth rate nor the death rate is affected by the class to which an individual belongs.

1.2.

Model Equations

Let S(t), I(t), R1 (t), R2 (t) be the respective sizes at time t of the Susceptibles, Infectives, Removed and Resistant. Everyone is born into the susceptible class. The transition rates are non constant, but depend mostly on R1 (t). The rate of entering the removed class depends on I(t), R1 (t) and R2 (t). The endemic is not possible, but the corruption-free state is not likely to be stable, because the condition for stability is quite complicated. The population is assumed open and of total size N (t) at time t. Let the birth and death rates be b(N ) and γ(N ), respectively. The per capita rate a at which susceptibles flow into the resistant class is assumed to be a function of R1 alone. The reason is that the higher the likelihood of being removed, the more a susceptible is likely to become resistant. Thus, a is a monotone increasing function of R1 . However, it is possible that there are some who on their own will become resistant, hence, we assume a(0) > 0. For a similar reason, we assume that c = c(R1 ) and is monotone increasing but c(0) = 0. d is also a function of R1 alone, but is monotone decreasing, while e = e(R1 ) is monotone increasing. Since the removed Z class has age structure, we assume a density ρ, function of t and α ∞

ρ(t, α)dα. ρ(t, 0) is the rate at which infectives are removed. Let

such that R1 (t) =

0

∂h ∂h ∂h < 0, < 0 and > 0. The reason is as follows. ∂I ∂R1 ∂R2 The more infectives there are, the less the chances of their being sent to the R1 -class. The more infectives are sent to the R1 -class, the more wary those in the I-class will be. Finally, the more those in R2 -class, the more the infectives will be removed. ρ(t, 0) = h(I, R1 , R2 ), where

We set S(0) = S0 , I(0) = I0 , ρ(0, α) = ρ0 (α) and R2 (0) = R20 . We assume that the interaction between the S and I classes is saturable and is governed by a function F of S and I, and that F has the usual properties [1].

Epidemiology of Corruption and Disease Transmission as a Saturable Interaction

3

We therefore have the following governing equations: dS dt

= −aR2 − F (S, I) + bN − γS,

dI dt

= F (S, I) + dR1 − cR2 − γI − h(I, R1 , R2 ),

∂ρ ∂ρ + ∂t ∂α

= −(d + e + γ)ρ,

dR2 dt

= eR1 − (γ − a − c)R2

(1.1)

1.3. Model Analysis The model consists of the system of equations (1.1) together with the initial conditions and the assumptions on the parameters stated in subsection 1.2 above. It is not difficult to determine a set of analytical conditions which will ensure global existence and uniqueness of a solution. We shall therefore proceed to investigate the steady states and their stability. Integrating the third equation of model system (1.1) w.r.t. α over [0, ∞) yields dR1 − ρ(t, 0) = −(d + e + γ)R1 . dt Thus,

dR1 = h(I, R1 , R2 ) − (d + e + γ)R1 . dt

(1.2)

Now, dN dt

= bN − γS − ρ(t, 0) + dR1 − γI + ρ(t, 0) − (d + e + γ)R1 + eR1 − γR2 , = mN,

(1.3) where m = b − γ. Rather than considering the mixed system (1.1) of ordinary and partial differential equations, we will consider the equivalent system made of the first, second and fourth equations of (1.1) of ordinary differential equations. It is easily seen that (S, I, R1 , R2 ) = (0, 0, 0, 0) is a solution of the system. This is the trivial steady state. A more interesting steady state, if it exists, would be the corruption-free steady state (S, I, R1 , R2 ) = (S0 , 0, 0, R20 ). Let I = 0 = R1 in the system of equations. Therefore, a(R1 ) = a(0) 6= 0 by assumption. Set a(0) = a0 , b = b(S0 + R20 ) = b0 , say. c(R1 ) = c(0) = 0. γ(S0 + R20 ) = γ0 , say. d(0) = d0 6= 0 since d is monotone decreasing and positive. e(R1 ) = e(0) = 0. Finally, h(0, 0, R20 ) = 0. Substituting (S, I, R1 , R2 ) = (S0 , 0, 0, R20 ) in the system, together with the parameter values yields −a0 R20 + b0 (S0 + R20 ) − γ0 S0 = 0,

(1.4)

(a0 − γ0 )R20 = 0

(1.5)

4

C.O.A. Sowunmi

From equation (1.4), we can have R20 6= 0 provided a0 = γ0 , while from equation (1.3), we also have S0 6= 0 if b0 = a0 . Thus, for arbitrary S0 , R20 6= 0 and provided a0 = b0 = γ0 , there is a corruption-free steady state (S0 , 0, 0, R20 ). Next, we investigate the possibility of an endemic steady state i.e., I0 , R10 6= 0 when S0 = R20 = 0. We see that the last equation of system (1.1) implies e(R10 )R20 = 0. Since e(R1 ) is monotone increasing and non-negative, it follows that R10 = 0, contrary to the assumption that R10 6= 0. Hence, an endemic steady state is not possible. We now investigate the stability of a corruption-free steady state by perturbing the system about the point (S0 , 0, 0, R20 ), given that S0 , R20 6= 0. Set S = S0 + s, I = 0 + i, R1 = 0 + r1 and R2 = R20 + r2 . Hence, ds dt

= (β − γ0 )s + (β − F2 )i + (β − a0 R20 )r1 + (β − a0 )r2 + M1 ,

di dt

= (F2 − γ0 − h1 )i + (d0 − c0 R20 )r1 + M2 ,

dr1 dt

= h1 + (γ0 + h1 − d0 )r1 + M3 ,

dr2 dt

= −γ0′ R20 s − γ0′ R20 i + {(a′0 + c′0 − γ0′ )R20 }r1 + (a0 − γ0′ R20 − γ0 )r2 + M4 ,

(1.6) where a′0 = a′ (0), c′0 = c′ (0), d′0 = d′ (0), γ0′ = γ ′ (S0 + R20 ), b′0 = b′ (S0 + R20 ), ∂h h1 = (0, 0, R20 ) and β = b(S0 + R20 ) + b′ (S0 + R20 )(S0 + R20 ) − γ ′ (S0 + R20 ). ∂I M1 , M2 , M3 , M4 are the nonlinear remainder terms: Written in matrix form, the perturbed system takes the form    l11 l12 l13 l14 s s      d  i   0 l22 l23 0   i = dt  r1   0 l32 l33 0   r1 l41 L42 43 l44 r2 r2 





 M1   M2  +    M3  M4

If L denotes the community matrix in the last equation, then, its characteristic equation det(L − λI) when expanded becomes 

λ2 − (l11 + l44 )λ + l11 l44 − l14 l41

 2 λ − (l22 + l32 )λ + l22 l33 − l23 l32 = 0. (1.7)

The Routh-Hurwitz conditions for the quartic are easily obtained. Thus, l11 + l44 < 0 > l11 l44 − l14 l41 ,

(1.8)

l22 + l33 < 0 > l22 l33 − l23 l32 . In terms of the parameters of the system, necessary and sufficient conditions for local

Epidemiology of Corruption and Disease Transmission as a Saturable Interaction

5

asymptotic stability of the corruption-free steady state are l11 + l44

= b′ N0 − r0′ (R20 + 1) < 0,

l11 l44 − l14 l41 = 0 > 0, l22 + l33

= F2 (s, 0) − 2γ0 − d0 < 0,

l22 l33 − l23 l32 = (F2 (S, 0) − γ0 )(γ0 + h1 − d0 ) − hi (γ0 + hi − c′0 R20 ) > 0. (1.9) Of the last four inequalities, only the first one is patently inconsistent. The remaining three can be logically consistent. From these, we infer that under suitable assumption, three out of the four characteristic roots have negative real parts while the odd one equals zero. This is a critical case of the type discussed by Lyapunov [2], where linearisation is inadequate for the resolution of the question of stability of a steady state. We will therefore apply the result of Lyapunov [2]. For this, it is convenient first to redesignate the dependent variables as s = y1 , i = y2 , r1 = y3 , r2 = y4 . Thus, the system of perturbed equations can be written as y = Ly˙ + M (y), (1.10) where y = (y1 , y2 , y3 , y4 )T , L = [lij ] and M (y) = (M1 (y), M2 (y), M3 (y), M4 (y))T . Next step is to rotate the coordinate frame in order to transform equation (1.10) into an appropriate form, such that one axis lies in the eigenspace of the zero root. We assume this subspace is 1-dimensional and the three non-zero roots are distinct. Let X0 be the kernel of L and X the orthogonal complement of X0 . Thus, any vector y can be written as y = x0 + x where x0 ∈ X0 and x ∈ X. Hence, y˙ = x˙ 0 + x˙ = L(x0 + x) + M(x0 + x) where L now denotes the linear operator which is represented by the matrix [lij ] in the original coordinate system. But, Lx0 = 0. Hence, x˙ 0 + x˙ = Lx + M(x0 + x). Obviously, Lx 6= 0, hence, it belongs to X. Let M(x0 + x) = M0 (x0 + x) + MX (x0 + x), where M0 ( ) ∈ X0 andMX ( ) ∈ X. Hence, equation (1.10) is equivalent to the pair x˙ 0 = M0 (x0 + x), (1.11) x˙

= Lx + MX (x0 + x)

In this latter form, we are almost ready for the final transformation preparatory to the application of the result of Lyapunov in [2]. The next step is to express the terms in system (1.11) in terms of the coordinates in the new basis. From equations (1.9), we see that we can take as basis for X0 , the vector  T  T −l14 l14 −l44 = , 0, 0, 1 , 0, 0, 1 . For convenience, we shall write ν for − . Hence, l11 l41 l11 the basis for X0 is (ν, 0, 0, 1)T denoted by a0 . As basis of the orthogonal complement X, we shall take the eigenvectors of the non-zero eigenvalues of L. From equation (1.7), since one root is zero, there must be another real root. The remaining pair may be real or complex

6

C.O.A. Sowunmi

conjugates. There is no loss of generality in assuming there are three real distinct roots. Denote these by λ1 , λ2 , λ3 and their corresponding eigenvectors by a1 , a2 , a3 . In terms of the basis {a0 , a1 , a2 , a3 } the first equation of (1.11) takes the form 3 X

x˙ 0 a0 = M0

xi ai

i=0

!

a0 .

Hence, x˙ 0 = M0 (x0 , x1 , x2 , x3 ).

(1.12)

Similarly, equation (1.12) takes the form 3 X i=1

x˙ i ai = L

3 X

xi ai

i=1

!

+

 3  X i=1

But, L

3 X i=1

xi ai

!

=





Mi 

3 X

3 X j=0

  xj aj  ai . 

xi λi ai .

i=1

From the linear independence of {a1 , a2 , a3 }, it follows that x˙ i = λi xi + Mi (x0 , x1 , x2 , x3 ).

(1.13)

We now adopt Lyapunov’s technique in [2] to ensure that in the power series expansions of M1 , M2 , M3 there is no term in x0 having a lower degree than the x0 -term in M0 . Consider the homogeneous system λ1 x1 + M1 (x0 , x1 , x2 , x3 ) = 0, λ2 x2 + M2 (x0 , x1 , x2 , x3 ) = 0,

(1.14)

λ3 x3 + M3 (x0 , x1 , x2 , x3 ) = 0. (x0 , x1 , x2 , x3 ) = (0, 0, 0, 0) is a solution. Furthermore, λ1 , λ2 , λ3 6= 0. Assuming M1 , M2 , M3 are smooth enough, it follows that the Implicit Function Theorem applies and so we have x1 = X1 (x0 ), x2 = X2 (x0 ), x3 = X3 (x0 ) in an open neighbourhood of zero. Still continuing in the technique of Lyapunov [2], we substitute x1 = X1 + z1 , x2 = X2 + z2 , x3 = X3 + z3 in the equations (1.12 - 1.13) to obtain the system in x0 , z1 , z2 , z3 , viz: x˙ 0 = M0 {x0 , X1 + z1 , X2 + z2 , X3 + z3 ), (1.15) z˙i = λi (Xi + zi ) + Mi (x0 , X1 + z1 , X2 + z2 , X3 + z3 ) −

dX0 dt ,

= λi zi − Mi (x0 , X1 , X2 , X3 ) + Mi (x0 , X1 + z1 , X2 + z2 , X3 + z3 ), i − dX dx0 M0 (x0 , X1 + z1 , X2 + z2 , X3 + z3 ),

i = 1, 2, 3.

(1.16)

Epidemiology of Corruption and Disease Transmission as a Saturable Interaction

7

In equation (1.16), the terms in x0 alone are obtained by putting z1 = z2 = z3 = 0 in the dXi M0 (x0 , X1 , X2 , x3 ), from which it is clear nonlinear terms. The terms are given by − dx0 −dXi that the factor ensures that the terms in Xi cannot be of degrees less than that in M0 . dx0 We now have to compute the power series expansion of M0 (x0 , X1 +z2 , X2 +z2 , X3 + z3 ) about the point (0, 0, 0, 0). As a first step, we need to compute the remaining eigenvalues of L and their associated eigenvectors. −l44 −l14 = . We already have (ν, 0, 0, 1) associated with λ = 0, where ν = l11 l41 −l14 −l44 For λ = l11 + l44 = l11 (1 − ν) = λ1 we have (σ, 0, 0, 1), where σ = = . For l44 l41 p l22 + l33 + (l22 − l33 )2 + 4l23 l32 + λ2 = λ = , (1.17) 2   + + λ − l33 + , 1, w4 , where we have w1 , l32 w1+ =

(λ+ − l44 ){l13 l32 + l12 (λ+ − l33 )} + l14 {(λ+ − l33 )(l42 + l23 l32 )} , λ+ {λ+ − (l11 + l44 )}

and w4+ =

(λ+ − l11 ){l43 l32 + l42 (λ+ − l33 )} + l41 {(λ+ − l33 )l12 + l13 l32 } . l32 λ+ {λ+ − (l11 + l44 )}

(1.18)

Lastly, for −

λ =

(l22 + l33 ) −

p (l22 + l33 )2 + 4l23 l32 = λ3 , 2

(1.19)

  − − − λ − l33 , 1, w4 , where w1− and w4− are obtained by replacing λ+ by λ− we have w1 , l32 in the expressions for w1+ and w4+ , respectively. Let {e1 , e2 , e3 , e4 } denote the standard basis for R I 4 while {a0 , a1 , a2 , a3 } denotes the basis made up of the eigenvectors which we have just obtained. The change of bases from standard to the newly obtained can be represented by the matrix   ν 0 0 1      σ  0 0 1       +  A=  w+ λ − l33 1 w+  .  1 4  l32       −   − λ − l33 − w1 1 w4 l32

Hence, AT is the matrix for the change of coordinates from (y1 , y2 , y3 , y4 ) to (x0 , x1 , x2 , x3 ). With reference to basis {e1 , e2 , e3 , e4 }, the nonlinear vector valued function M has components (M1 , M2 , M3 , M4 )T . Thus, with reference to basis

8

C.O.A. Sowunmi

{a0 , a1 , a2 , a3 }, the components (M0 , M1 , M2 , M3 )T are given by   M0 ν σ w1+ w1−  M 1   0 0 λ+ λ−     M2  =  0 0 1 1 1 1 w4+ w4− M3 

−1    

 M1  M2     M3  . M4

If B denotes the matrix in the above equation, then, B = (AT )−1 . We are interested in the coefficient of x20 . Each Mi consists of a homogeneous quadratic form in (y1 , y2 , y3 , y4 ) plus a remainder of higher degree terms. That is, each Mi consists of a term y T Qi y plus higher degree terms. 4 X

M0 =

B1s Ms = B1s yiT Qsij yj + higher degree terms

s=1 yiT B1s Qsij yj + higher degree terms 4 X yiT B1s Qsij yj + higher degrees i,j,s=1

= =

terms.

Since yα = ... ...

yαT

=

M0 =

3 X

r=0 3 X

ATαr+1 xr xTr Aαr+1

r=0 4 X

3 X

(xTr Air+1 )B1s Qsij yj + higher degree terms

i,j,s=1 r=0

3 4 X X

=

(xTr Air+1 )B1s Qsij

=

ATjα+1 xα + higher degree terms

α=0

i,j,s=1 r=0 4 X

3 X

3 3 X X

xTr Air+1 B1s Qsij ATjα+1 xα + higher degree terms

i,j,s=1 r=0 α=0

... term in x20 is

 

4 X

i,j,s=1

... coefficient of x20 is 4 X

i,j,s=1



Ai1 B1s Qsij ATj1  x20

Ai1 B1s Qsij ATj1 =

4 X

i,j,s=1

Ai1 Qsij B1s A1j .

(1.20)

Epidemiology of Corruption and Disease Transmission as a Saturable Interaction

9

To obtain the r.h.s. of equation (1.20), we must compute Qsij for all s, i, j. 

q1

q1

q1

q1

   q1 q1 − F22 q1 q1   =   q1 q1 q1 − a′′0 R20 q1 − a′0  

Q1ij

q1

q1

q1 − a′0

q1



    ,    

where q1 = b′′0 N0 + 2b′0 − γ0′′ S0 

Q2ij

0

0

   −γ0′ F22 − h11 − 2γ0′ −γ0′ −γ0′ − h13  =    0 −γ0′ −2d′0 − c′0 R20 − h22 −c′0   0

−γ0′ − h13

−γ0′

−c′0

0 0 0    0 h11 h12 − γ0′ h13  =    −γ ′ h12 − γ ′ h22 − q2 h22 − γ ′ 0 0 0   

Q3ij

−γ0′′

0

0

h13

h22 − γ0′

h33

a′0 − γ0′

        

    ,    

where q0 = 2d′ (0) + e′ (0) + 2γ0′ . Finally,  −γ0′′ R20 −γ0′′ R20 −γ0′′ R20 −γ0′′ R20    −γ0′′ R20 −γ0′′ R20 −γ0′′ R20 −γ0′′ R20  4 Qij =    −γ ′′ R20 −γ ′′ R20 q3 a′0 − γ0′ 0 0   −γ0′′ R20 −γ0′′ R20

−h33





−γ0′′ R20



    ,    

where q3 = 2e′0 + (a′′0 + c′′0 + c′0 − γ0′′ )R20 , a′′0 = a′′ (0), b′′0 = b′′ (S0 + R20 ), γ ′′ = ∂2F ∂2h (S0 + R20 ), F22 = (S0 , 0), h12 = (0, 0, R20 ) and h11 , h22 , h13 , h33 are ∂I∂R1 ∂I∂R1 similarly defined.

10

C.O.A. Sowunmi

Hence, X

Ai1 Q1ij Aij

= q1 (ν + 1){ν + σ + w1+ + w1− } − w1+ a′0 ,

P2

Ai1 Q2ij A1j

= −{νσγ0′ + σ(γ0′ + h13 ) + w1+ c′′0 − wi− h13 },

Ai1 Q3ij A1j

= σh13 + w1+ {h22 − γ0′ (1 + ν)},

Ai1 Q4ij A1j

= −γ0′′ R20 {(ν + 1)(ν + σ + w1− ) + νw1+ w1− }.

i,j

ij

P

ij

P

ij

(1.21)

Thus, the coefficient of x20 = B11

X

Ai1 Q′ij A1j +B12

ij

X

Ai1 Q2ij A1j +B13

X

Ai1 Q3ij A1j +B14

X

Ai1 Q4ij A1j .

Now, B = (A−1 )T . Since A is known, A−1 is easily computed using MathCAD. Therefore, 1  (ν − σ)     1   (ν − σ)  B=    0     0 

(w4+ − w4− ) + w1− − w1+ (λ+ − λ− )(ν − σ)

σ(λ+ w4− − w4+ λ− ) + w1+ λ− − w1− λ+ (λ+ − λ− )(ν − σ)

ν(w4+ − w4− ) + w1− − w1+ (λ+ − λ− )(σ − ν)

ν(w4+ − w4+ ) + w1− − w1+ (λ+ − λ− )(σ − ν)

λ+

1 − λ−

1 + λ − λ−

λ−

λ− − λ+

λ+ λ+ − λ−

 σ σ−ν      ν  ν−σ       0     0

Hence, the coefficient of x20

=

(ν − σ)−1 [q1 (ν + 1)(ν + σ + w1+ + w1− ) −a′0 w1+ + σγ0′′ R20 {(ν + 1)(ν + σ + w1− ) + νw1+ w−1 } +(λ− − λ+ )−1 {(σ(w4+ − w4− ) − (w1+ − w1− ))(νσγ0′ + σ(γ0′ + h13 ) − w1+ c′′0 + w1− h3 ) −λ+ (σw4− − w4+ ) − λ− (σw4+ − w1− )(σh13 + w1+ (h22 − γ0′ (1 − ν)))}].

(1.22)

If the r.h.s. of the above equation is not zero, then, by the result of Lyapunov [2] the corruption-free steady state is not stable, while if it vanishes, we will have to compute the coefficient of x30 . This is likely to be more cumbersome than the r.h.s. of this equation. Rather than get involved in this, we choose the case when the r.h.s. does not vanish and seek to stabilize the corruption-free steady state by means of a feedback control. Hence, we assume that the coefficient of x20 in M0 is not zero so that the corruption-free steady state is unstable.

Epidemiology of Corruption and Disease Transmission as a Saturable Interaction 11

1.4.

Stabilizing the Corruption-Free Steady State through Control

We begin with the linear parts of equations (2.24-2.25). Thus, z˙0 = 0, z˙1 = λ1 z1 , (1.23) z˙2 = λ2 z2 , z˙3 = λ3 z3 . The last three equations can be written in vector form as z˙ = Qz, where Q = diag[λ1 , λ2 , λ3 ]. Adopting the procedure in [3], we introduce a scalar valued function f of the real feedZ ±∞

back variable ρ, such that ρf (ρ) > 0 for ρ 6= 0, f (0) = 0 and

f (ρ)dρ is divergent.

0

f is called the characteristic of the servo motor or feedback mechanism. Let k, η be two constant 3-vectors, while u, v are real numbers. Finally, ρ is defined as a linear combination of z0 , z1 , z2 , z3 . Thus, ρ = vz0 + k · z. Our control problem therefore is to obtain sufficient conditions for the complete stability of the trivial solution of the system. z˙ = Qz + f (ρ)η, (1.24) z˙0 = f (ρ)u, where ρ = vz0 + k · z. We define the function T

V (z0 , z) = z Bz +

Z

(1.25)

ρ

f (τ )dτ,

(1.26)

0

where B is a positive definite, symmetric 3 × 3 matrix. V (z0 , z) is positive definite for all z0 , z. Reason: V (0, 0) implies ρ = 0 from equation (1.24). Hence, V (0, 0) = 0 by equation (1.26). On the other hand, V (z0 , z) = 0 implies z = 0 and ρ = 0 since the r.h.s. of (1.25) is the sum of positive terms. But, ρ = vz0 +k·z and v 6= 0, so that z0 = 0. Define C = −(QT B + BQ). Thus, C = C T . From equations (1.22 - 1.24), we have ρ˙ = v z˙0 + k · z˙ = vuf (ρ) + k · {Qz + f (ρ)η} = vuf (ρ) + K T Qz + f (ρ)k · η. Therefore, K T Qz + f (ρ)(k · η + vu) = ρ. ˙ Now, −V˙ = −(z˙ T Bz + z T B x˙ + f˙(ρ)ρ). ˙ Substituting for ρ from equation (1.25) above yields −V˙ = [z T Cz − f (ρ){η T Bz + z T Bη + k T Qz + f (ρ)(k · η + vu)}].

(1.27)

12

C.O.A. Sowunmi

Put K T Q = K ∗ and r = −(k · η + vu) in the last equation to obtain −V˙ = 4[z T Cz − f (η){η T Bz + z T Bη + k ∗ · z − rf (ρ)}].

(1.28)

For each t, equation (1.28) defines −V˙ as a monotone increasing function of r subject to the values of the remaining terms in the equation. Hence, increasing r increases the rate at which V decreases along a trajectory in (z0 , z)-space. As shown in [3], given any positive definite matrix C, there exists a unique symmetric matrix B also positive definite, which satisfies the equation C = −(QT B + BQ). Hence, let us choose C to be positive definite, and there will be a unique, symmetric positive matrix B which will fit into (1.26). Our goal is that −V˙ also be a positive definite function of (z0 , z). In matrix form, the r.h.s. of equation (1.28) is   1 C −(Bη + k ∗ )  z  2    . [z T , f (ρ)]    1 ∗ T f (ρ) −(Bη + k ) r 2

For the above to be a positive definite form in [z T , f (ρ)], it can be shown as in [3] that neces1 sary and sufficient conditions are that C be positive definite and r > (Bη + k ∗ )C −1 (Bη + 2 1 ∗ k ). Now, V˙ (0, 0) implies ρ = 0 = f (ρ), and conversely (z, ρ) = (0, 0) implies 2 (z0 , z) = (0, 0). Hence, −V˙ is a positive definite function of (z0 , z), provided 1 1 r > (Bη + k∗ )C −1 (Bη + k ∗ ), 2 2 since C is already chosen to be positive definite. Note also that r 6= −k · η.

(1.29)

(1.30)

(1.29 - 1.30) are the two fundamental control inequalities for the problem (cf. [3]. The next step is to apply (1.29) to our problem. Take C = diag(α1 , α2 , α3 ), αi > 0 for i = 1, 2, 3. Hence, B satisfies the equations bij = 0, i 6= j 2λi bii = −αi , ... bii =

i = 1, 2, 3

αi , i = 1, 2, 3 2λi

bij = 0, i 6= j. ...

B = diag

... Bη =

1 2





−α1 −α2 −α3 , , 2λ1 2λ2 2λ3

α1 η1 α2 η2 α3 η3 , , |λ1 | |λ2 | |λ3 |



T

Epidemiology of Corruption and Disease Transmission as a Saturable Interaction 13   Since C −1 = diag α11 , α12 , α13 , it follows that 3

1 1X 1 (Bη + k∗ )C −1 (Bη + k ∗ ) = 2 2 4 i=1



πi ηi ki∗ + |λi | πi

2

,

√ where πi = αi > 0. Hence, the fundamental control inequality (1.29) is 3

1X r> 4 i=1



πi ηi ki∗ + |λi | πi

2

(1.31)

We have seen that −V˙ is a monotone increasing function of r and when r satisfies (1.29), −V˙ becomes a positive definite function of (z0 , z). So far, our choice of (α1 , α2 , α3 ) has been arbitrary but for the positivity restriction. We now choose the α’s to minimize the r.h.s of (1.29) (cf [3]). This will give us the sharpest condition on r for −V˙ to be positive πi ηi k∗ definite, taking η, λi ’s and ki∗ ’s as fixed. + i is the sum of two numbers whose |λi | πi product is fixed. If they are both positive, then, their sum is minimum when they are equal. The case when both are negative can be converted to the preceding one. When they are of opposite signs there is no minimum, but since the sum is squared, the minimum is zero. Hence, 3 X πi ki∗ r> H(ηi ki∗ ), (1.32) |λi | i=1

where H is the Heaviside Unit Function. Next is the interpretation of (1.32) in terms of the parameters of the model.

1.5.

Discussion

We have already seen that the larger r is, the more rapid is the stabilization of the corruptionfree state. For a given r, efficiency of the feedback mechanism depends on how small we can make the r.h.s. of (1.32), i.e., how large we can make |λi | for all i = 2, 3. λ1 = l11 + l44 = β − γ0 + β − a0 = 2β − a0 − γ0 . Obviously, as a0 → ∞, λ1 → −∞. Now, 2λ2

= l22 + l33 +

2∂λ2 ∂h1

= −1 + 1 +

∂λ2 ∂h1

=

p (l22 + l33 )2 + 4l23 l32 ,

2(l22 +l33 )(−1+1)+4l23 +4l32 √ , (l22 +l33 )2 +4l23 l32 ′

0 R20 ) √2(h1 +d0 −c . 2

(l22 +l33 ) +4l23 l32

(1.33)

14

C.O.A. Sowunmi

Hence,

∂λ2 < 0, provided ∂h1

Correspondingly, for λ3 , we have

h1 + d0 − c′0 R20 < 0,

(1.34)

h1 + d0 − c′0 R20 > 0.

(1.35)

∂λ2 < 0, provided ∂h1

These are mutually exclusive. We can only have one or the other. This will be achieved by making use of the Heaviside function in (1.31). For (1.32), it would be enough to ensure that c′0 is large enough, whilst for (1.33), we would require that hi be large enough. The interpretation of the foregoing is as follows: If the corruption-free steady state is unstable, it is nonetheless possible to stabilize it absolutely through feedback control, by as much as possible encouraging direct movement of people from the Susceptible class to the Resistant class. In addition, those who are infective should be moved into the removed class. The efficiency of this feedback process depends on a number of factors, one of which is a0 : the rate at which people in the susceptible class become resistant purely on their volition, rather than through deterrence. Other factors are h1 and c′0 . If η2 k2∗ < 0, then, increasing c′0 so that (1.32) holds will increase the feedback efficiency. On the contrary, if the choice is that η3 k3∗ < 0, then, increasing h1 so that (1.33) holds, will increase the feedback efficiency.

2. 2.1.

Disease Transmission as a Saturable Interaction: The SIS Case Introduction

In the study of 2-sex population dynamics published in [4, 5], we began with the minimal analytical conditions on the interaction function; the function which encapsulates the dynamics of the interaction between the reproductive males and the reproductive non-gestating females. These minimal conditions were adequate for the proof of global existence and uniqueness of the solution of the model equations. One of the conditions is essentially a growth condition on the interaction function, and it is hardly surprising that one needed such a condition in order to prove global existence or uniqueness of solution. In paper [6], in order to prove boundedness and stability of steady states, additional conditions had to be imposed. Paper [7] was on stability, but of a time-discrete model, and the interaction function likewise needed additional conditions. The foregoing suggests the following approach to the modelling of any dynamical process which is driven by the interaction between some or all of its components, provided the interaction is saturable. We begin with what we consider as the minimal analytical conditions, and leave additional conditions on the interaction function to emerge from the qualitative study, other than existence and uniqueness of solution of the model equations. Disease transmission is a dynamical process driven by the interaction between the susceptibles and the infectives. It seems safe to assume that all such interactions are saturable. In a SIS model, there are just the two components, the susceptibles and the infectives. This

Epidemiology of Corruption and Disease Transmission as a Saturable Interaction 15 situation is analogous to that of 2-sex population dynamics or prey-predator interactions and therefore fits into the general formulation developed in [1]. We shall therefore proceed to the study of SIS models, which require a slight modification of the formulation in [1]. In the first SIS model, the function F (S, I) will be the incidence rate of new infectives when the susceptibles and infectives number S and I, respectively. Since F (S, 0) = F (0, I) = 0 ∀ S, I, it follows that F (S, I) = φ(S, I)SI for some function φ. The generalised law of the minimum [1] when applied to F yields the inequality F (S, I) ≤ min{k1 S, k2 I} for some positive constants k1 , k2 . k1 is therefore an upper estiF (S,I) mate of F (S,I) . Both are indicators of the ease or difficulty with S , whilst k2 is that of I which a disease is transmitted. In the second SIS model, the infective class has an age structure. Following the idea which was used in [4, 5] for the rate of new births, we assume the existence of a density Z ∞

π(α)F(S(t), J(t, α), α)dα,

function F such that the incidence rate at time t equals

0

where π(α) is the infectivity at class age α, and J(t, α) is the density of infectives aged α at time t. Obviously, F(0, x, α) = F(y, 0, α) = 0 for all positive x, y, α. We therefore have F(S(t), J(t, α), α) = Ψ(S(t), J(t, α), α)S(t)J(t, α) for some function Ψ. Thus, the incidence rate at time t also equals Z ∞ Ψ(S(t), J(t, α), α)π(α)J(t, α)dα. S(t) 0

Still following the pattern in [4, 5], we assume that the density function F satisfies the generalized law of the minimum in the form F(S(t), J(t, α), α) ≤ min{k1 S(t), k2 (α)J(t, α)}, where k1 , k2 (α) > 0. The function F is to a certain extent analogous to the interaction function in [4, 5]. Hence, by analogy, we expect that the functions such as min{k1 S(t), k2 (α)J(t, α)} and

k1 k2 (α)S(t)J(t, α) , k1 S(t) + k2 (α)]J(t, α)

which certainly satisfy all the conditions are admissible. In the next subsection, we shall formulate the governing equations of two SIS models, taken essentially from Brauer [8], but without his type of interaction functions.

2.2.

Formulation of Model Equations

Let Z ∞the total population at time t be N (t). Then, N (t) = S(t) + I(t) where I(t) = J(t, s)ds. We assume that there is a natural death rate D(N ), dependent on the popu0

lation size. There is also an additional mortality rate q among the infectives (a measure of the deadliness of the disease). P (s) is the Z fraction of infectives of class age s . P (0) = 1, ∞

and P is monotone decreasing such that

0

P (s)ds = τ < ∞ . P is also independent of

time. Hence, we are dealing with an autonomous case. Individuals are born into the S class

16

C.O.A. Sowunmi

  0 J(t + s, s) ∂0 ∂ only. If we exclude death, then, P (s) = J is the derivative . If + J(t, 0) ∂t ∂s following a cohort, where change is due to recovery alone, then,   0 ∂0 d ∂ J = J(t, s) ln(P (s)). + ∂t ∂s ds Hence, 

∂ ∂ + ∂t ∂s



  d J(t, s) = − q + D(N ) − ln(P (s)) J(t, s). ds

(2.36)

When infectively is not age-dependent, J(t, 0) = F (S(t), I(t)).

(2.37)

When infectivity is age-dependent, J(t, 0)

R∞

=

0

π(α)F(S(t), J(t, α), α)dα, (2.38)

J(0, s) = J0 (s). Furthermore,

DN Dt

= (B(N ) − D(N )N − qI,

(2.39)

N (0) = N0 . The entire population is assumed to have a carrying capacity K, say. Hence, B(K)

= D(K),

B(N )

> D(N ), if N < K,

(2.40)

B ′ (N ) < D′ (N ), if N ≥ K. We further assume that B ′ (N ), D′ (N ) > 0. We can deduce the equation for S from the fact that S = N − I. However, we would rather focus on the subsystem of equations in N and I. Thus, equations (2.37) and the first equation of (2.38) will be written respectively as J(t, 0) = F (N (t) − I(t), I(t)), J(t, 0) =

2.3.

R∞ 0

(2.41)

π(α)F(N (t) − I(t), J(t, α), α)dα.

Age-independent Infectivity and Existence of Steady States

dN = First, we consider the case without age-dependent infectivity. At a steady state, dt dS ∂J = = 0 ∀ t ≥ 0. dt ∂t Let N (t) = N∞ , J(t, s) = J∞ (s), and I(t) = I∞ at a steady state. So, (B(N∞ ) − D(N∞ ))N∞ − qI∞ = 0,

(2.42)

Epidemiology of Corruption and Disease Transmission as a Saturable Interaction 17 and

  dI∞ d = − q + D(N∞ ) − ln(P (s)) J∞ (s). ds ds

(2.43)

Thus, from equation (2.43), we obtain J∞ (s) = exp {−(q + D(N∞ ))} P (s)J∞ (0). This can be written as J∞ (s) = J∞ (0)P (s)e−λs ,

(2.44)

λ = q + D(N∞ ).

(2.45)

where

Since I∞ =

Z

∞

J∞ (s)ds, equation (2.44) yields

0

I∞ = J∞ (0)Pˆ (λ), where Pˆ (λ) =

Z

∞

P (s)e−λs ds.

(2.46)

(2.47)

0

Therefore, from equation (2.36), we have

B(N∞ ) − D(N∞ ))N∞ = q Pˆ (λ)F (N∞ − I∞ , I∞ ).

(2.48)

From equation (2.37) at a steady state, J∞ (0) = F (N∞ − I∞ , I∞ ). Substituting this into equation (2.46), we have I∞ = Pˆ (λ)F (N∞ − I∞ , I∞ ).

(2.49)

qI∞ = (B(N∞ ) − D(N∞ ))N∞ .

(2.50)

From equation (2.42), Equations (2.49) and (2.50) define the steady states (N∞ , I∞ ). If I∞ = 0, equation (2.49) is satisfied, while (2.50) is satisfied provided N∞ = K. Hence, (K, 0) is a steady state. This is the non-trivial disease-free steady state and there is only one such since B(N ) = D(N ) only if N = K. N = 0 is trivial. At an endemic state I∞ > 0. If we can solve equation (2.50) for N∞ as a function of I∞ , then, we shall substitute for N∞ in equation (2.49) to obtain an equation in I∞ alone. Define f (N ) = (B(N ) − D(N ))N , then, f ′ (N ) = B(N ) − D(N ) + N (B ′ (N ) − D′ (N )), and f ′ (K) < 0, but, f ′ (N ) > 0 for sufficiently small values of N . Hence, f ′ (N ) = 0 for some N < K, equal to K0 say. We assume there are at least two branches of the inverse function f −1 . Specifically, let ′ f (N ) < 0 ∀ N ∈ (K0 , K). Let g denote the branch of f −1 defined on (0, I0 ) and extended by continuity to [0, I0 ] such that g(I0 ) = K0 . Thus, N∞ = g(qI∞ ) for I∞ ∈ [0, I0 ],

(2.51)

18

C.O.A. Sowunmi

where qI0 = K0 (B(K0 ) − D(K0 )). In equation (2.51), we note that λ is a function of N∞ . Hence, substituting for N∞ from equation (2.51) in equation (2.49) yields I∞ = Pˆ (q + Dog(qI∞ ))F (g(qI∞ ) − I∞ , I∞ ),

(2.52)

where I∞ ∈ [0, I0 ]. An endemic state will exist iff equation (2.52) has a non-trivial solution. This will happen iff the straight line Y = X and the curve Y = Pˆ (q + D o g)(qX))F (g(qX) − X, X)

(2.53)

dY (0) intersect over (0, I0 ). Suppose > 1, then, equation (2.52) has non-trivial solution dX if, but not only if, there is X1 , 0 < X1 < I0 such that Pˆ (q + Dog(qX1 )F (g(qX1 ) − X1 , X1 ) < X1 To obtain a sufficient condition with epidemiological meaning, we bring in the generalised law of the minimum. Thus, suppose Pˆ (q + D o g(qX1 ))k2 X1 < X1 . Therefore, k2 Pˆ (q + Dog(qX1 )) < 1.

(2.54)

Recall that λ = q + Dog(qX1 ). We therefore have the following result. dY (0) > 1 and there is a value of X = X1 > 0, such that k2 Pˆ (q + dX Dog(qX1 )) < 1, then, an endemic steady state X∞ < X1 exists. Lemma 1 If

For an interpretation of condition (2.54), we note that Z ∞ J(t + s, s) −λs ˆ e ds. P (λ) = J(t, 0) 0 The first factor in the integrand is the fraction of those infected at time t that stays infected till s units of time later, barring death due to natural causes or fatality of the infection, while the second is the probability of surviving that interval when the infective population equals X1 . The integrand is therefore the fraction that actually stays infective for s units of time. Pˆ (λ) has the dimension of TIME. It can be taken as a measure of the mean time spent in the infective class. The higher the fatality of an infectious disease, other factors being equal, the lower its Pˆ (λ) at a given population of infectives. k2 is an upper bound of the rate at which susceptibles are infected by an infective. Equation (2.54) is therefore the condition that at X1 , infectives do less than replace themselves. dY (0) It can be shown that = Pˆ (q + D(K)).F2 (K, 0) where F2 stands for the partial dX derivative of F w.r.t. its second variable. F1 is likewise defined. dY (0) < 1, there may be no endemic steady state. A sufficient but not necessary If dX condition for an endemic steady state is that there is X2 , 0 < X2 < I0 such that Pˆ (q + Dog(qX2 ))F (g(qX2 ) − X2 , X2 ) > X2 .

(2.55)

Epidemiology of Corruption and Disease Transmission as a Saturable Interaction 19 But, unlike in the preceding case, we cannot make use of the generalised law of the minimum to obtain a sufficient condition. In fact, Pˆ (q + Dog(qX2 )k2 > 1,

(2.56)

does not guarantee (2.55).

2.4.

Stability of Equilibria

Let (N (t), I(t)) = (N∞ , I∞ ) be an equilibrium. Let its perturbation be N (t) = N∞ +n(t), I(t) = I∞ + i(t), J(t, Z J(0, s) = J∞ (s) + j(0, s) = J∞ (s) + j0 (s). Z s) = J∞ (s) + j(t, s), ∞

∞

j(t, s)ds = i(t). It can be shown, using

J∞ (s)ds = I∞ ,

By definition,

0

0

equations (2.36-2.39) that j satisfies the system ∂j ∂j + ∂t ∂s

= (−λ +

j(t, 0)

= {F2 (N∞ − I∞ , I∞ ) − F1 (N∞ − I∞ )} i(t) + R2 ,

j(0, s)

= j0 (s),

∂ ∂s

ln(P (s)))j(t, s) − D′ (N∞ )J∞ (s)n(t) + R1 , (2.57)

where F1 and F2 denote the partial derivatives of F w.r.t. its first and second variables, respectively. R1 , R2 , . . . represent higher order remainder terms in the perturbations. By the usual method of characteristics, the system of equations (2.57) eventually yields Z t e−λs P (s)i(t − s)ds. i(t) = {F2 (N∞ − I∞ , I∞ ) − F1 (N∞ − I∞ , I∞ )} 0

−D′ (N∞ )F (N∞ − I∞ , I∞ ) +

Z

0

∞

Z

∞

e−λσ dσ

Z

0

0

t

e−λs P (s + σ)n(t − s)ds

P (s + t) −λs e j0 (s)ds + R3 . P (s)

(2.58)

Likewise from the pair dn dt

= {(B(N∞ − D(N∞ )) + N∞ (B ′ (N∞ ) − D′ (N∞ ))} n(t) − qi(t) + R4 ,

n(0) = n0 , (2.59) we obtain ρt

n(t) = n0 e − q where

Z

0

t

eρs i(t − s)ds + R5 ,

ρ = B(N∞ ) − D(N∞ ) + N∞ (B ′ (N∞ ) − D′ (N∞ )).

(2.60)

(2.61)

We recall from subsection 2.3 that ρ < 0. From the pair of equations (2.58) and (2.60), we will now obtain estimates of i(t) and n(t). Our interest is not in the rigorous proof of

20

C.O.A. Sowunmi

estimates of the asymptotic behaviour of i(t) and n(t), which can be achieved, but in the estimates themselves. For brevity in writing the terms, we introduce the following notations. a11

= F2 (N∞ − I∞ , I∞ ) − F1 (N∞ − I∞ , I∞ ),

a12

= D′ (N∞ )F (N∞ − I∞ , I∞ ),

b1 (t)

=

b2 (t)

= n0 eρt + R5 ,

R∞ 0

P (s+t) −λs j0 (s)ds P (s) e

+ R3 , (2.62)

P1 (s) = e−λs P (s), P2 (s) =

R∞ 0

e−λσ P (s + σ)σ,

P3 (s) = e−λs P2 (s). The system of equations in (2.58) and (2.60) can be viewed as a perturbed pair of linear Volterra integral equations. Taking its Laplace transform, with ξ as transform variable yields ˆi(ξ) = a11 Pˆ1 (ξ)ˆi(ξ) − a12 Pˆ3 (ξ)ˆ n(ξ) + ˆb1 (ξ), (2.63) q ˆ i(ξ) + ˆb2 (ξ). n ˆ (ξ) = ξ−ρ From system (2.63), we obtain the characteristic equation for the system as qa12 Pˆ3 (ξ) = 0. 1 − a11 Pˆ1 (ξ) + ξ−ρ

(2.64)

At the disease-free steady state a11 = F2 (K, 0) and a12 = 0. Hence, equation (2.64) reduces to 1 − F2 (K, 0)Pˆ1 (ξ) = 0. (2.65)

dY (0) Suppose > 1, i.e., Pˆ (q + D(K))F2 (K, 0) > 1, and condition (2.54) holds, so that dX an endemic steady state exists also. Set ξ = u + iv, so that equation (2.65) is equivalent to the pair R∞ F2 (K, 0) 0 e−us cos vs P1 (s)ds = 1, (2.66) R ∞ −us e sin vsP (s)ds = 0. 1 0 Let v = 0, then, the pair reduces to

F2 (K, 0) Z

∞

Z

∞

e−us P1 (s)ds = 1,

(2.67)

0

e−us P1 (s)ds is a monotone decreasing function of u. When u = 0, where 0 Z ∞ −us e P1 (s)ds reduces to Pˆ (q+D(K)). Having assumed that Pˆ (q+D(K))F2 (K, 0) > 0

Epidemiology of Corruption and Disease Transmission as a Saturable Interaction 21 1, it follows that equation (2.65) has a solution which is greater than zero. Hence, a positive characteristic root exists. If Pˆ (q + D(K))F2 (K, 0) > 1 and the condition (2.54) holds, then, an endemic steady state exists and the disease-free steady state is unstable. If Pˆ (q + D(K))F2 (K, 0) < 1, there may be no endemic steady state. Consider again the pair in (2.66). Since we are interested in u being positive or negative, we will consider the first equation of (2.66) alone, regarding v as a parameter of the equation. And here we ask: Can we find a sufficient condition on P1 that guarantees that all solutions of the second equation of (2.66) are negative for all v? We look for an equation in u which is independent of v such that if its solutions are negative, then, all solutions of the first equation of (2.66) are necessarily negative. Since | cos vs| Z ≤ 1, it follows that Z ∞

e−us cos vsP1 (s) ≤ e−us P1 (s). Hence, F2 (K, 0)

Z

0

∞

e−us cos vsP1 (s)ds ≤

∞

e−us P1 (s)ds.

0

e−us P1 (s)ds = 1,

(2.68)

0

is an equation in u such that any solution of the first equation in (2.66) for any v is less than Z ∞

or equal to its solution. F2 (K, 0)

e−us P1 (s)ds is a monotone decreasing function of

0

u. When u = 0, it takes the value Z ∞ P1 (s)ds = F2 (K, 0)Pˆ (q + D(K)) < 1, F2 (K, 0) 0

by our assumption. It follows that equation (2.67) has a negative solution which is also unique. Hence, all characteristic roots of equation (2.65) have negative real parts provided F2 (K, 0)Pˆ (q + D(K)) < 1. Thus, the disease-free steady state is asymptotically stable if F2 (K, 0)Pˆ (q + D(K)) < 1, and an endemic steady state may not exist. Next is to consider the stability of an endemic steady state. We assume that Pˆ (q + D(K))F2 (K, 0) > 1 and that condition (2.19) holds. The characteristic equation (2.64) for an endemic state can be written as  Z s Z ∞ ρ(s−τ ) P3 (τ )e dτ e−ξs ds = 1. (2.69) a11 P1 (s) − qa12 0

0

Define W (s) = a11 P1 (s) − qa12 be written as

Z

∞

Z

0

s

P3 (τ )eρ(s−τ ) dτ,

s ≥ 0. Then, equation (2.69) can

W (s)e−(u+iv)s ds = 1 where ξ = u + iv.

0

Hence, equation (2.69) is equivalent to the pair R∞ −us cos vs ds = 1, 0 W (s)e R∞ 0

(2.70)

W (s)e−us sin vs ds = 0.

Let v = 0, then, the first equation in (2.70) is satisfied and the second equation reduces to Z ∞ W (s)e−us ds = 1. (2.71) 0

22

C.O.A. Sowunmi

Solving Z ∞equation (2.71) for u amounts to finding intersection of the curve W (s)e−us ds with the straight line z = 1. z= 0 Z ∞ Z ∞ −us W (s)e ds is a monotone decreasing function of u, thus, if W (s)ds > 1, there 0 0 Z ∞ W (s)ds > 1 an will be a unique intersection at a point (u, 1) where u > 0. Hence, if 0

endemic state if it exists will be unstable. For stability of a possible endemic state, we consider again the pair of equations in (2.70). We are interested in conditions which ensure that all solutions (u, v) have u < 0, regardless of v. Let us therefore consider only the first equation of (2.70) as an equation in u with v as parameter. We look for an equation in u which is independent of v and is such that if its solutions are negative, then, all solutions of the first equation of (2.70) as an equation in u, are necessarily negative. Since | cos vs| < 1, therefore, W (s) cos vs ≤ |W (s)|, Z ∞ |W (s)|e−us ds = 1, (2.72) 0

is such an equation. Since the left hand side of equation (2.72) is a monotone decreasing Z ∞ |W (s)|ds < 1 is a sufficient condition for the asymptotic function of u, it follows that 0

stability of an endemic state if it exists. We therefore have the following result.

Theorem 2 Given the model whose governing equations are (2.36 - 2.37), the second equation of (2.38) and (2.39) together with the conditions on the functions P , B and D, there always exists a disease-free steady state. An endemic steady state exists if Pˆ (q + D(K))F2 (K, 0) > 1 and condition (2.54) also holds. It may otherwise not exist. The disease-free steady state is stable if Pˆ (q + D(K))F2 (K, 0) < 1 and unstable if Pˆ (q + D(K))F2 (K, 0) > 1. Z s def P3 (τ )eρ(s−τ ) dτ, s ≥ 0, then, an endemic steady state if If W (s) = a11 P1 (s) − qa12 0 Z ∞ Z ∞ W (s)ds > 1. |W (s)|ds < 1, and unstable if it exists, is stable if 0

0

2.5.

Age-dependent Infectivity and Existence of Steady States

When infectivity is age-dependent, the governing equations are the same except that equation (2.37) is replaced by the second equations of (2.41) and (2.38), respectively. The analysis in subsection 2.3 remains the same up till equation (2.47). From equation the first equation of (2.38) at a steady state, Z ∞ π(α)F(N∞ − I∞ , J∞ (α), α)dα. (2.73) J∞ (0) = 0

Set x=

Z

∞ 0

π(α)F(N∞ − I∞ , J∞ (α), α)dα.

(2.74)

Epidemiology of Corruption and Disease Transmission as a Saturable Interaction 23 Then, equations (2.44) and (2.49) can be written respectively as J∞ (s) = xP (s)e−λs , (2.75) I∞

= xPˆ (λ).

Hence, equation (2.73) becomes Z ∞ π(α)F(N∞ − xPˆ (λ), xP (α)e−λα , α)dα. x=

(2.76)

0

Equation (2.42) becomes (B(N∞ ) − D(N∞ ))N∞ = qxPˆ (λ).

(2.77)

Equations (2.76) and (2.77) are a simultaneous pair in x and N∞ . We observe that λ = q + D(N∞ ) which therefore depends on N∞ . At a disease-free equilibrium, I∞ = 0, whence J∞ (s) = 0. Thus, x = 0. When x = 0, the r.h.s. of equation (2.76) vanishes by a condition on F. Hence, equation (2.76) is satisfied by x = 0, for any N∞ . The l.h.s. of equation (2.77) vanishes when N∞ = 0 or N∞ = K. The first is trivial, hence, (N∞ , I∞ ) = (K, 0) is a solution of the pair (2.76) and (2.77). Therefore, the disease-free steady state always exists. An endemic steady state exists if the pair of equations has a solution (N∞ , x), where x > 0. As in subsection 2.3, we try to solve equation (2.42) for N∞ as a function of x, then, substitute for N∞ on the r.h.s. of equation (2.76) so that we have an equation in x alone. Equation (2.77) can be solved for N∞ as a function of x, provided the function f∗ (N∞ ) =

(B(N∞ ) − D(N∞ ))N∞ , Pˆ (q + D(N∞ ))

is invertible in some open interval. = Pˆ (q + D(N ))[{B(N ) − D(N ) + N (B ′ (N ) − D′ (N )}−

f∗′ (N )

−(B(N ) − D(N )N Pˆ ′ (q + D(N ))D′ (N )]/{Pˆ (q + D(N ))}2 , F∗′ (K) =

(2.78)

K{B ′ (K) − D′ (K)} < 0. Pˆ (q + D(K))

For sufficiently small N , f∗′ (N ) > 0. Hence, ∃ a maximal open interval (K00 , K) where f∗′ < 0, and f∗′ (K00 ) = 0. By the Inverse Function Theorem, f∗ is invertible over (K00 , K). Hence there exists an open interval (0, I00 ) such that f∗−1 maps (0, I00 ) onto (K00 , K) continuously. We now extend f∗−1 to [0, I00 ] by continuity and denote it by g∗ . Since K is the carrying capacity, N∞ ∈ (K00 , K). Equation (2.76) can now be written as Z ∞ π(α)F(g∗ (x) − xPˆ (q + D · g∗ (x), xP (α)e−α(q+Doh(α)) , α)dα. (2.79) x= 0

24

C.O.A. Sowunmi

Solving equation (2.77) amounts to finding the non-zero intersections of the line Y = X with the curve Z ∞ π(α)F{g∗ (X) − X Pˆ (q + Dog∗ (X)), XP (α)e−α(q+Dog∗ (X)) , α}dα. Y = 0

Suppose

dY (0) > 1, i.e., dX Z ∞

π(α)F2 (K, 0, α)P (α)e−α(q+D(K)) dα > 1,

(2.80)

0

then, equation (2.77) has non-zero solution, if, but not only if, there is X2 , 0 < X2 such that Z ∞ π(α)F{g∗ (X2 ) − X2 Pˆ (q + Dog∗ (X2 )), X2 P (α)e−α(q+Dog∗ (X2 )) , α}dα < X2 . 0

(2.81) To obtain a sufficient condition with epidemiological significance, we involve the generalised law of the minimum which now takes the form: F(S(t), J(t, α), α) ≤ min{k1 S(t), k2 (α)J(t, α)},

(2.82)

for k1 , k2 (α) > 0, α ≥ 0. Set λ∗ = q + Dog∗ (X2 ). We assume that π(α) and k2 (α) are bounded and define K2 = sup k2 (α), Π = sup π(α). α

α

Suppose ΠK2

Z

∞

P (α)eαλ∗ dα < 1.

(2.83)

0

Then, by condition (2.82), Z ∞ π(α)J(g∗ (X2 ) − X2 Pˆ (q + Dog∗ (X2 )), X2 P (α)e−αλ∗ , α)dα 0

≤ X2 ΠK2

Z

∞

P (α)e−αλ∗ dα < X2 .

0

Thus, (2.83) implies (2.81). The interpretation of condition (2.83) is similar to that of (2.54). dY (0) Hence, if > 1 and the infectives numbering X2 do less than reproduce themselves, dX there will exist an endemic steady state. Otherwise, there may be no endemic steady state.

2.6.

Stability of Steady States

We proceed as in subsection 2.4, however, in place of the second equation of system (2.57), we shall have, from the second equation of (2.41) Z ∞ π(α)F2 (· · · , α)j(t, α)dα + FI 1 (n(t) − i(t)) + R6 , j(t, 0) = 0

Epidemiology of Corruption and Disease Transmission as a Saturable Interaction 25 where F2 (· · · , α) = F2 (N∞ − I∞ , J∞ (α, α), FI 1

R∞

=

0

π(α)F1 (N∞ − I∞ , J∞ (α), α)dα,

and the suffices attached to F denote partial derivatives of F as explained in subsection 2.4. For reasons which will soon be obvious, we prefer to write the boundary condition for j in terms of j as j(t, 0) =

Z

∞

{π(α)F2 (· · · , α) − FI 1 }j(t, α)dα + FI 1 n(t) + R6 .

0

(2.84)

We now have to obtain estimates of the solutions of the system of equations in j and n, namely, the first equation of (2.57), (2.84), the third equation of (2.57) and (2.59). From the first equation of (2.57), (2.84) and the last equation in (2.57) when s < t, we have, j(t, s) = P1 (s) − When s > t,

R ∞ 0

 {π(α)F2 (· · · , α) − FI 1 }j(t − s, α)dα + FI 1 n(t − s) −

P1 (s) ′ 0 P1 (σ) D (N∞ )n(t

Rs

P1 (s) j(t, s) = j0 (s − t) − P1 (s − t)

(2.85)

− s + σ)J∞ (σ)dσ + R7 .

t

P1 (s) D′ (N∞ )n(σ)J∞ (s + σ − t)dσ + R8 , P (s − t + σ) 1 0 (2.86) and from system (2.59), we obtain as in subsection 2.4, Z

n(t) = n0 eρt − q

Z

∞

dσ

Z

t

0

0

eρs j(t − s, σ)dσ + R9 .

(2.87)

It can be shown that K0 ≤ K00 . We note that the value of N∞ for the endemic steady state, where infectivity is age-dependent, may not be the same as we had when infectivity was not age-dependent. Hence, the value of ρ in equation (2.87) may differ from its value in subsection 2.4. Since (K0 , K) ⊇ (K00 , K), it follows that ρ in equation (2.87) is negative. Recall that ρ = f ′ (N∞ ). The Laplace transform of equation (2.87), using ξ as the transform variable yields n ˆ (ξ) =

q n0 − ξ−ρ ξ−ρ

Z

∞

ˆ9. ˆj(ξ, σ)dσ + R

(2.88)

0

Likewise from equations (2.85) and (2.86), we obtain ˆj(ξ, s) = e−ξs P1 (s)

hR

+P1 (s)e−ξs

∞ 0 {π(α)F2 (· · ·

Rs 0

j0 (t)eξt P1 (t) dt

i , α) − FI 1 }ˆj(ξ, α)dα + FI 1 n ˆ (ξ) +

− P1 (s)D′ (N∞ )ˆ n(ξ)

Rs 0

(s−t) ˆ 10 . e−ξt JP∞1 (s−t) dt + R (2.89)

26

C.O.A. Sowunmi

We substitute for n ˆ from equation (2.88) in equation (2.89) to obtain an equation whose linear part can be regarded as a Fredholm integral equation in ˆj(ξ, ·). For brevity, we introduce the following notation before writing out the integral equation A1 (α)

= π(α)F2 (· · · , α) − FI 1 ,

C(ξ, s) = P1 (s)

Rs 0

e−ξt j0 (s−t) P1 (s−t) dt,

G(ξ, s) = D′ (N∞ )P1 (s) Thus, ∞

Rs 0

(2.90)

(s−t) e−ξt JP∞1 (s−t) dt.

 q F I q 1 ˆ 11 . ˆj(ξ, s) = − G(ξ, s) ˆj(ξ, α)dα+C(ξ, s)+R e P1 (s)A1 (α) − e P1 (s) ξ−ρ ξ−ρ 0 (2.91) We further introduce the following notations: Z

−ξs

−ξs

A2 (α) = 1; B1 (s) = e−ξs P1 (s)., B2 (s) = −

{e−ξs P1 (s)I F1 q + qG(ξ, s)} . ξ−ρ

Hence, equation (2.91) can be written as Z ∞ ˆ 11 . ˆj(ξ, s) = {B1 (s)A1 (α) + B2 (s)A2 (α)}ˆj(ξ, α)dα + C(ξ, s) + R 0

Clearly, the linear part of the equation has a degenerate kernel, Z ∞ and applying a {B1 (s)A1 (s) + now familiar technique yields the following characteristic equation: 0

B2 (s)A2 (s)}ds = 1, i.e.,

L1 (ξ) − where L1 (ξ) =

Z

∞

L2 (ξ) = 1, ξ−ρ

(2.92)

e−ξs P1 (s)A1 (s)ds,

0

and L2 (ξ) = qI F1

Z

∞

e−ξs P1 (s)ds + qD′ (N∞ )

0

Z

∞

P1 (s) 0

Z

s 0

e−ξα J∞ (s − α) dα. P1 (s − α)

Equation (2.92) is of the same form as equation (2.64), and in order to apply the same method to it, we rewrite this equation in the form: Z

0

∞

−ξs

e

  Z ′ P1 (s)A1 (s) − q FI 1 P1 (s) + D (N∞ )

We can now state the following Theorem.

s

ρτ

e dτ 0

Z

∞ 0

 P1 (s + α − τ ) J∞ (α)dα ds. P1 (α) (2.93)

Epidemiology of Corruption and Disease Transmission as a Saturable Interaction 27 Theorem 3 Given the model whose governing equations are (2.36), the second equations of (2.38) and (2.41), and system (2.39) together with the conditions on the functions P , B and D, there always exists a disease-free steady state. An endemic steady state exists if Z ∞ π(α)F1 (K, 0, α)P (α)e−α(q+D(K)) dα > 1, 0

and there is X2 , 0 < X2 , such that ΠK2

Z

∞

P (α)e−αλ∗ dα < 1.

0

Otherwise, there may be no endemic steady state. The disease-free steady state is stable if Z ∞ π(α)F2 (K, 0, α)P (α)e−α(q+D(K)) dα < 1, 0

and unstable if Z

∞

π(α)F2 (K, 0, α)P (α)e−α(q+D(K)) da > 1.

0

If  Z def W(s) = P1 (s)A1 (s)−q F1 P1 (s) + D′ (N∞ )

s 0

s ≥ 0, then, an endemic state if it exists, is stable if Z

eρ(s−τ ) dτ Z

Z

∞ 0

 P1 (α + τ ) J∞ (α)dα , P1 (α)

∞

0

|W(s)|ds < 1, and unstable if

∞

W(s)ds > 1.

0

2.7.

Discussion

The idea of extending the framework of saturable interactions for the modelling of 2-sex population dynamics - to that of disease transmission, has worked for two SIS cases. It needs extending further to the cases with more than just the susceptible and infectives. The characteristic equation for the steady states becomes quite complicated owing to one single factor or parameter in particular, that is the fatality of infection. However, in the disease-free steady state, the characteristic equation (2.93) is Z ∞ e−ξs P1 (s)A1 (s)ds = 1, 0

where P1 (s) = e−(q+D(K))s P (s), is the only function of q. Let us write the equation as Z ∞ e−us cos vsP1 (s)A1 (s)ds = 1, X(u, v, q) = 0

28

C.O.A. Sowunmi

and Y (u, v, q) =

Z

∞

e−us sin vsP1 (s)A(s)ds = 0.

0

This pair defines u and v as implicit functions of q, provided the conditions of the Implicit Function Theorem are satisfied. In that case, if u = U (q) and v = V (q), it is easily shown dU dV that = −1, while = 0. Thus, near a disease-free steady state, increasing fatality dq dq tends to stabilize the steady state. If fatality is interpreted as removal of infectives, it is easy to see why this should be so.

2.8.

Summary of Results

2.8.1.

Age-independent Infectivity

The disease-free steady state exists always. It is stable if F2 (K, 0)Pˆ (q + D(K)) < 1, and unstable if F2 (K, 0)Pˆ (q + D(K)) > 1. The endemic steady state exists if F2 (K, 0)Pˆ (q+D(K)) > 1 and k2 Pˆ (q+Dog(qX1 )) < 1 for some X1 > 0. If F2 (K, 0)Pˆ (q + D(K)) < 1 and k2 Pˆ (q + Dog(qX2 )) > 1 for some X2Z > 0, an endemic steady state mayZnot exist. An existing endemic steady state is stable ∞

∞

if

0

|W (s)|ds < 1, and unstable if

W (s)ds > 1, where

0

W (s) = a11 P1 (s) − qa12

Rs 0

P3 (τ )eρ(s−τ ) dτ, s ≥ 0,

a11

= F2 (N∞ − I∞ , I∞ ) − F1 (N∞ − I∞ , I∞ ),

a12

= D′ (N∞ )F (N∞ − I∞ , I∞ ),

b1 (t)

=

b2 (t)

= n0 eρt + R5 ,

R∞ 0

P (s+t) −λs j0 (s)ds P (s) e

+ R3 ,

P1 (s) = e−λs P (s), P2 (s) =

R∞ 0

e−λσ P (s + σ)dσ,

P3 (s) = e−λs P2 (s). 2.8.2.

Age-dependent Infectivity

The disease-free steady state always exists. It is stable if Z ∞ π(α)F2 (K, 0, α)P (α)e−α(q+D(K)) dα < 1 0

and unstable if

Z

∞ 0

π(α)F2 (K, 0, α)P (α)e−α(q+D(K)) dα > 1.

Epidemiology of Corruption and Disease Transmission as a Saturable Interaction 29 The endemic steady state exists if Z ∞ π(α)F2 (K, 0 < α)P (α)e−α(q+D(K)) dα > 1, 0

and ∃ X2 > 0 ∋ πK2

Z

∞

P (α)e−αλ∗ dα < 1 where K2 = sup k2 (α), π = sup π(α) and α

0

α

λ∗ = q + Dog∗ (X2 ). g∗ is defined as the extension of f∗−1 by continuity to [0, I00 ]. The endemic steady state may not otherwise exist. If   Z ∞ Z s P1 (α + τ ) ρ(s−τ ) ′ e dτ W(s) = P1 (s)A1 (s) + q FI 1 P1 (s) + D (N∞ ) J∞ (α)dα , P1 (α) 0 0 Z ∞ |W(s)|ds < 1, and unstable s ≥ 0, then, an existing endemic steady state is stable if 0 Z ∞ W(s)ds > 1. if 0

P1 (s)

= e−λs P (s), where λ = q + D(K),

A1 (s)

= π(α)F2 (· · · , α) − FI 1 ,

FI 1

=

R∞ 0

π(α)F1 (N∞ − I∞ , J∞ (α), α)dα,

F2 (· · · , α) = F2 (N∞ − I∞ , J∞ (α), α).

Acknowledgments I am grateful to Prof. A.U. Afuwape of the Department of Mathematics, Obafemi Awolowo University, Ile-Ife, for a loan of the English translation of Lyapunov’s book.

References [1] C.O.A. Sowunmi, Saturation Processes, Math. Comput. Modelling 11 (1988), 250252. [2] A.M. Lyapunov, The General Problem of the Stability of Motion (Translated and edited by A.T. Fuller) Taylor and Francis (1992). [3] L. La Salle and S. Lefschetz, Stability by Liapunov’s Direct Method (with applications) Academic Press, N.Y., London (1961). [4] C.O.A. Sowunmi, A model of heterosexual population dynamics with age structure and gestation period. J. Math. Anal. Appl. 172(2), (1993), 390-411. [5] C.O.A. Sowunmi, An age-structured model of polygamy with density dependent birth and death moduli. J. Nig. Math. Soc. 11(3), (1992), 123-138.

30

C.O.A. Sowunmi

[6] C.O.A. Sowunmi, Stability of steady state and boundedness of a 2-sex population model. Nonlinear Analysis 39(2000), 693-709. Erratum 51, (2002), 903-920. [7] C.O.A. Sowunmi, Time-discrete 2-sex population model with gestation period. Mathematical modelling of population dynamics. Banach Center Publications vol. 63 (2004), 259-266. [8] F. Brauer, Models for the spread of universally fatal diseases. J. Math. Biol. 28 (1990), 451-462.

In: Infectious Disease Modelling Research Progress ISBN 978-1-60741-347-9 c 2009 Nova Science Publishers, Inc. Editors: J.M. Tchuenche, C. Chiyaka, pp. 31-84

Chapter 2

A M ATHEMATICAL A NALYSIS OF I NFLUENZA WITH T REATMENT AND VACCINATION 1

H. Rwezaura1,∗, E. Mtisi2 and J.M. Tchuenche1,† Department of Mathematics, University of Dar es Salaam, Box 35062, Dar es Salaam, Tanzania. 2 Dar es Salaam Institute of Technology, Tanzania.

Abstract The recent outbreaks of highly pathogenic avian influenza and associated human infections, arising primarily from direct contact with poultry in several regions of the world have highlighted the urgent need to prepare for the next influenza pandemic. Although measures such as closing schools, using face-masks, and keeping infected persons away from those susceptible (known as social distancing) may slow the effects of pandemic influenza, only vaccines and antiviral drugs are clearly efficacious in preventing infection or treating illness. Faced with the H5N1 pandemic threat and due to the lack of facilities for quarantine and isolation in some resource-poor countries (even though the availability of antiviral and the affordability of vaccine is also a challenging task), we construct a deterministic mathematical model with vaccination and treatment only in order to analyze their joint effect in curtailing an influenza epidemic. The results are interpreted in terms of the vaccination, treatment and vaccination and treatment-induced reproduction numbers, RV , RT , and RV T , respectively. We observe that vaccinating and treating individuals concurrently is more effective in slowing down the epidemic than concentrating on a cohort vaccination campaign or treatment campaign only. Population-level perversity cannot occur if the fitness ratio 0 < Hj < 1. Also, positivity and boundedness of solutions as well as persistence of the model are analyzed. We investigate the local and global stability of the steady states and observe that the treatment only sub-model likewise the vaccination-only sub-model undergoes the phenomenon of backward bifurcation, consequently, the full model also exhibits this phenomenon. Also, when RV T < 1, the model with inflow of infectives has a tri-stable equilibria where the disease-free equilibrium coexists with two stable endemic equilibria. Sensitivity analysis on the key parameters that drive the disease dynamics is performed in order to determine their relative importance to disease transmission. Numerical simulations are carried out to validate the model. ∗ †

E-mail address: [email protected] E-mail address: jmt [email protected]

32

H. Rwezaura, E. Mtisi and J.M. Tchuenche

MSC: 92B05, 92D30, 92C60, 93D05, 93D20. Keywords: Influenza, treatment, vaccination, basic reproduction number, stability.

1.

Introduction

Influenza, commonly known as flu, is an infectious disease of birds and mammals caused by Ribonucleic acid (RNA) virus of the family of Orthomyxoviridae. In humans, common symptoms of influenza infection are fever, sore throat, muscle pains, severe headache, coughing, weakness and fatigue. In more serious cases, influenza causes pneumonia, which can be fatal, particularly in young children and the elderly. It is transmitted from infected mammals through the air by coughs or sneezes, and from infected birds through their droppings (Wikipedia, 2007). The flu viruses are designated as types A, B, and C. Only influenza A and B cause major outbreaks and severe disease; influenza C is associated with common cold-like illness, principally in children. Influenza B virus usually causes a minor illness, but it does have the potential to cause more severe disease in older persons. Influenza A virus, however, causes pandemic (Parker et al., 2001). Historically, the number of deaths during a pandemic has varied greatly. Death rates are largely determined by four factors: the number of people who become infected, the virulence of the virus, the underlying characteristics and vulnerability of affected populations, and the effectiveness of preventive measures. Accurate predictions of mortality cannot be made before the pandemic virus emerges and begins to spread (URT, 2006). An influenza pandemic is a global epidemic caused by an especially virulent virus, newly infectious for humans, and for which there is no preexisting immunity (SPI, 2007). A pandemic occurs when a new influenza virus emerges and starts spreading as easily as normal influenza by coughing and sneezing. New influenza subtypes emerge as a result of a process called antigenic shift, which causes a sudden and major change in influenza A viruses. Because the virus is new, the human immune system will have no pre-existing immunity. This makes it likely that people who contract pandemic influenza will experience more serious disease than that caused by normal influenza and this is why pandemic strains have such potential to cause severe morbidity and mortality (SPI, 2007; WHOFS, 2007). Every year, influenza infects up to 20% of the US population, hospitalizes 200,000 people and kills 36,000 (CDC, 2007). Influenza pandemic is a rare but recurrent event. Three pandemic occurred in the previous century as shown in Table 1.1 (WHOFS, 2007). Table 1. Details of the previous three pandemics Name of Pandemic Spanish influenza Asian influenza Hong Kong influenza

Date 1918 1957 1968

Estimated deaths 40-50 million 2 million 1 million

Subtype involved H1N1 H2N2 H3N2

A Mathematical Analysis of Influenza with Treatment and Vaccination

33

The reason for the recurrent outbreaks is that the virus undergoes periodic antigenic shifts in its two outer membrane glycoproteins-hemagglutinin (H) and neuraminidase (N) for example, from H1N1 to H2N2 in 1957 and from H2N2 to H3N2 in 1968, thus, introducing a new virus into a population that has no protective serum antibody. No different subtypes of H and N have been identified for influenza B and C (Parker et al., 2001). Nowadays, the H5N1 subtype of influenza virus, also known as avian flu, has the potential to develop into a pandemic strain (ACP, 2006). The World Health Organization (WHO) has reported human cases of avian influenza A (H5N1), arising primarily from direct contact with sick or dead poultry or wild birds, or visiting a live poultry market, in Asia, Africa, the Pacific, Europe and the Near East. Infection in humans has occurred in three distinct waves of activity, since late December 2003. Overall, mortality in reported H5N1 cases is approximately 60% (CDCa, 2007). While the H5N1 virus does not yet infect people easily, infection in humans is very serious when it occurs, and so far, more than half of the people reported infected have died. Rare cases of limited human-to-human spread of H5N1 virus may have occurred, but there is no evidence of sustained human-to-human transmission. Nonetheless, because all influenza viruses have the ability to change, scientists are concerned that H5N1 viruses one day could be able to infect humans more easily and consequently spread from person-to-person. Since H5N1 viruses have not infected many humans worldwide, there is little or no immune protection against them in the human population and an influenza pandemic could begin if sustained H5N1 virus transmission occurs (CDCa, 2007). Prevention of influenza, particularly when it becomes pandemic, may be attempted by many measures, such as closing schools, using face masks, and keeping infected persons away from those susceptible. However, none of these measures are of clear value in preventing infection, even if they could be accomplished, but only vaccines and antiviral drugs are clearly efficacious in preventing infection or treating illness (Monto, 2006). They are the two most important medical interventions for reducing illness and deaths during a pandemic. However, until there is influenza pandemic, there is no evidence that vaccines or antiviral used in the treatment or prevention of such an outbreak would decrease morbidity and mortality. Since a pandemic vaccine is unlikely to be available during the first 4 to 6 months of the pandemic, appropriate use of antiviral may play an important role to limit mortality and morbidity, minimize social disruption, and reduce economic impact (HHS, 2007). The efficacy and effectiveness of influenza vaccines depend primarily on the age and immunocompetence of the vaccine recipient, the degree of similarity between the viruses in the vaccine and those in circulation, and the outcome being measured (CDCb, 2007). The impact of any vaccination strategy and antiviral drug use depends on how soon the pandemic starts. If it starts when there is no vaccine available and only limited supplies of antiviral drugs, it is more likely that targeted strategies for vaccine and antiviral drug use will be the only potential options. Flu viruses come in different strains that constantly mutate (making influenza quite unpredictable), until one that few people have immunity against emerges and is able to spread widely. Therefore, the flu vaccine must be reformulated every year to keep up with the fastevolving influenza virus. There is a potential danger, especially for developing countries where adequate health facilities are not generally available. Due to the lack of facilities in a country such as Tanzania, and the fact that most individuals live on less than a dollar a

34

H. Rwezaura, E. Mtisi and J.M. Tchuenche

day, affordability of vaccine in case of an epidemic is a challenging task. Consequently, the epidemic might spread and the only combating measure is the use of antiviral drugs. Therefore, we consider both vaccination and treatment in order to analyze their joint effect in curtailing an epidemic. This study is worth in its own right and we do hope the outcome will be useful to public health decision makers in the event of any epidemic/pandemic in the developing world.

1.1.

Motivation and Objectives

The motivation for writing this chapter comes from declared facts by the WHO that the world is as close as ever to the next pandemic (Democratis et al., 2006). Faced with the H5N1 pandemic threat, strategies designed to contain an emerging pandemic should be considered a public health priority. A number of mathematical models, using stochastic as well as deterministic formulations have been carried out to quantify the burden of a potential flu pandemic and assess various interventions (Alexander et al., 2007; Longini et al. 2004; Nu˜ no et al., 2006; Longini et al., 2005; Chowell et al., 2005 and Arino et al., 2008, to name but a few). Although findings in these studies seem reassuring, they assume the availability of an unlimited supply of antiviral drugs, vaccine, adequate health facilities for isolation and quarantine. These assumptions may of course not be realistic, especially in most resourcepoor countries in Sub-Saharan Africa. Given these limitations, whether a strategic use of vaccination and treatment can control the spread of influenza within a certain population is of great public health interest (Alexander et al., 2004). Since the failure of current influenza vaccines to protect all vaccine recipients warrants the determination of conditions necessary for a substantial reduction, approaching eradication of influenza infection in a population, this chapter therefore aims at exploring via mathematical modeling, the combined effects of both immunization (with a partially effective vaccine) and treatment on the transmission dynamics of influenza infection. Alexander et al., (2004) addressed the question of whether such a vaccine could ever completely stop the spread of infection and determines the minimal vaccine efficacy and vaccination rate required to control or eradicate infection in a population. Thus, the vaccination-only sub-model will not be analyzed herein. Understanding and analyzing the spread and control of an influenza epidemic in order to assess the ability to control the epidemic will be helpful in guiding intervention and policy decision. In addition to providing conceptual results such as the basic reproduction numbers, graphical representations of the model are illustrated.

1.2.

Methodology

We construct a deterministic mathematical model using a system of ordinary differential equations where the total population, denoted by N (t), is divided into a number of mutually exclusive sub-populations according to their epidemiological (or disease) status: susceptible (S), Infected and infectious (I), vaccinated (V ), Treated (T ), fully protected via vaccination (C) and recovered (R). We analyze the model qualitatively to determine the criteria for

A Mathematical Analysis of Influenza with Treatment and Vaccination

35

containing an epidemic/pandemic influenza in the presence of treatment and vaccination and use it to compute epidemic threshold numbers necessary for community-wide control of the disease. A sensitivity analysis on the key parameters that drive the disease dynamics is also performed. We evaluate the possibility that the disease may take-off in the absence of antiviral and vaccine via the classical basic reproduction number denoted by R0 . With this threshold, the qualitative mathematical properties of the model are studied, and the epidemiological consequences are discussed. If the parameter R0 is less than unity (R0 < 1), then the disease cannot spread in the population but, if R0 > 1, then, the spread of the disease in the population is always possible (Hethcote, 2000). R0 is calculated using the next generation operator method (Diekmann et al., 1990; van den Driessche and Watmough, 2002). This epidemiological quantity measures the average number of new cases generated by an infectious individual, for the duration of his/her infectiousness in a completely susceptible population. When vaccination and treatment strategies are applied, the aim is to reduce the threshold parameter to below unity in that strategy in order to prevent further spread of the disease. Alexander et al., (2004) analyzed a vaccination-only model with mass action incidence. We believe their results will hold in the standard incidence model considered herein, and for this reason, the vaccination-only sub-model will not be analyzed. The associated reproduction number for the vaccination-only sub-model denoted by RV will be computed for the purpose of analyzing the effects of the combination of vaccination and treatment on therapeutic measures taken one at a time. Similarly, for the antiviral-only scenario, the corresponding antiviral reproduction number is denoted by RT . For the case where vaccination and treatment are administered concurrently, the corresponding combined reproduction number is denoted by RV T . The epidemiological significance of the combined reproduction number, which represents the average number of new cases generated by a primary infectious individual in a population where the combined interventions are implemented, is that the pandemic may effectively be controlled (owing to the phenomenon of backward bifurcation) if the combined interventions can bring this threshold quantity to a value less than unity (the pandemic would persist otherwise). We use Descartes’ Rule of signs to analyze the local stability of the model and the the Korobeinikov-Maini’s (2004) type Lyapunov function for the persistence of the mass action incidence model. We investigate the possibility of periodic solutions by applying the classical Bendixon-Dulac criterion. A brief survey on previous works provides the context of the present study.

1.3.

Brief Review of Previous Studies

Mathematical models have long been recognized as useful tools in exploring complicated relationships underlying infectious disease transmission processes. Usually, the structure of the model is based on a set of causal hypotheses that describe current understanding of how different processes are interrelated (Spear, 2002). Unlike statistical models, their parameters generally have physical or biological meaning that allows their values to be estimated from literature as well as from field and experimental data. Like statistical mod-

36

H. Rwezaura, E. Mtisi and J.M. Tchuenche

els, mathematical models can be used to test competing hypotheses underlying research questions. Longini et al., (2004) used stochastic epidemic simulations to investigate the effectiveness of targeted antiviral prophylaxis to contain influenza and compare it with that of vaccination, if vaccine were available. They showed that although vaccination would be the best means for controlling influenza, 80% targeted antiviral prophylaxis for a possible 6 to 8 weeks is almost as effective as vaccinating 80% of the entire population, and thus targeted antiviral prophylaxis has potential as an effective measure for containing influenza until adequate quantities of vaccine are available. On the other hand, Lipsitch et al., (2007) designed and analyzed a deterministic compartmental model of the transmission of oseltamivir-sensitive and -resistant influenza infections during a pandemic and found that in the event of a future influenza pandemic for which antiviral drugs are used, there is a risk of resistance emerging and resistant strains causing illness in a substantial number of people. This would counteract the benefits of antiviral drugs but not eliminate those benefits entirely. Their model relies on realistic assumptions and it is hard to know how closely the model will mimic a real-life situation until the properties of an actual pandemic strain are known. They suggested that even if the benefits of antiviral drug use to control an influenza pandemic may be reduced, although not completely offset by drug resistance, the risk of resistance should be considered in pandemic planning and monitored closely during a pandemic. The use of vaccines to prevent human diseases is one of the major successes of modern medicine (Moghadas, 2004a), but it is well-known that, although vaccines can reduce or eliminate the incidence of infection, not all vaccines are 100% effective (McLean and Blower, 1993; Gandon et al., 2001). Some recent clinical studies have focused on the effect of partially effective (or imperfect) vaccines (waning and/or incomplete immunity) in controlling the transmission of infectious diseases. Alexander et al., (2004) evaluated the impact of a partially effective preventive vaccine on the control of influenza infection, using a new deterministic mathematical model. Their results showed that the disease can be controlled if infected individuals are not continuously introduced into the population and the eradication of the disease may not be feasible if infected individuals are continuously recruited. Therefore, increasing the level of vaccination will always reduce the level of epidemicity of the disease and vaccination can still be used to prevent a severe epidemic. However, since influenza can also be introduced into the population by recruitment of infected individuals this poses a challenge to the developing countries as they might have no access to vaccines throughout the duration of a pandemic. The use of partially effective preventive vaccine to prevent influenza infections in case of a pandemic will still be a challenging task for many developing countries like Tanzania. Longini et al., (2005) assessed the combined role of targeted prophylaxis, quarantine and pre-vaccination in containing an emerging influenza strain at the source and found that combinations of targeted antiviral prophylaxis, pre-vaccination, and quarantine could contain the strains. Nu˜ no et al., (2006) analyzed a more complex compartmental model to study the role of hospital and community control measures, antiviral and vaccination in combating a potential flu pandemic in a population of high and low-risk individuals. Their results suggested that countries with limited antiviral stockpiles should emphasize their use therapeutically (rather than prophylactically).However, countries with large antiviral stockpiles can achieve

A Mathematical Analysis of Influenza with Treatment and Vaccination

37

greater reductions in disease burden by implementing them both prophylactically and therapeutically. Their study showed that the prospect of combating the next flu pandemic is promising, provided a number of control measures (especially the use of a combined intervention strategies) are put in place in an efficient manner. Likewise Chowell et al., (2005) used a compartmental epidemic model to describe the transmission dynamics of pandemic influenza. Their results indicated that containment of the next influenza pandemic could require the simultaneous implementation of multiple component interventions that include effective isolation of hospitalized cases and reductions in the susceptibility of the general population through, increasing hygiene, using protective devices (e.g., face masks), prophylactic antiviral use and vaccination. The impact of targeted influenza vaccination and antiviral prophylaxis/treatment (oseltamivir) in high risk groups (elderly, chronic diseases), priority (essential professionals), and total populations was compared by Doyle et al., (2006) who suggested that if available initially, vaccination of the total population is preferred but if not, for priority populations, seasonal prophylaxis seems the best strategy. For high risk groups, antiviral treatment, although less effective, seems more feasible and cost effective than prophylaxis and should be chosen, especially if the availability of drugs is limited. Their results suggested a strong role for antiviral in an influenza pandemic. Alexander et al., (2007) developed a mathematical model to evaluate the effect of delay (which is not considered herein) in initiating a course of antiviral treatment on containing a pandemic. Their model assumed the availability of an unlimited supply of antiviral drugs for the entire course of a pandemic. Their findings showed that, due to uncertainty in the attack rate of a pandemic strain, initiating therapy immediately upon diagnosis can significantly increase the likelihood of disease control and substantially reduce the required community-level of treatment. This suggests that reliable diagnostic methods for influenza cases should be rapidly implemented within an antiviral treatment strategy. Recently, Arino et al., (2008) analyzed a similar model to ours. This study was almost completed before the authors learned about their work. Even though the titles have the same keywords, there are significant differences in the approach and content. Nevertheless, the common results agree on the point. They considered treated -susceptible, -latent infective and -asymptomatic classes, respectively, derived analytic expression for the final size of the epidemic and compared their numerical computations with those of recent stochastic simulation influenza models, which have great potential for predictions of outcomes and design of control strategies. However, some of the stochastic model parameters have considerable uncertainty and are not very amenable to sensitivity analysis. Nevertheless, compartmental models effectively allow a complete analysis of parameter space. For more details, see Arino et al., (2008) and the references therein. While we investigate local and global stability of steady states and observe that the treatment only sub-model exhibits the phenomenon of backward bifurcation which has important public health implications, they do not. An economic analysis of influenza vaccination and anti-retroviral treatment for healthy working adults has been carried out by Lee et al., (2002), and it was found that vaccination in a variety of settings is cost beneficial in most influenza seasons. The model is robust enough to provide great insights in curtailing an outbreak by analysing the impact of both vaccination and treatment in combating the spread of the disease. The rest of this chapter is organised as follows: In the next Section, we formulate the

38

H. Rwezaura, E. Mtisi and J.M. Tchuenche

model and carry out its complete analysis. Section 3 presents the sensitivity analysis and numerical simulations, while in Section 4, we briefly discuss and conclude the chapter.

2. 2.1.

Model Framework and Analysis Model Framework

The model consists of six ordinary differential equations which specify the rate of change of six categories of individuals in the population. The total population, denoted by N (t) consisting of a class of susceptible individuals (S), a class of vaccinated individuals (V ), a class of infected and infectious individuals (I), a class of treated individual (T ), a class of individuals who recover with temporary immunity (R) and a class of fully protected individuals via vaccination (C). The susceptible population is increased by recruitment of individuals (either by birth or immigration), and by loss of immunity. In this class, individuals can incur the disease but are not yet infected. Individuals in this class are reduced through vaccination, infection and by natural death. The population of vaccinated is increased by vaccination of susceptible. Vaccinated individuals may become fully protected or may become infected at lower rate than unvaccinated (those in class S) because the vaccine does not confer immunity to some vaccine recipients. The vaccinated class is thus diminished by this infection (moving to class I), by moving to a class of fully protected individuals (C) and further decreased by waning of vaccine-based immunity (moving to class S), and by natural death. G E U/

1  U /

S P

K

Z

1  H E

V

P

N

I P

J

T

WS

P

OD

C P

Figure 1. Model flowchart.

R P

1  O D

A Mathematical Analysis of Influenza with Treatment and Vaccination

2.2.

39

Descriptions of Variables and Parameters

The population of infected individuals is increased by recruitment of infected individuals from outside the population, as well as by infection of susceptible individuals including those who remain susceptible despite being vaccinated. It is diminished by natural death, by disease-induced death and by treatment (moving to class T ). The population of treated individuals is increased by treating the infected individuals and is decreased by individuals recovering from their infection (moving to class R), by disease-induced death and by natural death. Since the immunity acquired by infection wanes with time, the recovered individuals become susceptible to the disease again. Thus, recovered class is increased by individuals recovering from their infection through treatment and is decreased as the natural immunity wanes (moving back to class S) and by natural death. The transfer diagram for these processes is shown in Figure 1. The notations used in the flow diagram are described below. S(t) V(t) I(t) T(t)

: : : :

C(t) R(t) η ω γ κ τ ǫ π λ

: : : : : : : : : :

β Λ

: :

δ ρ µ

: : :

α

:

Susceptible individuals at time t Vaccinated group at time t Infected/infectious class at time t Individuals receiving treatment at time t (may also represent the hospitalized class) Protected class at time t Recovered class at time t Rate at which susceptible individuals are vaccinated Rate at which the vaccine-based immunity wanes Rate of acquiring protective antibodies Treatment rate Recovery rate Vaccine efficacy (ǫ ∈ [0, 1]) Measures the drug efficacy in increasing the recovery rate (π ≥ 1) Effectiveness of the drug as a reduction factor in disease-induced death (0 < λ ≤ 1) Contact rate and βI N the force of infection Recruitment rate of individuals into the population, a fraction ρ of which are infective Rate at which the immunity from previous infections wanes Fraction of recruited individuals who are already infected Natural death (or emigration) rate which is assumed to be the same for all sub-populations Disease induced death rate.

Note that if ǫ = 0, then, the vaccine is useless and if ǫ = 1, the vaccine is 100% effective, while if 0 < ǫ < 1, the vaccine is leaky. Also, in the I-class, λ(≥ 1) may

40

H. Rwezaura, E. Mtisi and J.M. Tchuenche

represent compliance (those who refuse to seek medical support).

2.3. The Model βSI dS = (1 − ρ)Λ − + δR − (µ + η)S + ωV, dt N dV βV I = ηS − (1 − ǫ) − (γ + ω + µ)V, dt N dI βSI βV I = ρΛ + + (1 − ǫ) − (κ + µ + λα)I, dt N N

(2.1)

dT = κI − (µ + τ π)T − (1 − λ)αT, dt dR = τ πT − (µ + δ)R, dt dC = γV − µC, dt with initial conditions S(0) = S0 , V (0) = V0 , I(0) = I0 , T (0) = T0 , R(0) = R0 , C(0) = C0 and N (t) = S(t)+V (t)+I(t)+T (t)+R(t)+C(t), where N (t) is the total population at time t.

2.4.

Model Analysis

Lemma 1 The feasible set of system (2.1) is given by 6 : S+V +I +T +R+C ≤ Ω = {(S, V, I, T, R, C) ∈ R I+

Λ µ }.

Proof. Adding the differential equations in the model system (2.1) gives dN = Λ − µN − λαI − (1 − λ)αT ≤ Λ − µN. dt

(2.2)

The feasible region for (2.1) is 6 Ω = {(S, V, I, T, R, C) ∈ R I+ : S+V +I +T +R+C ≤

Λ }. µ

(2.3)

Since from (2.2) lim sup N (t) ≤ t→∞

Λ , µ

(2.4)

then, the global attractor of (2.1) is contained in Ω. Thus, the dynamics of the model will be considered in Ω. Lemma 2 The set Ω is positively invariant.

A Mathematical Analysis of Influenza with Treatment and Vaccination

41

Proof. On the side of Ω, we have S=0⇒

dS = (1 − ρ)Λ + δR + ωV, dt

V =0⇒

dV = ηS, dt

I=0⇒

dI = ρΛ, dt

T =0⇒

dT = κI, dt

R=0⇒

dR = τ πT, dt

C=0⇒

dC = γV. dt

(2.5)

Since all the parameters are positive and any vector field on ∂Ω, the boundary of Ω, is tangent or has inward direction, the region Ω of biological interest is positively invariant, 0

and any solution of system (2.1) with initial point on ∂Ω enters Ω, the interior of Ω and remains there. Lemma 3 Existence: A solution of model system (2.1) is feasible in Ω. Proof. Since system (2.1) is dissipative, that is, all feasible solutions are uniformly bounded 6 , then, any solution with initial value in Ω attains its maximum in a proper subset Ω ⊂ R I+ and remain in there. Hence, Ω is compact and positively invariant.

2.5.

Positivity of Solutions

Model system (2.1) describes the dynamics of a human population, therefore, it is important to prove that the susceptible, vaccinated, infected (infectious), treated and recovered individuals are non-negative for all time. In other words, we want to prove that all solutions of the system with positive initial data will remain positive for all t > 0. Theorem 4 Positivity: Let the initial data be S(0) > 0, V (0) ≥ 0, I(0) ≥ 0, T (0) ≥ 0, R(0), C(0) ≥ 0 ∈ Ω. Then, solutions of S(t), V (t), I(t), T (t), R(t), C(t) of system (2.1) are positive for all t > 0. dR Proof. For t ∈ [0, T¯] , = τ πT − (µ + δ)R ≥ −(µ + δ)R, and by freshman integration, dt we obtain R(t) ≥ R(0)e−(µ+δ)t ≥ 0, µ + δ < +∞. (2.6) A similar argument on the remaining variables yields: T (t) ≥ T (0)e−[µ+τ π+(1−λ)α]t ≥ 0, −(µ+γ+ω)t−(1−ǫ)β

V (t) ≥ V (0)e

C(t) ≥ C(0)e−µt ≥ 0

I(s) 0 N (s) ds

Rt

I(t) ≥ I(0)e−(µ+κ+λα)t ≥ 0, ≥ 0,

−(µ+η)t−β

S(t) ≥ S(0)e

I(ζ) 0 N (ζ) dζ

Rt

≥ 0,

.

(2.7) This shows that the solution of (2.1) is such that min{S(t), V (t), I(t), T (t), R(t), C(t)} ≥ 0 in its interval of existence Ω.

42

H. Rwezaura, E. Mtisi and J.M. Tchuenche 2000 1800 1600

Susceptible

1400 1200 1000 800 600 400 200 0

0

1000

2000

3000 4000 Treated

5000

6000

7000

Figure 2. Evolution of S and T classes towards equilibrium. Lemma 5 The ω-limit set of any orbit of the system (2.1) with initial point in Ω is a rest point. Proof. Since the vector field related to (2.1) is inward on ∂Ω and Ω is compact, then, the 0

omega-limit set of each orbit with initial point in Ω is a non-empty subset of Ω. By the Poincar´ e-Bendixon Theorem (Perko, 2000), this set must be an equilibrium point. Lemma 5 is graphically illustrated in Figure 2. For simplicity, we show the phase portrait of S and T only, which converges to a positive equilibrium.

2.6.

The Model in the Absence of Inflow of Infectives (ρ = 0)

At an equilibrium point, the RHS of system (2.1) equals zero and substituting ρ = 0, we have βS ∗ I ∗ + δR∗ − (µ + η)S ∗ + ωV ∗ = 0, Λ− N∗ ηS ∗ − (1 − ǫ)

βV ∗ I ∗ − (γ + ω + µ)V ∗ = 0, N∗

βS ∗ I ∗ βV ∗ I ∗ + (1 − ǫ) − (κ + µ + λα)I ∗ = 0, N∗ N∗ κI ∗ − (µ + τ π)T ∗ − (1 − λ)αT ∗ = 0, τ πT ∗ − (µ + δ)R∗ = 0, γV ∗ − µC ∗ = 0.

(2.8)

A Mathematical Analysis of Influenza with Treatment and Vaccination

43

Thus, R∗ =

τ πT ∗ , µ+δ

C∗ =

T∗ =

κI ∗ , µ + τ π + (1 − λ)α

V∗ =

ηN ∗ S ∗ , (1 − ǫ)βI ∗ + (µ + ω + γ)N ∗

γV ∗ µ (2.9)

S∗ =

[Λ + δR∗ + ωV ∗ ]N ∗ βI ∗ + N ∗ (µ + η)

2.7. Non-existence of the Trivial Equilibrium Communicable diseases may be introduces into a community by the arrival of new infective individuals from outside the population. For example, travelers may return home from a foreign trip with an infection acquired abroad (Brauer and van den Driessche, 2001). For as long as the recruitment term Λ is not zero, the population will not be extinct. This implies that there is no trivial equilibrium point, i.e., (S ∗ , V ∗ , I ∗ , T ∗ , R∗ , C ∗ ) 6= (0, 0, 0, 0, 0, 0).

2.8.

Disease-Free Equilibrium (E0 )

At the disease-free equilibrium, we have I0 = 0 = T0 = R0 , therefore, Λη Ληγ Λ(µ + ω + γ) , , 0, 0, 0, ). (µ + γ)(µ + η) + µω (µ + γ)(µ + η) + µω µ(µ + γ)(µ + η) + µ2 ω (2.10) Remark: If β = 0 so that the only infectives are those who have entered the population from outside, this reduces system to a linear non-homogeneous system for which every ρΛ solution approaches a certain equilibrium E e , with I e = . We would expect a κ + µ + λα steady state with β > 0 to satisfy I ∗ ≥ I e . E0 = (

2.9.

Computation of the Reproduction Numbers R0 , RV , RT and RV T

Applying van den Driessche and Watmough technique (2002), the treatment and vaccination-induced reproduction number RV T is given by: RV T =

βµ[µ + γ + ω + (1 − ǫ)η] . [(µ + γ)(µ + η) + µω](µ + κ + λα)

(2.11)

The threshold quantity RV T can be interpreted as the number of infected people generated by one infected individual introduced into the population in the presence of vaccination and treatment. In the absence of treatment κ = 0, we have the vaccination-induced reproduction number given by: βµ[µ + γ + ω + (1 − ǫ)η] . (2.12) RV = [(µ + γ)(µ + η) + µω](µ + λα) In absence of vaccination η = 0 = γ = ω = ǫ, we have the treatment-induced reproduction number given by β . (2.13) RT = µ + κ + λα

44

H. Rwezaura, E. Mtisi and J.M. Tchuenche

In the absence of any intervention strategy (i.e., without vaccination and treatment) η = 0 = γ = ω = ǫ = κ, we have the basic reproduction number given by β . µ + λα

R0 =

(2.14)

We note that in the I-class, λα ≃ α, that is λ = 1 in this case, since those who are administered drugs are in the T -class.

2.10.

Local Stability of the Disease-Free Equilibrium E0

One of the most important concerns in the analysis of epidemiological models is the determination of the asymptotic behaviour of their solutions which is usually based on the stability of the associated equilibria (Moghadas, 2004a). These models typically consist of a disease-free equilibrium and at least one endemic equilibrium. The local stability of the disease-free equilibrium is determined based on a threshold parameter RV T , known as the vaccination and treatment-induced reproductive number. The Jacobian matrix at E0 is given by

JE0



  =  

−K11 ω −βS0 0 δ η −K22 −(1 − ǫ)βV0 0 0 0 0 K33 0 0 0 0 κ −K44 0 0 0 0 τπ −K55

     

(2.15)

where

βS0 + (1 − ǫ)βV0 − (µ + κ + λα), N0 K44 = (µ + τ π) + (1 − λ)α, K55 = (µ + δ), and Λ(µ + ω + γ) Λη S0 = , V0 = . (µ + γ)(µ + η) + µω (µ + γ)(µ + η) + µω K11 = (µ + η), K22 = (µ + ω + γ), K33 =

The eigenvalues of JE0 are ς1 = −(µ + δ), ς2 =

βS0 +(1−ǫ)βV0 N0

− (µ + κ + λα),

ς3 = −(µ + τ π + (1 − λ)α), ς4 = − 12 {2µ + η + γ + ω + ς5 = − 21 {2µ + η + γ + ω −

p (η + ω)2 + γ(γ + 2ω − 2η)}, p (η + ω)2 + γ(γ + 2ω − 2η)}.

(2.16)

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45

The following eigenvalues ς1 , ς3 , ς4 , ς5 are negative. Thus, the local stability of E0 depends on the sign of ς2 . That is, ς2 =

βS0 + (1 − ǫ)βV0 − (µ + κ + λα), N0

= −(µ + κ + λα) + = (µ + κ + λα)[

βµ[µ + γ + ω + (1 − ǫ)η] [(µ + γ)(µ + η) + µω]

(2.17)

βµ[µ + γ + ω + (1 − ǫ)η] − 1] [(µ + γ)(µ + η) + µω](µ + κ + λα)

= (µ + κ + λα)(RV T − 1). Thus we have established the following result: Lemma 6 The DFE of the model system 2.1 (when ρ = 0) is locally asymptotically stable if RV T < 1 and unstable if RV T > 1. Proof. An equilibrium is asymptotically stable if all eigenvalues have negative real parts; it is unstable if at least one eigenvalue has positive real part. Since the eigenvalues of JE0 are negative except ς2 , it follows that the DFE is locally asymptotically stable if ς2 < 0 and unstable if ς2 > 0. Note that ς2 < 0 if and only if RV T < 1 (cf. 2.17). Since under conditions (a)(i) and (c)(i) of Lemma 10 the model system 2.1 has a unique endemic equilibrium, therefore, eliminating any possibility of backward bifurcation, we can show that under those assumptions, the DFE of system 2.1 will be GAS when RV T < 1.

2.11.

Global Stability of the Disease-Free Equilibrium E0

Lemma 7 The DFE of the model system 2.1 (when ρ = 0) is globally asymptotically stable (GAS) whenever RV T < 1. Proof. The proof is based on using a comparison Theorem (cf. Lakshmikantham et al., 1989; p.31). Note first of all that we can write the equations of (I, T, R) for the system 2.1 as follows;   dI      dt  I I          dT        = ϕ(t)  T  ≤ ϕ(0)  T  , (2.18)  dt                R R dR dt

46

H. Rwezaura, E. Mtisi and J.M. Tchuenche

where  βS (1 − ǫ)βV + − (µ + κ + λα) 0 0   N ϕ(t) =  N . κ −(µ + τ π + (1 − λ)α) 0 0 τπ −(µ + δ) (2.19) and ϕ(0) is the value of ϕ(t) evaluated for the variables S(t), V (t), and N (t) when t = 0. The eigenvalues of ϕ(0) are 

ς1 = −(µ + δ), ς2 =

βS0 +(1−ǫ)βV0 N0

− (µ + κ + λα) = (µ + κ + λα)(RV T − 1),

ς3 = −(µ + τ π + (1 − λ)α), Using the fact that the eigenvalues of the matrix ϕ(0) all have negative real parts if ς2 < 0 i.e.,if RV T < 1, it follows that the differential inequality system (2.18) is stable whenever RV T < 1. Consequently, (I(t); T (t); R(t)) → (0; 0; 0) as t → ∞. Thus, by a comparison Theorem (cf. Lakshmikantham et al., 1989; p.31), (I(t); T (t); R(t)) → (0; 0; 0) as t → ∞. Substituting I = 0 = T = R in the first two equation of (2.1) gives S(t) → S ∗ , V (t) → V ∗ and C(t) → C ∗ as t → ∞. Thus, (S(t), V (t), I(t), T (t), R(t), C(t)) → (S ∗ , V ∗ , 0, 0, 0, C ∗ ) as t → ∞ for RV T < 1 and hence, E0 is GAS if RV T < 1 (for ρ = 0).

2.12.

Effects of Public Health Measures (Treatment and Vaccination)

In order to study the effects of public health measures in slowing down the spread of the influenza epidemic in a community, we investigate the role of the basic reproductive number (which is a measure of the power of a disease to invade a population under conditions that facilitates maximal growth) on influenza eradication. We rewrite the treatment-induced reproductive number, the vaccination-induced reproductive number and the vaccination and treatment-induced reproductive number as follows: The quantity RT can be rewritten as 1

βR0 , β + κR0

(2.20)

µ[µ + γ + ω + (1 − ǫ)η] , [(µ + γ)(µ + η) + µω]

(2.21)

RT = R0 [

1+

κ µ+λα

]=

while RV can be rewritten as RV = R0

A Mathematical Analysis of Influenza with Treatment and Vaccination

47

and finally, RV T can be rewritten as RV T = R0

µ[µ + γ + ω + (1 − ǫ)η] = HV T R0 , κ [(µ + γ)(µ + η) + µω][1 + µ+λα ]

(2.22)

β µ + λα RV T µ[µ + γ + ω + (1 − ǫ)η] . := = κ ] R0 [(µ + γ)(µ + η) + µω][1 + µ+λα

where R0 = and HV T

Since µ[µ + γ + ω + (1 − ǫ)η] < [(µ + γ)(µ + η) + µω][1 + HV T =

κ ] , then, µ + λα

µ[µ + γ + ω + (1 − ǫ)η] < 1. κ ] [(µ + γ)(µ + η) + µω][1 + µ+λα

(2.23)

HV T is the factor by which public health measures (treatment and vaccination) reduce the number of secondary infections. It is also referred to as fitness ratio (Smith? and Blower, 2004). Since 0 < HV T < 1, then, population level perversity is not possible and influenza vaccine combined with treatment will always have a beneficial impact. 1 We also note that in the presence of treatment only, HT = < 1, HV = κ 1 + µ+λα 1 − ǫηµ M γη < 1, where M = µ(µ + γ + ω + η). Therefore, vaccination and treatment 1+ M taken separately are always beneficial, but with minimal impact compared to the combined strategy as shown below (cf. equations 2.24 and 2.25). If R0 < 1, it is possible that influenza will not spread into an epidemic (and therapeutic measures may not be necessary) and for R0 > 1, we now determine the necessary condition for slowing down the spread of influenza. Following Hsu Schmitz (2000), we have ∆V T := R0 − RV T = R0 [1 − HV T ] for which ∆V T > 0 is expected to slow down the spread of the influenza epidemic in a community. This condition is satisfied for all 0 < η, ǫ, κ < 1. We note that under this condition, the factor H multiplying R0 is less than unity (HV T < 1), indicating that public health measures have the capability of reducing the number of secondary infections. Now, we determine the threshold level of vaccination and treatment coverage that guarantee disease eradication. It is well-known that, although vaccines can reduce or eliminate the incidence of infection, not all vaccines are 100% effective. Also, treatment efficacy is of paramount importance in curtailing an epidemics. This section addresses the following questions: What proportion of susceptible people must be immunized and/or treated in order to prevent an endemic spread of influenza? By addressing this question, one would expect to explore the impact of three major parameters associated with a vaccination program on disease transmission (Moghadas, 2004a): - vaccination coverage level (η); - efficacy of

48

H. Rwezaura, E. Mtisi and J.M. Tchuenche

vaccine (ǫ); - and treatment rate (κ). Differentiating RV T partially with respect to ǫ, η and κ, we obtain, ∂RV T µηR0 =− ∂ǫ [(µ + γ)(µ + η) + µω][1 +

κ µ+λα ]

< 0,

µR0 (µ + γ + ω)(ǫµ + γ) ∂RV T =− < 0, κ ∂η [(µ + γ)(µ + η) + µω]2 [1 + µ+λα ]

(2.24)

βµ[(µ + γ + ω) + (1 − ǫ)η] ∂RV T =− < 0. ∂κ [(µ + γ)(µ + η) + µω](µ + κ + λα)2 From (2.24), the necessary conditions ∆V T > 0,

∂RV T ∂RV T ∂RV T <0 < 0 and < 0, ∂ǫ ∂η ∂κ

(2.25)

for slowing down the epidemic are satisfied for all 0 < η, ǫ, κ < 1. Setting RV = 1, RT = 1 and RV T = 1 and solving for η, κ, ǫ gives the threshold proportion of the vaccinated rate for susceptible individuals, the treatment rate and the efficacy of vaccine respectively as follows, RV = 1 ⇒ ηc =

µ(µ + γ + ω)(R0 − 1) , µ + γ − µR0 (1 − ǫ)

RT = 1 ⇒ κc =

β(R0 − 1) , R0

RV T = 1 ⇒ η¯c =

µ(µ + γ + ω)[R0 (β − κ) − β] , µR0 [κ − (1 − ǫ)β] + β(µ + γ)

RV T = 1 ⇒ κ ¯ c = β[

(2.26)

1 β µ[(µ + γ + ω) + (1 − ǫ)η] ]= (RV − 1), − [(µ + γ)(µ + η) + µω] R0 R0

RV T = 1 ⇒ ǫ¯c =

1 1 (µ + γ + ω + η) − [(µ + γ)(µ + η) + µω](κR0 + β), η µβηR0

where η¯c and κ ¯ c are the values of ηc and κc in the presence of vaccination and treatment, respectively. Public health measures (treatment and vaccination) used to reduce the number of secondary infection on influenza will succeed in controlling the epidemic (RV T < 1) if η > ηc , κ > κc , η > η¯c , κ > κ ¯ c and ǫ > ǫ¯c . In conclusion, if the vaccine efficacy is low, influenza may not be controlled using vaccination alone because the corresponding value of ηc required is large (Alexander et al., 2004) and perhaps not feasible in rural and poor communities settings.

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49

Now, comparing RV T and RV , RV T and RT , we have RV T = R0

µ[µ + γ + ω + (1 − ǫ)η] βRV , = κ [(µ + γ)(µ + η) + µω][1 + µ+λα ] β + κR0

(2.27)

which implies that RV T < RV . Also RV T = R0

since 0 <

µ[µ + γ + ω + (1 − ǫ)η] µ[µ + γ + ω + (1 − ǫ)η] = RT κ [(µ + γ)(µ + η) + µω][1 + µ+λα ] [(µ + γ)(µ + η) + µω]

(2.28)

µ[µ + γ + ω + (1 − ǫ)η] < 1, then, RV T < RT . [(µ + γ)(µ + η) + µω]

RV T < RV and RV T < RT suggest that vaccinating and treating individuals concurrently is likely more effective in controlling (slowing down) the epidemic than concentrating on a cohort vaccination campaign or treatment campaign only (i.e., therapeutic measures taken one at a time is less effective).

2.13.

The Role of RV T on Disease Eradication

Here, we discuss the role of the vaccination and treatment-induced reproduction number on the global stability of E0 . That is, we are interested in determining the threshold area of the quantity RV T for which the disease can be eradicated from the population (in the absence of inflow of infectives). More details on this approach can be found in Moghadas (2004b). Writing RV T as RV T (η, κ), we have RV T (0, 0) = R0 , and µR0 [µ + γ + ω + 1 − ǫ](µ + λα) RV T (1, 1) = . Since RV T (η, κ) > RV T (1, 1), the (1 + µ + λα)[(µ + γ)(µ + 1) + µω] threshold for disease eradication is given by RV T (1, 1) < RV T < RV T (0, 0) = R0 .

2.14.

(2.29)

Endemic Equilibrium and Its Stability

The model system (2.1) has an endemic equilibrium point (EE), where the infected components I is non-zero, given by Ee = (S ∗ , V ∗ , I ∗ , T ∗ , R∗ , C ∗ ), with R∗ =

τ πκI ∗ , (µ + δ)[µ + τ π + (1 − λ)α]

T∗ =

κI ∗ , µ + τ π + (1 − λ)α

V∗ =

ηN ∗ S ∗ , (1 − ǫ)βI ∗ + (µ + ω + γ)N ∗

S∗ =

[(1 − ρ)Λ + δR∗ + ωV ∗ ]N ∗ , βI ∗ + N ∗ (µ + η)

C∗ =

γV ∗ µ

I∗ = −

ρΛN ∗ βS ∗ + (1 − ǫ)βV ∗ − (µ + κ + λα)N ∗

(2.30)

50

2.15.

H. Rwezaura, E. Mtisi and J.M. Tchuenche

Stability Analysis when RV T > 1

Theorem 8 The model system (2.1) has no periodic orbits and its unique EE is GAS when RV T > 1.

Proof. The proof is based on reducing the model (2.1) into a 2-dimensional one by consecutively eliminating C, T, R and S respectively, to obtain: dV Λ βµV I = η( − V − I) − (1 − ǫ) − (γ + ω + µ)V, dt µ Λ (2.31) βµI Λ βµV I dI = ρΛ + ( − V − I) + (1 − ǫ) − (κ + µ + λα)I. dt Λ µ Λ Let X and Y denote the right hand sides of the first and second equations of (2.31), respec1 tively. Consider the Dulac function D = for V > 0, I > 0. Then, VI ∂(DX) ∂(DY ) η Λ 1 Λρ µβ + = −{ 2 ( − 1) + ( 2 + )} < 0, ∂V ∂I V µI V I Λ

(2.32)

Λ > 1. Thus, by Dulac’s criterion, there are no periodic orbits in Ω|Ω∗ where µI Ω∗ = {(S, V, I, T, R, C) ∈ Ω : I = 0 = T = R}. Since Ω is positively invariant, and the endemic equilibrium exists whenever RV T > 1, then it follows from the Poincar´ e-Bendixon Theorem (Perko, 2000), that all solutions of the limiting system originating in Ω remain in Ω for all t. Further, the absence of periodic orbits in Ω implies that the unique endemic equilibrium of the special case of the model system (2.1) is GAS whenever RV T > 1. since

2.16.

Endemic Equilibria when ρ > 0

Conditions under which none or multiple endemic equilibria exists are explored below (see alternative proof in Appendix D(2) when it is assumed that the C-class plays no further part in the disease transmission or when γ = 0). In order to find these conditions, we use the first equation of (2.31) to express the variables V in terms of the variable I when I 6= 0. This gives (at an arbitrary equilibrium)

V∗ =

Λ η(Λ − µI) · . µ Λ(µ + γ + ω + η) + (1 − ǫ)µβI

(2.33)

A Mathematical Analysis of Influenza with Treatment and Vaccination

51

Substitution of equation (2.33) into the second equation of (3.47) gives the following polynomial P (I, ρ) = −

(1 − ǫ)β 2 µ2 3 I − µβ[µ + γ + ω + (1 − ǫ)(µ + η + κ + λα − β)]I 2 Λ

−Λ[(µ + γ + ω + η)(µ + κ + λα − β) + β(ǫη − (1 − ǫ)µρ)]I +Λ2 ρ(µ + γ + ω + η) = 0, =

(1 − ǫ)β 2 µ2 3 I + µβ[µ + γ + ω + (1 − ǫ)(µ + η + κ + λα − β)]I 2 Λ +Λ[(µ + γ + ω + η)(µ + κ + λα − β) + β(ǫη − (1 − ǫ)µρ)]I −Λ2 ρ(µ + γ + ω + η) = 0,

= AI 3 + BI 2 + [C0 − (1 − ǫ)Λµρ]I + Λ2 ρ(µ + γ + ω + η) = 0, = IQ(I) − ρ{(1 − ǫ)ΛµI − Λ2 (µ + γ + ω + η)} = 0, = AI 3 + BI 2 + CI + D, (2.34) where

Q(I) = AI 2 + BI + C0 , A=

(1 − ǫ)β 2 µ2 , Λ

B = µβ[µ + γ + ω + (1 − ǫ)(µ + η + κ + λα − β)],

(2.35)

C0 = Λ[(µ + γ + ω + η)(µ + κ + λα − β) + βǫη], C = C0 − Λβµρ(1 − ǫ), D = −Λ2 ρ(µ + γ + ω + η). Since all the model parameters are nonnegative, it follows from (2.34) that A ≥ 0 and D ≤ 0. Furthermore, if RV T < 1 or RT < 1, then C0 > 0. That is, C0 = Λ[(µ + γ + ω + η)(µ + κ + λα − β) + βǫη], = Λ[(µ + γ + ω + η)(µ + κ + λα) − β(µ + γ + ω + (1 − ǫ)η)], γη Λ [µ(µ + γ + ω + η + )(µ + κ + λα) − µβ(µ + γ + ω + (1 − ǫ)η)] µ µ Λγη (µ + κ + λα), − µ =

52

H. Rwezaura, E. Mtisi and J.M. Tchuenche = Λ(µ + γ + ω + η(1 + µγ ))(µ + κ + λα)[1 − − Λγη µ (µ + κ + λα), = Λ(µ + γ + ω + η(1 +

βµ[µ+γ+ω+(1−ǫ)η] [(µ+γ)(µ+η)+µω](µ+κ+λα) ]

Λγη γ ))(µ + κ + λα)(1 − RV T ) − (µ + κ + λα), µ µ

Λγη γ (µ + κ + λα) + Λ(µ + γ + ω + η(1 + ))(µ + κ + λα)(RV T − 1)]. µ µ (2.36) Alternatively, C0 can be written in terms of RT as follows = −[

C0 = Λ(µ + κ + λα)[(µ + γ + ω + η) − RT (µ + γ + ω + (1 − ǫ)η)], = Λ(µ + κ + λα)(µ + γ + ω + η)[1 − RT (1 −

ǫη (µ+γ+ω+η) )]

(2.37)

= Λ(µ + κ + λα)(µ + γ + ω + η)(1 − ϑRT ). where ϑ = 1 −

ǫη < 1. We now consider the following three case: (µ + γ + ω + η)

Case 1: C0 < 0 (RV T > 1 or RT > 1), this is equivalent to η < ηc < η¯c , κ < κc < κ ¯ c . In this case, by Descartes Rule of Signs, IQ(I) = P (I, 0) = 0 has at most one positive root regardless of the sign of B, and one zero root. In this case, Q(I) has a unique positive root, denoted by I ∗ . Since P (I, ρ) is a decreasing function of ρ for positive I, it follows that P (I ∗ , ρ) < 0 for ρ > 0. Furthermore, P (I, ρ) → ∞ as I → ∞. Thus, P (I, ρ) has a unique positive root for all ρ ≥ 0, and this unique positive root must be at I > I ∗ . That is, when the model has a unique endemic equilibrium with ρ = 0, recruitment of infected individuals introduces no new equilibria but serves to shift the existing (unique) equilibrium to a higher disease state. Case 2: C0 > 0 (RV T < 1 or RT < 1) this is equivalent to η > ηc > η¯c , κ > κc > κ ¯c. If B < 0 and B 2 − 4AC0 > 0, then, IQ(I) = P (I, 0) = 0 has two positive and one zero roots. Since A ≥ 0, C > 0 and D < 0, then, by Descartes Rule of Signs, P (I, ρ) = 0 has at most three positive roots (when the aforementioned threshold is less than unity). If B > 0 then, by Descartes Rule of Signs, Q(I) = 0 has no positive roots and P (I, ρ) = 0 has at most one positive root. Case 3: C0 = 0 (RV T = 1), or RT = 1 provided ǫ = 0. This is equivalent to κ = κc , and Q(I) reduces to (AI + B)I = 0. In this case, the model has a unique endemic equilibrium if B < 0 and no endemic equilibrium if B > 0 orA = 0. Since A ≥ 0, C < 0 and D < 0, then, by Descartes Rule of Signs, P (I, ρ) = 0 has at most one positive root regardless of the sign of B. We note here that Case 2 confirms that the model exhibits backward bifurcation (for which its existence is shown in Section 2.18). The results of this section are summarized in the following Theorem. Theorem 9 Suppose ρ > 0 in (2.1).

A Mathematical Analysis of Influenza with Treatment and Vaccination

53

(a) If C0 ≤ 0, then (i) the model has a unique endemic equilibrium if η ≤ ηc ≤ η¯c , κ ≤ κc ≤ κ ¯c. (b) If C0 > 0, then (i) the model has unique endemic equilibria if B > 0, (ii) the model has three equilibria if B < 0 (where the disease-free equilibrium coexists with two stable endemic equilibria).

2.17.

Equilibria when ρ = 0

Notice that when ρ = 0, P (I, ρ) = 0 is reduced to IQ(I) = 0 for which I ∗ = 0 is a solution corresponding to the the DFE of (2.1), and P (I, ρ) < IQ(I) for ρ > 0 and I > 0. From a similar argument as above, the following result which we state without proof can be established (see alternative proof in Appendix D(1) with same assumption as in D(2) above). Lemma 10 Suppose ρ = 0 in (2.1). (a) If C0 < 0, then (i) the model has a unique endemic equilibrium if η < ηc < η¯c , κ < κc < κ ¯c. (b) If C0 > 0, then (i) the model has no endemic equilibria if B > 0, (ii) the model has two endemic equilibria if B < 0 and B 2 − 4AC0 > 0. (c) If C0 = 0, then (i) the model has a unique endemic equilibrium if B < 0 (ii) the model has no endemic equilibrium if B > 0 orA = 0.

2.18.

Existence of Backward Bifurcation

It has been shown (Alexander et al., 2004) that the vaccination only sub-model undergoes the phenomenon of backward bifurcation (BB), that is, the classical requirement of having the associated reproduction number to be less than unity, although necessary, is not sufficient for the disease control and eradication. We note that I ∗ in system (2.30) exists provided RV T < 1, consequently, the full model will also exhibit this phenomenon (see Figure 3 for the co-existence of a DFE and an EE when RV T < 1). Hence, the following result: Lemma 11 The DFE of the model system (2.1) co-exists with a stable EE when RV T < 1.

54

H. Rwezaura, E. Mtisi and J.M. Tchuenche 1

0.8

Stable EE I

0.6

0.4

Unstable EE

0.2

Stable DFE 0 0.8

0.85

0.9

0.95

1

Reproduction number

Figure 3. Backward bifurcation diagram for the full model.

Proof. It is sufficient to show that I ∗ above is epidemiologically relevant if RV T < 1 as shown below.

βS ∗ + (1 − ǫ)βV ∗ − (µ + κ + λα)N ∗ < βS0 + (1 − ǫ)βV0 − (µ + κ + λα)N0 = =

βΛ(µ + ω + γ) (1 − ǫ)βΛη Λ + − (µ + κ + λα), (µ + γ)(µ + η) + µω (µ + γ)(µ + η) + µω µ

=

βµ[µ + γ + ω + (1 − ǫ)η] Λ (µ + κ + λα)[ − 1], µ [(µ + γ)(µ + η) + µω](µ + κ + λα)

=

Λ (µ + κ + λα)[RV T − 1], µ (2.38)

The last expression is positive provided RV T < 1. Hence the proof.

A Mathematical Analysis of Influenza with Treatment and Vaccination

2.19.

55

Local Stability of the Endemic Equilibrium E1

Consider the following reduced model dS βSI = (1 − ρ)Λ − + δ(N − S − V − I − T ) − (µ + η)S + ωV, dt N dV βV I = ηS − (1 − ǫ) − (γ + ω + µ)V, dt N

(2.39)

dI βSI βV I = ρΛ + + (1 − ǫ) − (κ + µ + λα)I, dt N N dT = κI − (µ + τ π)T − (1 − λ)αT, dt defined in 4 Φ = {(S, V, I, T ) ∈ R I+ } ⊂ Ω.

For existence and uniqueness of endemic equilibrium E1 = (S ∗ , V ∗ , I ∗ , T ∗ ), its coordinates should satisfy the condition S ∗ > 0, V ∗ > 0, I ∗ > 0, T ∗ > 0. From (2.39) at equilibrium, we have (1 − ρ)Λ −

βS ∗ I ∗ + δ(N ∗ − S ∗ − V ∗ − I ∗ − T ∗ ) − (µ + η)S ∗ + ωV ∗ = 0, N∗

ηS ∗ − (1 − ǫ) ρΛ +

βS ∗ I ∗ N∗

βV ∗ I ∗ − (γ + ω + µ)V ∗ = 0, N∗

+ (1 − ǫ)

∗I ∗

βV N∗

(2.40)

− (κ + µ + λα)I ∗ = 0,

κI ∗ − (µ + τ π)T ∗ − (1 − λ)αT ∗ = 0 At the steady states of the reduced model (2.39), the Jacobian matrix J(E1 ) is given by

J(E1 )

βI ∗ ω−δ −  N ∗ − X11  (1−ǫ)βI ∗ η − N ∗ − X22 =  (1−ǫ)βI ∗ βI ∗  N∗ N∗ 0 0 

∗ − βS N∗ − δ ∗ − (1−ǫ)βV N ∗∗ ∗ βS +(1−ǫ)βV − N∗

κ

−δ X33

0 0 −X44

X11 = (µ + δ + η),

X22 = (µ + γ + ω),

X33 = (µ + κ + λα),

X44 = (µ + τ π + (1 − λ)α).



   , (2.41)   (2.42)

We use the additive compound matrices (or geometric) approach of Li and Moldowney (1996) to analyze the stability of the endemic equilibrium. From the Jacobian matrix J(E1 ) ,

56

H. Rwezaura, E. Mtisi and J.M. Tchuenche

the second additive compound matrix is given by 

[2]

J(E1 )

    =    

K11

(1−ǫ)βI ∗ N∗

0

∗

− βI N∗ 0 0

− (1−ǫ)βV N∗ K22 κ η 0 0

∗

0 0 K33 0 η

βS ∗ N∗

βI ∗ N∗

+δ ω−δ 0 K44 κ 0

δ 0 ω−δ 0 K55

(1−ǫ)βI ∗ N∗

0 δ



   βS ∗ − N∗ − δ  ,  0  (1−ǫ)βV ∗  − N∗  K66

(2.43)

where

∗

− (2µ + δ + η + γ + ω), K11 = − (2−ǫ)βI N∗ K22 =

βS ∗ +(1−ǫ)β(V ∗ −I ∗ ) N∗

− (2µ + δ + η + κ + λα),

∗

K33 = − βI N ∗ − (2µ + δ + η + τ π + (1 − λ)α), K44 =

βS ∗ +(1−ǫ)β(V ∗ −I ∗ ) N∗

(2.44)

− (2µ + γ + ω + κ + λα),

∗

K55 = − (1−ǫ)βI − (2µ + γ + ω + τ π + (1 − λ)α), N∗ K66 =

βS ∗ +(1−ǫ)βV ∗ N∗

− (2µ + κ + α + τ π).

The Local stability of the endemic equilibrium point E1 is demonstrated by the following Theorem of Li and Wang (1998). Theorem 12 An n × n real matrix A is stable if and only if A[2] is stable and (−1)n det(A) > 0. Proof. In order to determine det(J(E1 ) ), the following simplified equations of system (2.40) are used. (1 − ρ)Λ + δ(N ∗ − V ∗ − I ∗ − T ∗ ) + ωV ∗ βI ∗ = + µ + δ + η, S∗ N∗ ηS ∗ βI ∗ = (1 − ǫ) + (γ + ω + µ), V∗ N∗ ρΛ βS ∗ βV ∗ − ∗ = ∗ + (1 − ǫ) ∗ − (κ + µ + λα), I N N κI ∗ = µ + τ π + (1 − λ)α. T∗

(2.45)

A Mathematical Analysis of Influenza with Treatment and Vaccination

57

Then, from the Jacobian J(E1 ) and the simplified expressions (2.45), we have βI ∗ − ω−δ N ∗ − X11 ∗ − X22 η − (1−ǫ)βI ∗ det(J(E1 ) ) = N (1−ǫ)βI ∗ βI ∗ N∗ N∗ 0 0 (1−ρ)Λ+δ(N ∗ −V ∗ −I ∗ −T ∗ )+ωV ∗ − S∗ η = βI ∗ N∗ 0

=

κ

S ∗ V ∗ T ∗ (N ∗ )2

∗

− βS N∗ − δ

− (1−ǫ)βV N∗

βS ∗ +(1−ǫ)βV ∗ N∗

κ

ω−δ ∗ − ηS V∗

∗

− X33 ∗

(1−ǫ)βI ∗ N∗

0

−δ

− βS N∗ − δ (1−ǫ)βV ∗ − N∗ − ρΛ I∗ κ

0 0 −X44 −δ 0 0 ∗ − κI T∗





{[βI ∗ V ∗ (1 − ǫ)]2 [(1 − ρ)Λ + δ(N ∗ − V ∗ − I ∗ − T ∗ ) + ωV ∗ ]

+βηδI ∗ S ∗ N ∗ (I ∗ + T ∗ )[S ∗ + (1 − ǫ)V ∗ ] + (βI ∗ )2 S ∗ V ∗ (1 − ǫ)[(ω − δ)V ∗ + ηS ∗ ] +β 2 ηS ∗ 3 I ∗ 2 + ηρΛS ∗ N ∗ 2 [(1 − ρ)Λ + δ(N ∗ − I ∗ − T ∗ )]} > 0, = (−1)4 det(J(E1 ) ) > 0. (2.46) is

[2] J(E1 )

[2] J(E1 )

From the second additive compound matrix in (2.43), the stability of demonstrated as follows: For the endemic equilibrium point E1 = (S ∗ , V ∗ , I ∗ , T ∗ ), let P = [2] diag(1, −1, 1, −1, 1, −1) be the diagonal matrix. Then, the matrix J(E1 ) is similar to the matrix given by 

[2]

P J(E1 ) P −1

    =    

K11

(1−ǫ)βI ∗ N∗

0 ∗ − βI N∗ 0 0

− (1−ǫ)βV N∗ K22 κ η 0 0

∗

0 0 K33 0 η βI ∗ N∗

βS ∗ N∗

+δ ω−δ 0 K44 κ 0

δ 0 ω−δ 0 K55

(1−ǫ)βI ∗ N∗

0 δ



   βS ∗ − N∗ − δ  ,  0  ∗  − (1−ǫ)βV  N∗ K66 (2.47)

where K11 , K22 , K33 , K44 , K55 , K66 are as defined above. [2] Since similarity preserves the eigenvalues, then matrix J(E1 ) is stable if and only if the [2]

matrix P J(E1 ) P −1 is stable. Therefore, we apply the following Theorem due to McKenzie (1960) which establishes that if A has a negative dominant diagonal, then, A is stable. Theorem 13 (Sufficiency) If an n × n matrix A is dominant diagonally and the diagonal is composed of negative elements (aii < 0 ∀ i = 1, .., n), then, the real parts of all its eigenvalues are negative, i.e., A is stable.

58

H. Rwezaura, E. Mtisi and J.M. Tchuenche [2]

This can be done by examining if the matrix P J(E1 ) P −1 is diagonally dominant in rows, since its diagonal elements are negative ∗

h1 = − (2−ǫ)βI − (2µ + δ + η + γ + ω) + N∗

(1−ǫ)βV ∗ N∗

+0−

βS ∗ N∗

− δ + δ + 0,

= − Nβ∗ [S ∗ + (2 − ǫ)I ∗ − (1 − ǫ)V ∗ ] − (2µ + δ + η + γ + ω) < 0.

Using have,

βS ∗ +(1−ǫ)βV ∗ N∗

∗

βS ∗ +(1−ǫ)β(V ∗ −I ∗ ) N∗

h2 = − (1−ǫ)βI + N∗ ∗

= − 2(1−ǫ)βI + (κ + µ + λα) − N∗ ∗

+ = −[ 2(1−ǫ)βI N∗ h3 = −κ −

βI ∗ N∗

ρΛ I∗

= (κ + µ + λα) −

ρΛ I∗

from the third equation of (2.45) we

− (2µ + δ + η + κ + λα) + 0 + ω − δ + 0 + δ,

ρΛ I∗

− (2µ + δ + η + κ + λα) + ω,

+ µ + δ + η − ω] < 0.

− (2µ + δ + η + τ π + (1 − λ)α) + ω − δ +

βS ∗ N∗

+ δ,

= −[ Nβ∗ (I ∗ − S ∗ ) + 2µ + δ + η + κ + τ π + (1 − λ)α − ω] < 0.

Using have,

βS ∗ +(1−ǫ)βV ∗ N∗

h4 = =

βI ∗ N∗ ǫβI ∗ N∗

βS ∗ +(1−ǫ)β(V ∗ −I ∗ ) N∗

+ η + (κ + µ + λα) −

= −[ ρΛ I∗ + µ + γ + ω − For

ρΛ I∗

+µ+γ+ω >

h5 = η − κ − =

ǫβI ∗ N∗

ǫβI ∗ N∗

(2.49)

from the third equation of (2.45) we

− (2µ + γ + ω + κ + λα),

ρΛ I∗

− (2µ + γ + ω + κ + λα),

(2.50)

− η] < 0.

+ η,

(1−ǫ)βI ∗ N∗

∗ −V ∗ ) −[ (1−ǫ)β(I N∗

Finally, using

ρΛ I∗

= (κ + µ + λα) − +η+

(2.48)

βS ∗ +(1−ǫ)βV ∗ N∗

− (2µ + γ + ω + τ π + (1 − λ)α) +

(1−ǫ)βV ∗ , N∗

(2.51)

+ 2µ + γ + ω + κ + τ π + (1 − λ)α − η] < 0.

= (κ + µ + λα) −

ρΛ I∗

from the third equation of

A Mathematical Analysis of Influenza with Treatment and Vaccination

59

(2.45) we have, ∗

h6 = − βI N∗ −

(1−ǫ)βI ∗ N∗

+

βS ∗ +(1−ǫ)βV ∗ N∗

∗

= − (2−ǫ)βI + (κ + µ + λα) − N∗ ∗

= −[ (2−ǫ)βI + N∗

ρΛ I∗

ρΛ I∗

− (2µ + κ + α + τ π), (2.52)

− (2µ + κ + α + τ π),

+ µ + τ π + (1 − λ)α] < 0.

We have all values h1 , h2 , h3 , h4 , h5 , h6 < 0, therefore, all the diagonals are negative. Thus, from Theorems 12 and 13, the system has a local stability at the endemic equilibrium point.

2.20.

Global Stability of the EE E1 when RV T > 1

The global stability of endemic equilibrium point E1 is demonstrated by the following Lemma of Li and Muldowney (1993). ∂g Lemma 14 Let Ω ⊆ R I n be a simply connected region and ( ∂u (u(t, x0 )))[2] is the second additive compound matrix of the Jacobian matrix of vector function g(u) at u(t, x0 ). Assume that the family of linear systems

z(t) ˙ =



[2] ∂g z(t), x ∈ Ω, (u(t, x0 )) ∂u

(2.53)

is equi-uniformly asymptotically stable. Then, (i) Ω contains no simple closed invariant curves including periodic orbits, homoclinic, heteroclinic cycles, (ii) each semi-orbit in Ω converges to a single equilibrium. In particular, if Ω is positively invariant and contains a unique equilibrium x ¯, then x ¯ is globally asymptotically stable in Ω. Proof. The proof is based on the approach of Yan and Li (2005). From the second additive [2] compound matrix J(E1 ) (2.43) above which we reproduce here for convenience 

[2]

J(E1 )

    =    

K11 (1−ǫ)βI N

0 − βI N 0 0

− (1−ǫ)βV N K22 κ η 0 0

0 0 K33 0 η βI N

βS N

+δ ω−δ 0 K44 κ 0

δ 0 ω−δ 0 K55 (1−ǫ)βI N

0 δ



   βS −N −δ  ,  0   − (1−ǫ)βV  N K66

(2.54)

60

H. Rwezaura, E. Mtisi and J.M. Tchuenche

where − (2µ + δ + η + γ + ω), K11 = − (2−ǫ)βI N K22 =

βS+(1−ǫ)β(V −I) N

− (2µ + δ + η + κ + λα),

K33 = − βI N − (2µ + δ + η + τ π + (1 − λ)α), K44 =

βS+(1−ǫ)β(V −I) N

(2.55)

− (2µ + γ + ω + κ + λα),

− (2µ + γ + ω + τ π + (1 − λ)α), K55 = − (1−ǫ)βI N K66 =

βS+(1−ǫ)βV N

− (2µ + κ + α + τ π),

we have the linear system with respect to the solutions (S(t), V (t), I(t), T (t)) written as ˙ 1 (t) = K11 W1 (t) − (1 − ǫ)βV W2 (t) + ( βS + δ)W4 (t) + δW5 (t), W N N ˙ 2 (t) = (1 − ǫ)βI W1 (t) + K22 W2 (t) + (ω − δ)W4 (t) + δW6 (t), W N ˙ 3 (t) = κW2 (t) + K33 W3 (t) + (ω − δ)W5 (t) − ( βS + δ)W6 (t), W N

(2.56)

˙ 4 (t) = − βI W1 (t) + ηW2 (t) + K44 W4 (t), W N ˙ 5 (t) = ηW3 (t) + κW4 (t) + K55 W5 (t) − (1 − ǫ)βV W6 (t), W N ˙ 6 (t) = βI W3 (t) + (1 − ǫ)βI W5 (t) + K66 W6 (t). W N N Set V (W ) = max{Xl |Wl | : l = 1, 2, ..., 6}, where Xl > 0 (l = 1, 2, ..., 6) are constants. Then, direct calculation leads to the following inequalities: X1 (1 − ǫ)βV d+ X1 |W1 (t)| ≤ K11 X1 |W1 (t)| + | |X2 |W2 (t)| dt X2 N X1 βS 1 |( N + δ)|X4 |W4 (t)| + X +X X5 |δ|X5 |W5 (t)|, 4 d+ X2 (1 − ǫ)βI X2 |W2 (t)| ≤ K22 X2 |W2 (t)| + | |X1 |W1 (t)| dt X1 N X2 X2 + X4 |ω − δ|X4 |W4 (t)| + X6 |δ|X6 |W6 (t)|,

A Mathematical Analysis of Influenza with Treatment and Vaccination

d+ dt X3 |W3 (t)|

61

3 ≤ K33 X3 |W3 (t)| + X X2 |κ|X2 |W2 (t)| βS X3 3 + X5 |ω − δ|X5 |W5 (t)| + X X6 |( N + δ)|X6 |W6 (t)|,

X4 βI X4 d+ X4 |W4 (t)| ≤ K44 X4 |W4 (t)| + | |X1 |W1 (t| + |η|X2 |W2 (t)|, dt X1 N X2 (2.57)

d+ X5 X5 |W5 (t)| ≤ K55 X5 |W5 (t)| + |η|X3 |W3 (t)| dt X3 X5 X5 (1−ǫ)βV + X4 |κ|X4 |W4 (t)| + X6 | N |X6 |W6 (t)|, X6 βI X6 (1 − ǫ)βI d+ X6 |W6 (t)| ≤ K66 X6 |W6 (t)| + |X5 |W5 (t)|. | |X3 |W3 (t)| + | dt X3 N X5 N

where

d+ denotes the upper right-hand derivative. Consequently, we have dt d+ V (W (t)) ≤ φ(t)V (W (t)), dt

(2.58)

with φ(t) = max{K11 +

X1 (1−ǫ)βV X2 | N

K22 +

X2 (1−ǫ)βI | X1 | N

K33 +

X3 X2 |κ|

K44 +

X4 βI X1 | N |

K55 +

X5 X3 |η|

K66 +

X6 βI X3 | N |

+

X3 X5 |ω

+

+

+

X2 X4 |ω

X1 βS X4 |( N

− δ| +

− δ| +

+ δ)| +

X1 X5 |δ|,

X2 X6 |δ|,

X3 βS X6 |( N

+ δ)|, (2.59)

X4 X2 |η|,

X5 X4 |κ|

+

|+

+

X5 (1−ǫ)βV X6 | N

|,

X6 (1−ǫ)βI |}. X5 | N

For convenience, we further assume the following hypothesis: [2] Since the matrix J(E1 ) is diagonally dominant in rows and its diagonal elements K11 , K22 , K33 , K44 , K55 , K66 are negative, then, there exist constants Xl > 0 (l = 1, 2, ..., 6) such that sup{K11 +

X1 (1−ǫ)βV X2 | N

K22 +

X2 (1−ǫ)βI | X1 | N

K33 +

X3 X2 |κ|

+

+

X3 X5 |ω

|+

X1 βS X4 |( N

+ δ)| +

X2 X4 |ω

− δ| +

X2 X6 |δ|,

− δ| +

X3 βS X6 |( N

+ δ)|,

X1 X5 |δ|,

62

H. Rwezaura, E. Mtisi and J.M. Tchuenche K44 +

X4 βI X1 | N |

K55 +

X5 X3 |η|

K66 +

X6 βI X3 | N |

+

+

X4 X2 |η|,

X5 X4 |κ|

+

+

X5 (1−ǫ)βV X6 | N

X6 (1−ǫ)βI |} X5 | N

|,

(2.60)

< 0.

Therefore, by the boundedness of solution of (2.39), there exists ̺ > 0 such that φ(t) = −̺ < 0, and thus V (W (t)) ≤ V (W (s))e−̺(t−s) , t ≥ s ≥ 0. This shows that the second compound system (2.56) is equi-uniform asymptotically stable, and from Lemma 14, the system (2.39) has no non-constant periodic solution and the unique endemic equilibrium E1 is globally asymptotically stable in R I 4. For mathematical convenience, the analysis of the persistence of the model is carried out below using the mass-action incidence model. Mathematically speaking, if inflow of infected individuals is allowed, then, the epidemiological implication is that the DFE will not exist and eradication of the disease may not be feasible. In this case the public health objective is to minimize the level of epidemicity.

2.21.

The Model with Mass-Action Incidence dS = (1 − ρ)Λ − βSI + δR − (µ + η)S + ωV, dt dV = ηS − (1 − ǫ)βV I − (γ + ω + µ)V, dt dI = ρΛ + βSI + (1 − ǫ)βV I − (κ + µ + λα)I, dt dT = κI − (µ + τ π)T − (1 − λ)αT, dt

(2.61)

dR = τ πT − (µ + δ)R, dt dC = γV − µC, dt

2.22.

Persistence of Solutions of the Model with Mass-Action Incidence (ρ = 0)

Permanent co-existence of a dynamical system for which none of the variables is zero at steady states is biologically meaningful for survival of competing species, or the persistence of a disease, or existence of the endemic steady state, even though for disease dynamics, the disease-free equilibrium is the goal of any intervention strategy. The permanence of

A Mathematical Analysis of Influenza with Treatment and Vaccination

63

the disease destabilizes the disease-free equilibrium and since RV T > 1, the interior or endemic equilibrium will exist.

Lemma 15 The solution of model system (2.61) is uniformly persistent.

Proof. The proof is based on the approach by McCluskey (2006). Consider the following Lyapunov function (Korobeinikov and Maini, 2004) which is defined and continuous for all S, V, I, T, R, C ≥ 0, VL = c1 (S − S ∗ lnS) + c2(V − V ∗ lnV ) + c3 (I − I ∗ lnI) +

+c4 (T − T ∗ lnT ) + c5 (R − R∗ lnR) + c6 (C − C ∗ lnC),

where c1 , c2 , c3 , c4 , c5 , c6 are constants. Differentiating VL with respect to time t gives S∗ ˙ V∗ ˙ I∗ T∗ ˙ V˙ L = c1 (1 − )S + c2 (1 − )V + c3 (1 − )I˙ + c4 (1 − )T + S V I T ∗ ∗ C ˙ R ˙ )R + c6 (1 − )C, +c5 (1 − R C S∗ )[Λ − βSI + δR − (µ + η)S + ωV ] + = c1 (1 − S V∗ )[ηS − (1 − ǫ)βV I − (γ + ω + µ)V ] + +c2 (1 − V ∗ I +c3 (1 − )[βSI + (1 − ǫ)βV I I T∗ )[κI − (µ + τ π)T − (1 − λ)αT ] + −(κ + µ + λα)I] + c4 (1 − T R∗ C∗ +c5 (1 − )[τ πT − (µ + δ)R] + c6 (1 − )[γV − µC], R C S∗ )[βS ∗ I ∗ − δR∗ + (µ + η)S − ωV − βSI + δR] − = c1 (1 − S S∗ ) − [(µ + η)S + ωV ] + −c1 (1 − S ∗ V )[ηS − (1 − ǫ)βV I − (γ + ω + µ)V ] + +c2 (1 − V I∗ +c3 (1 − )[βSI + (1 − ǫ)βV I I T∗ )[κI − (µ + τ π)T − (1 − λ)αT ] + −(κ + µ + λα)I] + c4 (1 − T ∗ R C∗ +c5 (1 − )[τ πT − (µ + δ)R] + c6 (1 − )[γV − µC], R C

64

H. Rwezaura, E. Mtisi and J.M. Tchuenche

S∗ )[β(S ∗ I ∗ − SI) + δ(R − R∗ ) + S S∗ )[(µ + η)(S ∗ − S) + ω(V − V ∗ )] +c1 (1 − S VI I V∗ )[η(S − S ∗ ∗ ∗ ) + (µ + γ + ω)( ∗ − 1)V ] +c2 (1 − V V I I V I∗ ∗ V +c3 (1 − )[βI(S − S ∗ ) + (µ + κ + λα)( ∗ − 1)I] I V V T∗ I +c4 (1 − )[(µ + τ π + (1 − λ)α)(T ∗ ∗ − T )] T I C∗ V R∗ ∗ T )[(µ + δ)(R ∗ − R)] + c6 (1 − )[µ(C ∗ ∗ − C)]. +c5 (1 − R T C V Without loss of generality, let c1 = 1 = c2 = c3 = c3 = c4 = c5 = c6 , and = c1 (1 −

(

S V I T R C , , , , , ) = (x1 , x2 , x3 , x4 , x5 , x6 ). S ∗ V ∗ I ∗ T ∗ R∗ C ∗

Then, ∗ )2 V˙ L = −(µ + η)( (S−S ) + (1 − S

S∗ ∗ ∗ S )[β(S I

− SI)+

+δ(R − R∗ ) + ω(V − V ∗ )] +(1 −

V∗ V )[η(S

+(1 −

I∗ I )[βI(S

+(1 −

T∗ T )[(µ

+ τ π + (1 − λ)α)(T ∗ II∗ − T )]

+(1 −

R∗ R )[(µ

+ δ)(R∗ TT∗ − R)] + (1 −

= −(µ + η)( (S−S S

− S ∗ VV∗ II ∗ ) + (µ + γ + ω)( II∗ − 1)V ] − S ∗ VV∗ ) + (µ + κ + λα)( VV∗ − 1)I]

∗ )2

C∗ ∗ V C )[µ(C V ∗

(2.62)

− C)]

) + f (x1 , x2 , x3 , x4 , x5 , x6 ),

where f (x1 , . . . , x6 ) = (1 −

1 ∗ ∗ x1 )[βS I (1

+(1 −

− x1 x3 ) + δR∗ (x5 − 1) + ωV ∗ (x2 − 1)]

1 ∗ x2 )[ηS (x1

− x2 x3 ) + (µ + γ + ω)(x3 − 1)x2 ]

+(x3 − 1)[βS ∗ I ∗ (x1 − x2 ) + I ∗ (µ + κ + λα)(x2 − 1)] +(1 −

1 ∗ x4 )[T (µ

+ τ π + (1 − λ)α)(x3 − x4 )]

+(1 −

1 ∗ x5 )[R (µ

+ δ)(x4 − x5 )] + (1 −

1 ∗ x6 )[µC (x2

− x6 )]. (2.63)

Collecting the like terms in βS ∗ I ∗ we have; f (x1 , · · · , x6 ) = βS ∗ I ∗ [(1 −

1 − x1 + x2 (1 − x3 ) + x3 )] + F (x1 , · · · , x6 ), x1

(2.64)

A Mathematical Analysis of Influenza with Treatment and Vaccination

65

where F (x1 , x2 , x3 , x4 , x5 , x6 ) are the remaining terms. Without loss of reality and for convenience, we assume that F = 0, that is, x2 = 1 = x3 = x4 = x5 = x6 , and we have, (S − S ∗ )2 1 V˙ L = −(µ + η) + βS ∗ I ∗ (2 − − x1 ), S x1

(2.65)

1 − x1 ≤ 0 by the arithmetic mean-geometric mean inequality (McCluskey, x1 2006), with equality if and only if x1 = 1. Hence, V˙ L ≤ 0 with equality iff S = S ∗ and consequently VL is a Lyapunov function. Let P (t) := (S(t), V (t), I(t), T (t), R(t), C(t)), then, rewriting V˙ L as follows, we can determine the constant ξ such that where 2 −

lim inf P (t) > ξ.

(2.66)

t→∞

That is, ˙ ˙ ˙ V˙ L = c1 (S − S ∗ ) SS + c2 (V − V ∗ ) VV + c3 (I − I ∗ ) II ˙

˙

˙

R +c4 (T − T ∗ ) TT + c5 (R − R∗ ) R + c6 (C − C ∗ ) C C, R V Λ ∗ = c1 (S − S )[ S − βI + δ S − (µ + η) + ω S ]

+c2 (V − V ∗ )[η VS − (1 − ǫ)βI − (γ + ω + µ)]

(2.67)

+c3 (I − I ∗ )[βS + (1 − ǫ)βV − (κ + µ + λα)] +c4 (T − T ∗ )[κ TI − (µ + τ π) − (1 − λ)α] T +c5 (R − R∗ )[τ π R − (µ + δ)] + c6 (C − C ∗ )[γ VC − µ].

Since V˙ L ≤ 0, constants c1 , c2 , c3 , c4 , c5 , c6 can be found such that V˙ L ≤ −(µ + η)(S − S ∗ ) − (γ + ω + µ)(V − V ∗ ) − (κ + µ + λα)(I − I ∗ ) −[µ + τ π + (1 − λ)α](T − T ∗ ) − (µ + δ)(R − R∗ ) − µ(C − C ∗ )] ≤ −ξ[(S − S ∗ ) + (V − V ∗ ) + (I − I ∗ ) + · · · + (C − C ∗ )],

(2.68)

where ξ = min{µ, µ + η, µ + δ, γ + ω + µ, κ + µ + λα, µ + τ π + (1 − λ)α}, ∴ 0 < ξ ≤ lim inf P (t) ≤ lim sup P (t) = t→∞

t→∞

Λ . µ

(2.69) (2.70)

Since all the variables in P (t) are continuous and bounded with derivatives in L∞ , then, [0,T¯] applying a slightly modified version of Barbalat’s Lemma (1959) which is stated without proof in Appendix B on equation (2.68) above yields

66

H. Rwezaura, E. Mtisi and J.M. Tchuenche

(S − S ∗ ) + (V − V ∗ ) + (I − I ∗ ) + (T − T ∗ ) + (R − R∗ ) + (C − C ∗ ) → 0 as t → ∞, and the only largest invariant subset in the set Ω1 ⊂ Ω given by Ω1 = {(S, V, I, T, R, C) : S = S ∗ , V = V ∗ , I = I ∗ , T = T ∗ , R = R∗ , C = C ∗ } (2.71) is E ∗ = (S ∗ , V ∗ , I ∗ , T ∗ , R∗ , C ∗ ). Therefore, by Lyapunov-LaSalle’s invariance properties, system (2.61) is uniformly persistent.

2.23.

Treatment-Only Submodel (with Mass-Action Incidence)

We assume that treatment is the only intervention adopted. In this case η = 0 = ω and the model reduces to the following system dS = (1 − ρ)Λ − βSI + δR − µS, dt dI = ρΛ + βSI − (κ + µ + λα)I, dt

(2.72)

dT = κI − (µ + τ π)T − (1 − λ)αT, dt dR = τ πT − (µ + δ)R, dt For simplicity of the mathematical analysis, we normalize the treatment-only submodel (2.72) by defining the new variables µ µ µ µ s = S, i = I, tˆ = T, r = R, and the parameters Λ Λ Λ Λ δ κ τ α Λ ˜ ˜ ˜ ˜ = , τ˜ = , α ˜ = , to obtain β = 2 β, t = µt, δ = , κ µ µ µ µ µ ds ˜ + δr ˜ − s, = (1 − ρ) − βsi dt˜ di ˜ − (˜ = ρ + βsi κ + 1 + λα ˜ )i, dt˜ (2.73) dtˆ =κ ˜ i − (1 + τ˜π)tˆ − (1 − λ)α ˜ tˆ, dt˜ dr ˜ = τ˜π tˆ − (1 + δ)r, dt˜ with invariant region given by 4 Ω2 = {(s, i, tˆ, r) ∈ R I+ : s + i + tˆ + r ≤ 1}.

(2.74)

System (2.73) can be decoupled and reduced to a 3-dimensional system with variables s, i and tˆ. Since the use of the dimensionless variables s = Λµ S,

A Mathematical Analysis of Influenza with Treatment and Vaccination µ i = Λ I, tˆ = replaced by

µ Λ T,

and r =

µ ΛR

67

leads to further simplifications, system (2.73) is now

ds ˜ + δ(1 ˜ − s − i − tˆ) − s, = (1 − ρ) − βsi dt˜ di ˜ − (˜ = ρ + βsi κ + 1 + λα ˜ )i, dt˜

(2.75)

dtˆ =κ ˜ i − (1 + τ˜π)tˆ − (1 − λ)α ˜ tˆ, dt˜ for which the disease-free equilibrium when (ρ = 0) is given by ET0 = (1, 0, 0). To determine the local stability of ET0 , the Jacobian of the normalized reduced (NR) model is evaluated at the DFE to yield   ˜ −(δ˜ + 1) −(β˜ + δ) −δ˜  JE 0 =  (2.76) 0 β˜ − (1 + κ ˜ + λα ˜) 0 T 0 κ ˜ −(1 + τ˜π + (1 − λ)α) ˜

The eigenvalues of JE 0 are ς1 = −(δ˜ + 1), ς2 = β˜ − (1 + κ ˜ + λα ˜ ), and ς3 = −(1 + τ˜π + T (1 − λ)α). ˜ The eigenvalues ς1 and ς3 are negative. Thus, the local stability of ET0 depends on the sign of ς2 = β˜ −(1+ κ ˜ +λα ˜ ) = (1+ κ ˜ +λα ˜ )(RT −1). Therefore, the DFE of the treatment-only submodel (2.73) is locally asymptotically stable if RT < 1 and unstable if RT > 1. Lemma 16 The treatment only submodel (2.73) has a unique endemic equilibrium (EE) ET∗ = (s∗ , i∗ , tˆ∗ ) ∈ Ω2 when ρ = 0 iff RT > 1.

Proof. Equating the RHS of the NR model to zero and solving for the variables, we obtain ˜ ˜ (RT − 1)(1 + δ) 1 ˜ (RT − 1)(1 + δ) ˆ∗ = κ , i∗ = s∗ = , where , t ˜ T (1 + κ˜ ) ˜ T (A + κ RT β˜ + δR Aβ˜ + δR ˜) A β˜ A = 1 + τ˜π + (1 − λ)α, ˜ RT = . Thus, ET∗ exists provided RT > 1. Hence 1+κ ˜ + λα ˜ the proof. From (2.75) we can further reduced the 3-dimensional system to a 2-dimensional system by replacing s by 1 − i − tˆ. That is, di ˜ − i − tˆ)i − (˜ = ρ + β(1 κ + 1 + λα ˜ )i, dt˜ dtˆ =κ ˜ i − (1 + τ˜π)tˆ − (1 − λ)α ˜ tˆ. dt˜

(2.77)

Lemma 17 The treatment-only sub-model (2.73) has no periodic solutions in Ω2 |Ω0 when RT > 1. Proof. The EE exists only when RT > 1 (see Lemma 16) and i + tˆ + r → 0 as t˜ → ∞. Since Ω0 = {(s, i, tˆ, r) ∈ Ω2 : i = 0 = tˆ = r; s = 1}, let G be the sum of the RHS of 1 system (2.77) and ψ = be a candidate Dulac’s function. It follows that itˆ 1 ρ div(ψG) = − ( 2 + β + κ) < 0. tˆ i

(2.78)

68

H. Rwezaura, E. Mtisi and J.M. Tchuenche 0.9 0.25

0.8 0.7

0.2

0.15

0.5 Treated

Infected

0.6

0.4

0.1

0.3 0.2 0.05

0.1 0

0

0.05

0.1

0.15 0.2 Treated

0.25

0.3

0.35

0

0

(a)

0.1

0.2

0.3 Infected

0.4

0.5

0.6

(b)

Figure 4. Phase portraits of i and tˆ showing the effects of treatment on infectives.

Hence, by Dulac’s criterion, there are no periodic orbits in Ω2 |Ω0 . Since Ω2 is positively invariant and the EE exists whenever RT > 1, then by the Poincar´ e-Bendixon Theorem (Perko, 2000), all solutions of the limiting system originating in Ω2 remain there for t˜. Further, the absence of periodic orbits in Ω2 implies that the unique EE of the special case of the treatment-only submodel (when ρ = 0) is GAS whenever RT > 1. Now, suppose ρ = 0, then, the following result holds.

Theorem 18 If the treatment only submodel has a unique endemic equilibrium, then, it is GAS and the disease persists within the population.

Proof. The two-dimensional simplex Ω2 is bounded and the submodel has no periodic orbits, homoclinic orbits, or polygons (see Lemma 17) and by the Poincar´ e-Bendixon Theorem (Perko, 2000), the omega-limit set of every solution in Ω2 is an equilibrium point. Since Ω2 is positively invariant, it follows from Lemma 17 that the omega-limit set in Ω2 |Ω0 must be the EE. Figure 4 illustrates the effect of treatment on the infective class. Treatment reduces the number of infectives to a lower lever, but does not eradicate the disease due to the flux of infectives. The parameter values used are (a): ˜ α ˜ = 0.01, δ = 0.15, κ ˜ = 0 : 0.1 : 1, λ = 0.5, π = 0.6, ρ = 0.1, τ˜ = 0.25, β˜ = 4.5, with initial values s0 = 0.9; i0 = 0.1; tˆ0 = 0; r0 = 0. (b): α ˜ = 0.01, δ˜ = 0.15, κ ˜ = 0.5, λ = 0.5, π = 0 : 0.1 : 1, ρ = 0.1, τ˜ = 0.25, β˜ = 4.5, ˆ and s0 = 0.9; i0 = 0.1; t0 = 0; r0 = 0 In Figure (a), κ ˜ varies from 0-1 with π constant, while in (b), π varies from 0-1 with κ ˜ constant.

A Mathematical Analysis of Influenza with Treatment and Vaccination

2.24.

69

Existence of Backward Bifurcation in the Treatment-Only Model

We investigate the existence of backward bifurcation (or subcritical bifurcation, i.e., positive endemic equilibria exist for RT < 1 near the bifurcation point) in the treatment-only model using the normalized reduced system (2.77) for which at steady state we have, ˜ − i∗ − tˆ∗ )i − (˜ ρ + β(1 κ + 1 + λα ˜ )i∗ = 0, ∗ ∗ ∗ κ ˜ i − (1 + τ˜π)tˆ − (1 − λ)α ˜ tˆ = 0.

(2.79)

˜ At steady state, system (2.79) can be expressed The force of infection is given by χ = βi. in terms of χ as follows: χ ˆ∗ χ − t ) − (˜ = 0, κ + 1 + λα ˜) ˜ β β˜ χ χ ρ + χ(1 − − tˆ∗ ) − = 0, ˜ R β T χ ˜ tˆ∗ = 0. κ ˜ − [1 + τ˜π + (1 − λ)α] ˜ β ρ + χ(1 −

(2.80)

Expressing the variables tˆ∗ in terms of the variable χ gives tˆ∗ =

κ ˜χ . ˜ β[1 + τ˜π + (1 − λ)α] ˜

(2.81)

Substituting (2.81) into the first equation of (2.80) gives the following quadratic equation: ˜ + τ˜π + (1 − λ)α)]( Q(i, ρ) = [1 + κ ˜ + τ˜π + (1 − λ)α)]χ ˜ 2 + β[1 ˜ R1T − 1)χ ˜ + τ˜π + (1 − λ)α)] −ρβ[1 ˜ = 0, 2 = aχ + bχ + c = 0. (2.82) ˜ + τ˜π + (1 − λ)α)]( where a = [1 + κ ˜ + τ˜π + (1 − λ)α)], ˜ b = β[1 ˜ R1T − 1) ˜ + τ˜π + (1 − λ)α)] and c = ρβ[1 ˜ = 0. Since all the model parameters are nonnegative, it follows from (2.82) that a > 0 and c ≤ 0. Furthermore, if 0 < RT ≤ 1, then b ≥ 0 and if RT > 1, then b < 0. Notice that when ρ = 0, the quadratic equation Q(χ, ρ) is reduced to a linear equation b aχ + b = 0, so that χ = − . If b ≥ 0, that is, 0 < RT ≤ 1, then, χ ≤ 0. Therefore, a no endemic equilibrium exists whenever 0 < RT ≤ 1. But for b < 0, i.e., RT > 1 then, χ > 0,and in this case, the treatment-only submodel has a unique endemic equilibrium if and only if b < 0, i.e., RT > 1, ruling out backward bifurcation in this case. If ρ > 0, then, since a > 0 and c < 0, the quadratic equation Q(χ, ρ) has at most one positive root regardless of the sign of b, i.e., RT ≤ 1 or RT > 1. Hence, the treatment only submodel has a unique endemic equilibrium regardless of RT ≤ 1 or RT > 1. This result indicates the possibility of backward bifurcation due to the existance of the endemic equilibrium when RT ≤ 1. If b = 0 (i.e., RT = q1) then the quadratic equation Q(χ, ρ) has two roots with opposite signs, namely, χ = ± −c a since a > 0 and c ≤ 0. In this case, the treatment-only submodel has a unique endemic if c < 0, and ρ > 0. Hence, the following result is established (an alternative proof of this Theorem is given in Appendix C):

70

H. Rwezaura, E. Mtisi and J.M. Tchuenche

Theorem 19 The treatment only submodel (2.73) has (i) a unique endemic equilibrium if c = 0 (for ρ = 0) and b < 0(i.e., RT > 1), (ii) tow equilibria if c < 0 for ρ > 0, RT ≤ 1, (iii) no endemic equilibrium otherwise. Case (ii) indicates the possibility of backward bifurcation in the the treatment only submodel (2.80) when RT ≤ 1. To check for this, the discriminant b2 − 4ac is set to zero and the result solved for the critical value of RcT , from which it can be shown that backward bifurcation occurs for values of RT such that RcT < RT < 1. But, since a > 0 and c ≤ 0, then, b2 −4ac = 0 when b2 = 0 = 4ac i.e., when (c = 0 ⇒ ρ = 0) and (b = 0 ⇒ RT = 1).

3. 3.1.

Sensitivity Analysis and Numerical Simulations Sensitivity Analysis

Sensitivity indices allow us to measure the relative change in a state variable when a parameter changes, while its analysis is commonly used to determine the robustness of model predictions to parameter values (since there are usually errors in data collection and presumed parameter estimations). The sensitivity indices to the parameters in the model are calculated in order to determine parameters that have a high impact on RV , RT , RV T , and that should be targeted by intervention strategies (Chitnis et al., 2008). These indices tell us how crucial each parameter is to disease transmission and prevalence. In order to avoid repetition, we omit the calculations of the sensitivity indices of the endemic equilibrium E ∗ . Nevertheless, it is noted that disease prevalence is directly related to the endemic equilibrium point, specifically to the magnitudes of I ∗ . The fraction of infectious humans, I, is especially important because it represents the people who may be clinically ill, and is directly related to the total number of influenza deaths. For a detail description of these concept, see (Chitnis et al., 2008). The most important parameter for equilibrium disease prevalence is the human-mosquito contact rate β. Other important parameters are the treatment rate κ, the vaccine efficacy ǫ and the fraction of recruited individuals who are already infected ρ. In determining how best to reduce human mortality and morbidity due to influenza, it is necessary therefore to know the relative importance of the different factors responsible for its transmission and prevalence. Initial disease transmission in the absence of any intervention strategy is directly related to R0 , and disease prevalence is directly related to the endemic equilibrium point, specifically to the magnitudes of I and R. In computing the sensitivity analysis, we use method described by Chitnis et al., (2008). The normalized forward sensitivity index of a variable to a parameter is the ratio of the relative change in the variable to the relative change in the parameter. When the variable is a differentiable function of the parameter, the sensitivity index may be alternatively defined using partial derivatives. Therefore, the normalized forward sensitivity index of a variable, u, that depends differentially on a parameter, p, is defined as: Υup =

∂u p × . ∂p u

(3.83)

A Mathematical Analysis of Influenza with Treatment and Vaccination

3.2.

71

Sensitivity Indices of RV T

The sensitivity indices of RV T =

βµR0 [µ + γ + ω + (1 − ǫ)η] with respect to its (κR0 + β)[(µ + γ)(µ + η) + µω]

(eight) parameters are given by VT ΥR µ

=

2µ + γ + ω + (1 − ǫ)η µ(2µ + γ + ω + η) − , µ + γ + ω + (1 − ǫ)η (µ + γ)(µ + η) + µω

VT ΥR γ

=

γ γ(µ + η) − , µ + γ + ω + (1 − ǫ)η (µ + γ)(µ + η) + µω

VT ΥR ω

=

ωµ ω − , µ + γ + ω + (1 − ǫ)η (µ + γ)(µ + η) + µω

VT ΥR η

=

η(µ + γ) η(1 − ǫ) − , µ + γ + ω + (1 − ǫ)η (µ + γ)(µ + η) + µω

VT = − ΥR ǫ

VT ΥR = β

ǫη , µ + γ + ω + (1 − ǫ)η

κR0 , κR0 + β

VT ΥR R0 =

(3.84)

β , κR0 + β

VT = − ΥR κ

κR0 . κR0 + β

Similarly, we can derive the expression for R0 , RV and RT . When RV T is expressed in µ[µ + γ + ω + (1 − ǫ)η] with respect to terms of RT , the sensitivity index of RV T = RT [(µ + γ)(µ + η) + µω] VT RT is given by ΥR RT = 1. βRV β + κR0 VT = 1. Here, we note that R can also be expressed with respect to RV is given by ΥR V T RV RV RT in terms of the three reproduction numbers RV , RT , and R0 as RV T = . R0 When RV T is expressed in terms of RV , the sensitivity index of RV T =

By evaluating the sensitivity indices of R0 , RV , RT and RV T using the parameter values in the Table 3.1, it can be seen (Table 3.2) that the sensitivity indices of R0 , RV , RT and RV T with respect to β does not depend on any parameter values. The most sensitive V = −1.6254 = ΥRV T , parameter for RV and RV T is the vaccine efficacy ǫ, since ΥR ǫ ǫ then, increasing (or decreasing) ǫ by 10% decreases (or increases) RV and RV T by 16%. Also the vaccination rate η is an important parameter for RV and RV T since by increasing (or decreasing) η by 10% decreases (or increases) RV and RV T by 5.5%. For R0 as well as RV the most sensitive parameter is the disease-induced death rate α, since by increasing the disease induced death rate α decreases the life expectancy which tends to reduce R0 and RV , consequently, treatment is absolutely necessary (since combined strategy applied concurrently is more beneficial) in order to curtail the epidemic. The rate of loss of immunity δ may be responsible for RV to be very sensitive

72

H. Rwezaura, E. Mtisi and J.M. Tchuenche Table 2. Parameters definition and values

η ω γ κ τ ǫ π λ β Λ ρ δ µ α

Definition vaccination rate (days−1 ) waning rate of vaccine-based immunity (days−1 ) rate of acquiring protective antibodies (days−1 )

treatment rate (days−1 ) recovery rate (days−1 ) vaccine efficacy drug efficacy effectiveness of the drug contact rate (days−1 /person) recruitment rate (days−1 ) fraction of recruited infectives (days−1 ) rate of loss of immunity (days−1 ) outflow rate of individuals (days−1 ) the disease induced death rate (days−1 )

Range 0.3 − 0.7 0.003 0.143 0.25 − 0.7 0.14 − 0.25 0.3 − 0.9 0.3 − 0.9 0<λ≤1 − − 0.05 − 0.1 0.1 − 0.2 0.0001 0.03 − 0.01

Average 0.5 0.003 0.143 0.5 0.25 0.8 0.6 0.5 0.019 20 0.1 0.15 0.0001 0.01

References Nu˜ no et al., (2006) Alexander et al., (2004) Nuno et al., (2006) Nu˜ no et al., (2006) Nu˜ no et al., (2006) Alexander et al., (2004) Nu˜ no et al., (2006) estimated estimated estimated estimated Nu˜ no et al., (2006) estimated Gani et al., (2005)

to disease-induced death rate α. The goal of any intervention strategy such as vaccination is to reduce the disease-induced death rate α. In order to reduce RV , the waning rate of vaccine-based immunity ω as well as the vaccine efficacy ǫ should be improved upon. The most sensitive parameter for RT and RV T is the treatment rate κ, since T = −0.9898 = ΥRV T then, increasing (or decreasing) κ by 10% decreases (or ΥR ǫ ǫ increases) RV and RV T by 9.9%.

3.3.

Numerical Simulations

In this section, we graphically show the evolution of the various classes using different parameters values. We note that treatment and vaccination quickly lower the level of epidemicity and all population classes progress to a positive steady state, with the number of individuals in the fully protected class far exceeding the numbers in other classes. The graphs showing the evolution of the various classes against time are shown below with the following parameters values: Figure 5: α = 0.01, δ = 0.15, ω = 0.003, γ = 0.143, ǫ = 0.8, η = 0.5, κ = 0.5, λ = 0.5, µ = 0.001, π = 0.6, ρ = 0.1, τ = 0.25, β = 4.5, Λ = 200 and initial values S0 = 1000; V0 = 500; I0 = 0; T0 = 0; R0 = 0; C0 = 0. We observe that the concavity in the susceptible, vaccinated and infective classes confirms that the model can exhibit bistable dynamics, which is further illustrated by the backward bifurcation Figure 7. The evolution of classes in Figure 5 are reproduced in the same phase plane in Figure 6. Due to inflow of infectives, none of the classes goes to extinction, since

A Mathematical Analysis of Influenza with Treatment and Vaccination

73

Table 3. Sensitivity indices of R0 , RV , RT and RV T with respect to each of their parameters. The sensitivity indices of RT and RV T with respect to R0 are 0.0102 when R0 = 3.7, and 0.0196 when R0 = 1.9. Parameters

R0

RV

RT

RV T

R0 η

-

1

0.0102(3.7) or 0.0196(1.9)

0.0102 (3.7) or 0.01961(1.9)

-

−0.5474

-

−0.5474

ω

-

0.0122

-

0.0122

γ

-

−0.4122

-

−0.4122

κ

-

-

−0.9898(3.7) or − 0.9804(1.9)

−0.9898(3.7) or − 0.9804(1.9)

ǫ

-

−1.6254

-

−1.6254

λ

−0.99

−0.9804

−0.0099

−0.0099

β

1

1

1

1

µ

−0.01

0.9995

−0.0002

0.9995

α

−0.99

−0.9804

−0.0099

−0.0099

the trivial equilibrium does not exist when ρ > 0. Thus, the public health objective in this case is to minimize the level of epidemicity as shown on Figure 8 (for the effects of treatment on infectives and vaccination on susceptibles). Figure 6 shows that vaccination and treatment applied concurrently quickly lower the level of epidemicity. Figure 7: The parameter values used here are the same as those in Figure 5, but with the following initial conditions S0 = 1000; V0 = 500; I0 = 0; T0 = 0; R0 = 0; C0 = 70. A similar trend to that in Figure 5 is observed due to the combine effects of vaccination and treatment. Figures 8 (a) and (b): These depict the effects of vaccination and treatment on susceptible and infected individuals, respectively.

4. 4.1.

Discussion and Conclusion Discussion

This work is based on the construction and use of a mathematical model for the transmission dynamics of influenza with treatment and vaccination in a human population. The main innovation of this work with respect to previous influenza models is that we explicitly consider both vaccination with an imperfect vaccine and treatment (with treatment efficacy), as well as disease induced death rate, where a fraction of susceptible individuals is vaccinated per unit time with inflow of infectives. Indeed, this model which includes only vaccination and treatment is more appropriate for developing countries where adequate health facilities are not generally available for mass hospitalization, isolation and quarantine. Our proposed model, likewise others, incorporates some essential parameters (such as vaccine and antiviral drugs efficacy, rate at which immunity wanes from previous infection, and

74

H. Rwezaura, E. Mtisi and J.M. Tchuenche Susceptible

Vaccinated

1000

2000

500

1000

0

0

50

100

0

0

50

Infected

Treated

1000

2000

500

1000

0

0

50

100

0

2

1000

1

0

0

50 4

Recovered 2000

0

100

50

100

0

x 10

0

100

Protected

50

100

Figure 5. Phase portrait of the evolution of each class against time.

5000 S V I T R C

4500 4000 3500 3000 2500 2000 1500 1000 500 0

0

50

100

150

Figure 6. Evolution of classes against time on the same phase plane.

A Mathematical Analysis of Influenza with Treatment and Vaccination Susceptible

Vaccinated

1000

2000

500

1000

0

0

50

0

100

0

50

Infected 2000

500

1000

0

50

0

100

2000

2

1000

1

0

0

50 4

Recovered

0

100

Treated

1000

0

75

50

0

100

100

Protected

x 10

0

50

100

Figure 7. Evolution of each class against time with different initial values. 1000

16000

900 14000

800 12000 Infected/infectious

Susceptible

700 600 500 400

10000 8000 6000

300 4000

200 2000

100 0

0

200

400

600

800 1000 1200 Vaccinated

(a)

1400

1600

1800

2000

0

0

1000

2000

3000 4000 Treated

5000

6000

7000

(b)

Figure 8. Effects of vaccination (a) and treatment (b).

the disease-induced death rate) of influenza transmission, which enable the assessment of various anti-influenza preventive strategies, and their epidemiological consequences. The model is given in the form of a non-linear ODE, and our analytical results show that:

76

H. Rwezaura, E. Mtisi and J.M. Tchuenche (i) In the absence of recruitment of infected humans into the community, the DFE exists and consequently, influenza can be controlled using treatment and vaccination.

(ii) The basic reproduction number R0 is reduced whenever treatment or vaccination is introduced as a therapeutic measure (i.e., RV < R0 and RT < R0 ). (iii) Also, the vaccination and treatment-induced reproduction number RV T reduces both RV and RT . Thus, concurrent administration of vaccination and treatment is more adequate in curtailing the epidemic. This is consistent with the fact that control measures are necessary to reduced the value of the basic reproduction number R0 , and if this quantity is less than unity, the disease can be eradicated. Thus, a combined control program is more effective in curtailing the epidemic. Since it is well-known that infected persons do migrate from one region to another, this is incorporated into our model by assuming that a proportion (ρ) of recruited individuals are infected. That is, a proportion (1 − ρ)Λ of recruited individuals are susceptible, while the remaining fraction ρΛ is infected. The consequence of this inflow of infectives is that the DFE equilibrium does not exist any longer. Thus, no amount of preventive measures can eradicate influenza from the population, but can lead to significant reductions in influenza infections. For ρ > 0, the disease remains endemic (persists) in the community, and the number of infectives can only be reduced by reducing ρ or Λ, or by increasing the treatment and vaccination rate κ and η (as well as the vaccine and drug efficacy ǫ, and π), respectively. Nevertheless, the drug efficacy π does not appear explicitly in the expression of RV T , and this could indicate that vaccination is more important in preventing an influenza outbreak (i.e., prevention is better than cure). We use a continuous vaccination program (where a fraction of susceptible individuals is vaccinated per unit time), and do hope that the result is independent of the type of vaccination program adopted (i.e., using cohort vaccination or a combination of both cohort and continuous vaccination). Therefore, the present model can be refined in various ways: For instance, it is instructive to determine whether or not the results obtained using continuous vaccination or a combined continuous and cohort vaccination (where a fraction of the newly-recruited members of the community are vaccinated) will hold. It is also worthwhile to investigate the effect of co-circulating influenza strains and the effect of seasonality (using time-dependent transmission coefficients) on the transmission dynamics of this disease. It is important to note that our model is relaxed by considering a model with mass action incidence and without inflow of infectives in order to study the persistence of solutions of the system. In general, results are given regarding invariant regions, existence, positivity and stability of equilibria. Basically, we have established the following results: (a) Local and global stability of the DFE of the model without inflow of infectives (ρ = 0). (b) Local and tri-stability stability analysis of the EE when ρ > 0. (c) Persistence of the model system (2.1) with ρ = 0.

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77

(d) Public health measures significantly reduce influenza infections. (e) The treatment-only sub-model with mass action incidence has no periodic orbits, and its EE if its exists, is GAS and the disease will persist. (f) The treatment-only sub-model with mass action incidence exhibits the phenomenon of backward bifurcation, likewise the vaccination-only sub-model (cf. Alexander et al., 2004), consequently, the full model also undergoes the same phenomenon when it reproductive threshold RV T < 1 as shown in Theorem 9, Lemma 10 and Section 2.18. Also, it is algebraically shown that the model with inflow of infectives has a tri-stable equilibria, where the disease-free equilibrium coexists with two stable endemic equilibrium when the aforementioned threshold is less than unity - a dynamical feature that has been observed in TB dynamics (Gumel and Song, 2008). (g) Sensitivity of the model parameters to determine their relative importance in disease transmission. (h) Low fitness ratio is beneficial and reduce the probability of population-level perversity. We have performed sensitivity analysis on a mathematical model of influenza transmission to determine the relative importance of model parameters to disease transmission and prevalence by computing sensitivity indices of the reproductive numbers, which measures initial disease transmission (Chitnis et al., 2008). Mathematical modeling of influenza can play a unique role in comparing the effects of control strategies. We begin such a comparison by determining the relative importance of model parameters in influenza transmission. The model is based on a continuous vaccination program . The same conclusion will hold if a continuous vaccination program only (where a fraction of susceptible individuals is vaccinated per unit time) is considered (Sharomi et al., 2007). The threshold fitness ratios Hj = 1, j = T, V indicates that above this value, infected treated, vaccinated, respectively, generate more secondary infections than infected untreated, and unvaccinated individuals (Smith? and Blower, 2004). Numerical simulations help examine the dynamics and suggest some properties of these models that we were unable to prove mathematically. One of them shows that our model may have a backward bifurcation where the DFE and EE co-exist.

4.2.

Conclusion

In summary, mathematical models are potentially useful tools to aid in the design of control programs for infectious diseases. We developed an epidemiological model of human influenza with vaccination and treatment and used it to predict trends in infection as well as possible control measures. The model incorporates several realistic features including vaccination of susceptible and drug treatment for infected individuals. The qualitative and quantitative mathematical properties of the models are studied, their biological consequences and some control strategies are discussed, and the results of the models are compared with previous ones. It is shown that the full model exhibits the phenomenon

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of backward bifurcation whereby two equilibria, namely the disease-free and the endemic equilibrium co-exist. Explicit thresholds of vaccination and treatment rates are established for which the infection will be controlled under certain levels. Realistic model parameters are used to validate the model.

Acknowledgments HR acknowledges with thanks the Eastern Africa Universities Mathematics Program for partial support and the Center for International Mobility for student exchange program at Lappeenranta University of Technology (Finland) within the framework of North-SouthSouth Network Program.

Appendix A C0

=

Λ[(µ + γ + ω + η)(µ + κ + λα − β) + βǫη],

=

Λ[(µ + γ + ω + η)(µ + κ + λα) − β(µ + γ + ω + (1 − ǫ)η)],

=

Λ γη [µ(µ + γ + ω + η + )(µ + κ + λα) − µβ(µ + γ + ω + (1 − ǫ)η)]− µ µ Λγη (µ + κ + λα), − µ

= Λ(µ + γ + ω + η(1 + µγ ))(µ + κ + λα)[1 − −

Λγη (µ + κ + λα), µ

βµ[µ + γ + ω + (1 − ǫ)η] ] [(µ + γ)(µ + η) + µω](µ + κ + λα)

= Λ(µ + γ + ω + η(1 + µγ ))(µ + κ + λα)(1 − RV T ) − = −[

Λγη µ (µ

+ κ + λα),

Λγη γ (µ + κ + λα) + Λ(µ + γ + ω + η(1 + ))(µ + κ + λα)(RV T − 1)]. µ µ

Appendix B Barbalat Lemma (1959) Lemma 20 Let x 7→ F (t) be a differentiable function with a finite limit as t → ∞. If F˙ is uniformly continuous, then F˙ → 0 as t → ∞.

Appendix C Alternative Proof of Theorem 19 In order to find the conditions for the existence of multiple equilibria, we use the second equation of (2.79) to express the variables tˆ∗ in terms of the variable i∗ when i∗ 6= 0. This

A Mathematical Analysis of Influenza with Treatment and Vaccination

79

gives (at equilibrium) tˆ∗ =

κ ˜ i∗ . (1 + τ˜π) + (1 − λ)α ˜

Substituting it into the first equation of (2.79) gives the following quadratic equation: ˜ + τ˜π + (1 − λ)α)]i ˜ + τ˜π + (1 − λ)α)]i Q(i, ρ) = [˜ κ + β(1 ˜ 2 + (1 + κ ˜ + λα ˜ − β)[1 ˜ −ρ[1 + τ˜π + (1 − λ)α)] ˜ = 0, = ai2 + bi + c = 0, ˜ + τ˜π + (1 − λ)α)], ˜ + τ˜π + (1 − λ)α)] where a = [˜ κ + β(1 ˜ b = (1 + κ ˜ + λα ˜ − β)[1 ˜ and c = ρ[1 + τ˜π + (1 − λ)α)]. ˜ Since all the model parameters are nonnegative, it follows that a > 0 and c ≤ 0. Furthermore, if RT < 1, then b > 0, that is, ˜ + τ˜π + (1 − λ)α)], b = (1 + κ ˜ + λα ˜ − β)[1 ˜ = (1 + κ ˜ + λα ˜ )[1 + τ˜π + (1 − λ)α)](1 ˜ −

β˜ 1+˜ κ+λα ˜ ),

= (1 + κ ˜ + λα ˜ )[1 + τ˜π + (1 − λ)α)][1 ˜ − RT ]. Notice that when ρ = 0, the quadratic equation Q(i, ρ) is reduced to a linear equation b ai + b = 0, so that i = − . If b ≥ 0, i.e., RT ≤ 1, then, i ≤ 0. Therefore, no endemic a equilibrium exists whenever RT ≤ 1. But, for b < 0, i.e., RT > 1, i > 0. Therefore, there exists a unique endemic equilibrium. If ρ > 0, and since a > 0 and c < 0, then, the quadratic equation Q(i, ρ) has two positive roots for b > 0 (RT ≤ 1). Hence, the treatment only sub-model has an endemic equilibrium regardless of RT ≤ 1 or RT > 1. This result indicates the possibility of backward bifurcation due to the existence of the endemic equilibrium when RT ≤ 1. Case (ii) of Theorem 19 thus implies that the model exhibits the phenomenon of backward bifurcation in the the treatment only sub-model (2.72) when RT ≤ 1, for b > 0. To check for this, we solve for β˜c when b = 0 which implies RT = 1) and obtain β˜c = 1 + κ ˜ + λα ˜, ˜ from which it can be shown that backward bifurcation occurs for values of β such that β˜c < β˜ < 1 + κ ˜ + λα ˜ and ρ > 0.

Appendix D (1) Endemic Equilibria when ρ = 0 The endemic equilibria of model system (2.1) with ρ = 0 (if they exist) cannot be cleanly expressed in closed form. In order to find the existence of these equilibria, we use (2.1) to express the variables S, V, T, R in terms of I when I 6= 0, with the assumption that the Cclass plays no further part in the disease transmission or γ = 0. Let the endemic equilibrium of system (2.1) without the fully protected class be denoted by E ∗ = (S ∗ , V ∗ , I ∗ , T ∗ , R∗ ).

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Then, expressing E ∗ in terms of I ∗ , we have  (−a11 − a12 I ∗ /(a7 − a5 I ∗ ))(a8 + a13 I ∗ /a4 )  ∗ =  S    ηω − (a9 + βa6 I ∗ /(a7 − a5 I ∗ ))(a11 + a12 I ∗ /(a7 − a5 I ∗ ))        −η(a8 + a13 I ∗ /a4 )   V∗ =    ηω − (a9 + βa6 I ∗ /(a7 − a5 I ∗ ))(a11 + a12 I ∗ /(a7 − a5 I ∗ ))

(4.85)

  κI ∗   T∗ =    µ + πτ + (1 − λ)α        κπτ I ∗   ,  R∗ = (µδ )(µ + πτ + (1 − λ)α)

where

a1 = µ + πτ a2 = (1 − λ)α a3 = µ + δ a4 = a3 (a1 + a2 ) a5 = a2 κ + (a1 + a2 )αλ a6 = µ(a1 + a2 ) a7 = Λ(a1 + a2 ) a8 = (1 − ρ)Λ a9 = (µ + η) a10 = (1 − ǫ) a11 = γ + ω + µ a12 = βa6 a10 a13 = δκρτ a14 = κ + µ + λα a15 = a9 (γ + µ) + µω. Substituting (4.85) into the third equation of system (2.1) gives (at equilibrium) P (I ∗ ) ≡ I ∗ (b3 I ∗2 + b2 I ∗ + b1 ) = 0,

(4.86)

where b3 = a12 (a4 a14 (βa6 − a5 a9 ) + a13 (a5 η − a6 )) +a5 (a4 a5 a14 a15 + β6 a11 (a13 − a4 a14 )),

b2 = a4 a7 a9 a14 (a12 − 2a5 a11 ) + βa6 a7 a11 (a4 a14 − a13 ) + βΛa4 a6 (a5 a11 − a12 ) +a12 η(a4 a5 Λ − a7 a13 ) + 2a4 a5 a7 a14 ηω,

b1 = a27 a214 µa15 (1 − RV T ).

(4.87)

Clearly I ∗ = 0 is a solution to the equation (4.86) which corresponds to the disease free equilibrium of system (2.1) with ρ = 0. The remaining quadratic equation will be analysed for the existence of a positive solution which corresponds to the endemic equilibrium. Since all parameters are positive it follows from (4.87) that b3 > 0 and the sign b1 corresponds to that of 1 − RV T . The existence of the equilibria are summarized in the following theorem. Theorem 21 Suppose ρ = 0 in (2.1). Then (i) a unique endemic equilibrium if b1 < 0 ⇐⇒ RV T > 1, (ii) a unique endemic equilibrium if b2 < 0 or b22 − 4b3 b1 = 0, (iii) two endemic equilibria if b1 > 0 ⇐⇒ RV T < 1, b2 < 0 and b22 − 4b3 b1 > 0, (iv) no endemic otherwise.

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81

(2) Endemic Equilibrium when ρ > 0 and γ = 0 To find the existence of the endemic equilibria of model (2.1) with the C-class omitted, we substitute (4.85) into the third equation of (2.1). The resulting polynomial equation is given by P (I ∗ ) + ρ(c2 I ∗2 + c1 I ∗ + c0 ) = 0, (4.88) where P (I ∗ ) is given in (infe1) and c2 = a4 a5 a9 Λ(a12 − a5 a11 ) + a4 a5 ηΛ(a5 ω − a6 a10 ), c1 = 2a5 a15 Λ − a6 a10 βΛµ, c0 = −a4 a27 a15 Λ.

We note that when ρ > 0, the disease free equilibrium does not exist. This is because the model (2.1) assumes a constant flow of new members into the population of which a specific fraction is infective. Therefore at any time the infected populated is not zero except when ρ = 0. From analysis of the coefficients in (4.88) we note that b3 is positive and c0 is negative. By Descartes’ Rule of signs, we note that there is at least one sign change one sign change in the sequence of coefficients. Hence the model (2.1) with ρ > 0 and γ = 0 has at least a positive root.

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[38] Nu˜ no M., Chowell G. and Gumel A.B. (2006) Assessing the role of basic control measures, antivirals and vaccine in curtailing pandemic influenza: Scenarios for the US, UK and the Netherlands, J. R. Soc. Interface 4(14), 505-521. [39] Parker A.S. Jr. and Bradley S.B. (2001) Influenza: A to B, http : //cme.uf l.edu/media/f lu/index.html, retrieved Sept. 19, 2007. [40] Perko, L. Differential Equations and Dynamical Systems. Springer, New York (1991). [41] Sanofi Pasteur’s Investigational H5N1 Influenza vaccine achieves high immune response at low dosage, http : //online.wsj.com/public/article/P R − CO − 20070918 − 900189.html?mod = crnews, retrieved Sept. 14, 2007. [42] Smith? R.J. and Blower S.M. (2004) Could disease-modifying HIV vaccines cause population-level perversity? Lancet Infect. Dis. 4, 636-639. [43] Spear R. (2002) Mathematical modeling in environmental health, Environ. Health Perspect., 110 A382. [44] United Republic of Tanzania, Ministry of Livestock Development: Avian influenza and human influenza pandemic, Bird Flu Communication Strategy, www.mifugo.go.tz/documents\_storage/ TanzaniaBirdFluCommunicationStrategy.pdf, retrieved Sept. 11, 2007. [45] van den Driessche P. and Watmough J. (2002) Reproduction numbers and sub- threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci. 180, 2948. [46] Wikipedia Influenza pandemic, http://www.en.wikipedia.org/wiki/Influenza_pandemic, trieved Sept. 14, 2007.

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[47] World Health Organization fact sheet on avian influenza emergency preparedness and response plan. [48] Yan X-P. and Li W-T. (2006) Global existence of periodic solutions in a simplified four-neuron bam neural network model with multiple delays, Discr. Dyn. Nature and Soc., 1-18.

In: Infectious Disease Modelling Research Progress ISBN 978-1-60741-347-9 c 2009 Nova Science Publishers, Inc. Editors: J.M. Tchuenche, C. Chiyaka, pp. 85-131

Chapter 3

A T HEORETICAL A SSESSMENT OF THE E FFECTS OF C HEMOPROPHYLAXIS , T REATMENT AND D RUG R ESISTANCE IN TB I NDIVIDUALS C O - INFECTED WITH HIV/AIDS C.P. Bhunu1,∗ and W. Garira1,2 Modelling Biomedical Systems Research Group, Department of Applied Mathematics, 1 National University of Science and Technology, P. O. Box 939 Ascot, Bulawayo, Zimbabwe 2 Department of Mathematics and Applied Mathematics, University of Venda, South Africa

Abstract Tuberculosis (TB) patients who do not complete TB treatment risk developing antibiotic resistant TB, one of the most serious health problems facing the society today. Effects of TB drug resistance are severe in individuals co-infected with HIV/AIDS. In this paper we develop a two strain TB model, two strain HIV/AIDS model and an HIV/AIDS-TB co-infection model for assessing the impact of treatment and drug resistance in controlling TB in settings with high HIV/AIDS prevalence. In the absence of HIV/AIDS, the TB-only model is shown to exhibit a phenomenon known as backward bifurcation where a stable disease-free equilibrium co-exists with the stable endemic equilibrium when the associated reproduction number is less than unity. On the contrary, the HIV/AIDS-only model shows a globally asymptotically stable disease-free equilibrium when the associated reproduction number is less than unity. The centre manifold theory is used to determine the local asymptotic stability of the endemic equilibria. From the study we conclude that chemoprophylaxis and TB treatment are more effective in controlling TB in the absence of drug resistant TB and HIV/AIDS as shown for some derived critical threshold values. The results of the study show that chemoprophylaxis and treatment of TB infectives are equally effective in controlling TB in co-infected individuals, but not so in cases involving drug ∗

E-mail address: [email protected], [email protected]. Corresponding author.

86

C.P. Bhunu and W. Garira resistant TB. Further, we conclude from the study that chemoprophylaxis and treatment of TB are more effective in TB infected individuals co-infected with HIV/AIDS who are not yet on antiretroviral therapy.

Key words: TB, HIV/AIDS, Co-infection, Chemoprophylaxis, Treatment, Antiretroviral therapy, Stability.

1.

Introduction

Tuberculosis (TB) is present in 1.8 billion people world wide (Mukherjee [34]). Its incidence and mortality in Sub-Saharan Africa have reached alarming levels and continue to rise (Cohen et al. [16]). In Sub-Saharan Africa, TB is the leading cause of mortality and in developing countries it accounts for an estimated 2 million deaths which is a quarter of avoidable adult deaths (Raviglione [42]). TB was assumed to be on its way “out” in developed countries until the number of TB cases began to increase in the 1980s. Its control in Sub-Saharan Africa has been hindered by emergence of the Human Immunodeficiency Virus (HIV) and the associated Acquired Immune Deficiency Syndrome (AIDS). HIV/AIDS significantly affects the progression of Mycobacterium tuberculosis(Mtb) infection. In immune-competent individuals Mtb infection rarely leads to disease and usually results in latent non-transmissible infection, but in individuals co-infected with HIV/AIDS the story is different. Mtb infected individuals dually infected with HIV/AIDS tend to develop active TB quickly due to immune suppression and malabsorption of rifampicin and isoniazid/ethambutol in advanced stages of HIV infection, such that it is perhaps counterproductive to conceive that TB can be managed in exactly the same way as immunocompetent people (Perlman [38]). TB treatment has been successful in low HIV/AIDS prevalence areas. Additional measures are necessary to reduce TB incidence in high burden regions (Nunn et al. [37]). In Sub-Saharan Africa the face of HIV/AIDS is TB, HIV/AIDS and TB fuel one another. Preventive therapy of TB in HIV/AIDS infected individuals is highly recommended (WH0 [47]) and could dramatically reduce the impact of HIV on TB epidemiology, but its implementation is limited in developing countries because of complex logistical and practical difficulties (Frieden [23]). HIV/AIDS infected patients given 6 months of anti-TB treatment had a higher relapse rate than those treated longer [39]. From a programme perspective, the ability to use the 6 months short-course anti-TB regimen in both HIV-infected and unifected patients is attractive especially in poor-limited settings. However, given the global estimate of 741 000 HIV-TB cases annually even a low rate of acquired rifampicin resistance can have significant consequences [38]. Apart from the effects of HIV on TB control, multi-drug resistant forms of TB have worsened the situation. Due to high number of TB individuals per health professional, Directly Observed Treatment Strategy (DOTS) has not been successfully implemented in most countries in Sub-Saharan Africa. This has led to incomplete treatment resulting in multidrug resistant cases of TB. Multi-drug resistance is defined as resistance to isoniazid and rifampicin whether there is resistance to other first line drugs (pyrazinamide, ethambutol) or not (Davies [17]). Resistance to isoniazid and streptomycin is the most common form of TB resistance by two drugs. Treatment of multi-drug resistant forms of TB is done us-

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ing second-line drugs (fluoroquinolone, capreomycin, kanamycin, and amikacin) which are more expensive and have more side effects. Unfortunately these second-line drugs are not available in most poor resource settings like those in Sub-Saharan Africa excluding South Africa. People with multi-drug resistant TB can be infectious and pass on the drug resistant bacteria to other people. For those individuals with access to second-line drugs misusing and/or mismanaging them results in creation of extensively drug resistant TB. Erratic supply of second line anti-TB drugs in areas of high HIV/AIDS prevalence has proven to be a fertile ground of extensively drug resistant TB. Extensively drug resistant TB is defined as resistance to any fluoroquine, and at least one injectable second-line drugs (capreomycin, kanamycin, and amikacin), in addition to isoniazid and rifampicin. Demands for the introduction of antiretroviral therapy in Africa have been growing over the past few years. On the face of it seems to be good news, but only a few individuals in need of them in Sub-Saharan Africa have access to these life saving drugs. Thus, like TB, HIV/AIDS has become primarily the disease of the poor (Farmer et al. [21]). If used optimally, HIV treatment could delay the onset of AIDS thus, reducing the burden of HIV/AIDS on TB epidemics. Virus strains with reduced sensitivity to Zidovudine, the first drug used against AIDS were observed in 1989 three years after it was introduced (Lader et al. [28]). Subsequently resistance to every currently licensed antiretroviral drug has been observed (Schinazi et al. [43]). Cross resistance between drugs of the same class is the rule rather than exception implying that drug resistance within an individual is not limited to a single compound (Harrington and Lader [24]). Drug resistance arises by natural selection, mutant HIV strains being selected when virus replicates in sublimiting drug concentrations (Steven et al. [45]). The only way to prevent drug resistance is to use a drug regimen that reduces virus replication to zero [45]. Most resistant strains of HIV are poor at replicating and do not persist in the absence of drugs (Quinones-Mateu and Arts [40]). People infected with drug resistant virus are more likely to have their treatment regimen fail allowing the virus to develop resistance to other drugs. Thus, in the absence of second line drugs for multi-drug resistant TB, HIV/TB co-infection with drug resistance of both infections is a double sword for the poor lives in Sub-Saharan Africa. To avert a looming disaster, there is an urgent need to strengthen both HIV/AIDS and TB control programmes in areas with high rates of HIV-related TB. A number of theoretical studies have been done on the mathematical modelling of coexistence of different pathogens (strains) in the same host [1–3, 5–10, 12, 13, 19, 30, 32, 35, 36]. Castillo-Chavez and Feng [12, 13] studied two strain TB model in the context of treatment, but we differ from [12,13] in that, in our TB submodel we considered exogenous re-infection as well. Naresh and Tripathi [35] did analyse the HIV-TB co-infection model, but we differ from [35] in that we have considered treatment of both infections as well as resistance of anti-retroviral therapy and anti-TB drugs. Bacaer et al. [5], Sharomi et al. [44] did analyse the HIV-TB co-infection model, but we differ from [5, 44] in that in addition to AIDS treatment and TB treatment we have considered resistance to anti-TB drugs and antiretroviral therapy. This paper is organised as follows. In Section 2 we have a two strain TB model description and its analysis. Section 3 presents a description and an analysis of two strain HIV/AIDS model. In Section 4 we present a description and an analysis of full model. Finally, we present the summary and concluding remarks.

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A Two Strain Tuberculosis Model

In this section we begin by presenting a two strain TB model with treatment for the drug sensitive strain. The model subdivides the human population into the following sub-population of susceptible individuals (S), those exposed to TB, ET1 (drug sensitive) and ET2 (drug resistant), individuals with symptoms of TB, IT1 (drug sensitive) and IT2 (drug resistant), and those who have recovered from drug sensitive TB (RT ). It is assumed susceptible humans are recruited into the population at per capita rate Λ through birth and migration. Susceptible individuals acquire TB following contact with an infectious case at rate λTj , j = 1, 2 where j = 1 and j = 2 correponds to the drug sensitive TB and drug resistant TB, respectively with, βj cT ITj (t) (1) λTj = . NT (t)

Table 1. Model parameters and their interpretations. Definition

Symbol

Estimate(Range)

Recruitment rate

Λ

−1

0.029yr

Mukandavire and Garira [33]

Natural mortality rate

µ

0.02yr−1

Mukandavire and Garira [33]

Contact rate

cT

3yr−1

dT 1 , d T 2

TB induced death rate

Source

Estimate −1

0.3,0.5yr

Dye and Williams [19] −1

Transmission rate

β1 , β 2

0.35 (0.1-0.6)yr

Dye and Williams [19]

Endogeneous reactivation rates

k1 , k2

0.00013

Dye and Williams [19]

(0.0001-0.0003)yr−1 Probability of successful treatment

q

0.2 (0.15-0.25)

Dye and Williams [19]

Treatment rate for the latently infected

r1

0.7yr−1

Dye and Williams [19]

r2

Treatment rate for the infectives

0.88yr

−1

Qing-Song Bao et al. [41]

−1

Estimate

Protective factor

δ1

0.7yr

Protective factor

δ2

0.9yr−1

Estimate

1 − p1

0.01

Estimate

1 − p2

0.1

Estimate

Probability of susceptibles developing drug sensitive fast TB Probability of susceptibles developing drug sensitive fast TB

In equation (1), cT is the per capita contact rate, βj is the average number of susceptible individuals infected by one infectious individual per unit contact of time and NT (t) is the total population size given by, NT (t) = S(t) +

2 X j=1


ETj (t) + ITj (t) + RT (t).

(2)

Susceptibles infected with Mtb enter exposed class, (ETj ) at rate pj λTj and develop fast TB at rate (1 − pj )λTj to enter the infectious class, (ITj ). Individuals in the latently

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infected class, (ETj ) develops active TB as a result endogenous reactivation and exogenous reinfection at rates kj and δj λTj respectively into the infectious class, (ITj ). Individuals infected with the drug sensitive strain in the latently infected class, (ET1 ) and the infectious class, (IT1 ) are treated at rates r1 and r2 respectively. Individuals in ET1 (t) treated will move into RT (t) at rate r1 and of the individuals in IT1 receiving treatment a proportion q responds well to treatment and move into RT (t) and the remainder 1 − q develops drug resistance due to poor administration of treatment and move into ET2 at rate (1 − q)r2 . Individuals in RT (t) are not immune to TB and are reinfected at a rate λTj and individuals in each subgroup have a per capita death rate µ. The per capita disease induced death rate in each infectious class, (ITj ) is given by dTj . Parameters described will assume values in Table 1. The model flow diagram is given in Figure 1.

Figure 1. Structure of model. Putting the formulations together gives the following system of differential equations de-

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C.P. Bhunu and W. Garira

scribing the model. ′

S =Λ−

2 X j=1

λTj S − µS,

ET′ 1 = p1 λT1 (S + RT ) − (k1 + r1 + µ)ET1 − δ1 λT1 ET1 − λT2 ET1 , IT′ 1 = (1 − p1 )λT1 (S + RT ) + k1 ET1 + δ1 λT1 ET1 − (µ + dT1 + r2 )IT1 , RT′ = r1 ET1 + qr2 IT1 − µRT −

2 X

(3)

λTj RT ,

j=1

ET′ 2 = (1 − q)r2 IT1 + p2 λT2 (S + RT ) + λT2 ET1 − δ2 λT2 ET2 − (µ + k2 )ET2 , IT′ 2 = (1 − p2 )λT2 (S + RT ) + k2 ET2 + δ2 λT2 ET2 − (µ + dT2 )IT2 , Model system (3) has initial conditions given by S(0) = S0 ≥ 0, ETj (0) = ETj0 ≥ 0, ITj (0) = ITj0 ≥ 0, RT (0) = RT0 .

(4)

Based on biological considerations, the model system (3) will be studied in the following region   Λ 6 ΩT = (S, ET1 , IT1 , RT , ET2 , IT2 ) ∈ R+ : NT ≤ . (5) µ Theorem 1 assures that model system (3) is well posed in the sense that all solutions with non-negative initial condition remains non-negative for all t ≥ 0, and therefore makes biological sense. Theorem 1. The region ΩT ⊂ R6+ is positively invariant with respect to model system (3). Proof. We prove that the vector field of model system (3) points to the boundary of ΩT and is given by positive S, ETj , ITj and RT -axes. At the start of the process, the vector field, (F1 , F2 , F3 , F4 , F5 , F6 ) of model system (3) restricted to the positive S-axes has the form F1 (S, 0, 0, 0, 0, 0) = Λ − µS, F2 (S, 0, 0, 0, 0, 0) = F3 (S, 0, 0, 0, 0, 0) = · · · = F6 (S, 0, 0, 0, 0, 0) = 0.

(6)

Since F1 (S, 0, 0, 0, 0, 0) > 0 for S > 0 and Λ > µS the vector field point towards the interior of ΩT . On the positive ETj , ITj , (j = 1, 2) and RT -axis we have, F1 (0, ET1 , 0, 0, 0, 0) = F1 (0, 0, IT1 , 0, 0, 0) = F1 (0, 0, 0, RT , 0, 0) (7) = F1 (0, 0, 0, 0, ET2 , 0) = F1 (0, 0, 0, 0, 0, IT2 ) = Λ, implying that the vector field point towards the interior of ΩT on each axis. Thus region ΩT is positively invariant with respect to system (3). Adding the equations in model system (3) together we obtain, 2 X dTj ITj NT′ (t) = Λ − µNT (t) − (8) j=1

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  P Λ − µ + 2j=1 dTj NT (t) ≤ NT′ (t) ≤ Λ − µNT (t).

(9)

so that, Thus NT (t) is bounded.

2.1.

Disease-Free Equilibrium and Stability Analysis

Model system (3) has the disease-free equilibrium given by   Λ , 0, 0, 0, 0, 0 . ET0 = (S0 , ET10 , IT10 , RT0 , ET20 , IT20 ) = µ

(10)

It can be shown that ET0 attracts the region ΩT0 = {(S, ET1 , IT1 , RT , ET2 , IT2 ) ∈ ΩT : ET1 = IT1 = RT = ET2 = IT2 = 0} . (11) The basic reproduction number R0 , defined as the expected number of secondary infections caused by an infective individual upon entering a totally susceptible population (Anderson and May [4], Diekman et al. [18], van den Driessche and Watmough [46]). The linear stability of ET0 is governed by the reproduction number RTT . Closely following [46], we have 

   F =   

p1 λT1 (S + RT ) (1 − p1 )λT1 (S + RT ) p2 λT2 (S + RT ) (1 − p2 )λT2 (S + RT ) 0 0

       



      and V =       

(k1 + r1 + µ)ET1 + (δ1 λT1 + λT2 )ET1 (µ + dT1 + r2 )IT1 − (k1 + δ1 λT1 )ET1 (k2 + µ + δ2 λT2 )ET2 − λT2 ET2 − (1 − q)r2 IT1 (µ + dT2 )IT2 − (k2 + δ2 λT2 )ET2 2 X λTj RT − qr2 IT1 − r1 ET1 µRT + j=1

µS +

2 X j=1

λTj S − Λ



      .      

(12)

The infected compartments are ET1 , IT1 , ET2 and IT2 . Thus matrices F and V for the new infection terms and the remaining terms are respectively given by   0 p1 β1 cT 0 0   0 (1 − p1 )β1 cT 0 0  and F =   0 0 0 p2 β2 cT 0 0 0 (1 − p2 )β2 CT (13)   k1 + r1 + µ 0 0 0   −k1 µ + dT1 + r2 0 0  V =   0 −(1 − q)r2 k2 + µ 0 0 0 −k2 µ + dT2 The dominant eigenvalues of F V −1 are RT2 =

(1 − p2 )β2 cT µ + β2 cT k2 , (µ + k2 )(µ + dT2 ) (14)

RT1

(µ + r1 )(1 − p1 )β1 cT + β1 cT k1 . = (µ + dT1 + r2 )(µ + k1 + r1 )

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Thus the reproduction number for model system (3) is given by ρ(F V −1 ) = RTT = max {RT1 , RT2 } ,

(15)

where RT1 and RT2 are respectively the reproduction numbers for the drug sensitive TB only and drug resistant TB only. Theorem 2 follows from Theorem 2 in [46]. Theorem 2. The disease-free equilibrium ET0 is locally asymptotically stable whenever RTT < 1 and unstable otherwise. In the following section we analyse the endemic equilibria states.

2.2.

Endemic Equilibria

For the model system (3), we have the following steady states. 2.2.1.

Drug Resistant TB-only Equilibrium

The drug resistant TB only equilibrium occurs when ET1 = IT1 = RT = 0 and is denoted by ET∗1 = (S ∗ , 0, 0, 0, ET∗2 , IT∗2 ) (16) where in terms of the equilibrium value of the force of infection λ∗T2 we have S∗ =

IT∗2 =

NT∗

p2 Λλ∗T2 Λ ∗ , , ET2 = ∗ µ + λ∗T2 δ2 λ∗2 T2 + (δ2 µ + k2 + µ)λT2 + µ(µ + k2 ) (k2 + µ + δ2 λ∗T2 )λ∗T2 Λ − µp2 λ∗T2 Λ ,  ∗ + µ(µ + k ) (µ + dT2 ) δ2 λ∗2 (δ µ + k + µ)λ + 2 2 2 T2 T2

(17)


∗ Λ δ2 λ∗2 T2 + (k2 + µ + p2 dT2 + δ2 µ + δ2 dT2 )λT2 + (k2 + µ)(µ + dT2 )   = . ∗ + µ(µ + k ) (µ + dT2 ) δ2 λ∗2 (δ µ + k + µ)λ + 2 2 2 T2 T2

Substituting equation (17) into expression for λ∗T2 in (1), we have

∗ λ∗T2 f (λ∗T2 ) = λ∗T2 (A1 λ∗2 T2 + B1 λT2 + C1 ) = 0,

(18)

where λ∗T2 = 0 corresponds to the disease free equilibrium and f (λ∗T2 ) = 0 corresponds to the existence of endemic equilibria which implies, p −B1 ± B12 − 4A1 C1 ∗ (19) λT2 = 2A1 where, δ2 k2 + µ + p2 dT2 + δ2 (µ + dT2 ) − β2 cT δ2 , B1 = , (µ + dT2 )(µ + k2 ) (µ + dT2 )(µ + k2 ) and C1 = 1 − RT2 .

A1 =

(20)

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By examining the quadratic equation we see that there is a unique endemic equilibrium if B1 < 0 and C1 = 0 or B12 − 4A1 C1 = 0, there are two if C1 > 0, B1 < 0 and B12 − 4A1 C1 > 0, and there is none otherwise. The coefficient A1 is always positive and C1 is positive or negative if RT2 is less than or greater than one respectively. We therefore rewrite these conditions in Lemma 1. Lemma 1. Model system (3) has (i) precisely one unique endemic equilibrium if C1 < 0 ⇔ RT2 > 1, (ii) precisely one unique endemic equilibrium if B1 < 0 and C1 = 0 or B12 − 4A1 C1 = 0, (iii) precisely two endemic equilibria if C1 > 0, B1 < 0 and B12 − 4A1 C1 > 0, (iv) otherwise there is none. To find the backward bifurcation point, we set the discriminant B12 − 4A1 C1 = 0 and make RT2 the subject of the formulae to obtain RcT2 = 1 −

B12 , 4A1

(21)

from which it can be shown that backward bifurcation occurs for values of RT2 in the range RcT2 < RT2 < 1.

Lemma 2. The endemic equilibrium point ET∗1 exists for RT2 > 1. p −B1 + B12 − 4A1 C1 ∗ Proof. Analysing expression, λT2 = from which is clear that the 2Ap 1 −B1 + B12 − 4A1 C1 disease is endemic only when λ∗T2 > 0 ⇒ > 0 ⇒ B12 − 4A1 C1 > 2A1 B12 ⇒ 4A1 (1 − RT2 ) < 0 ⇒ RT2 > 1 since A1 is positive. RT2 > 1.

Thus ET∗1 exists for

Figure 2 is a graphical representation showing the proportion of drug resistant TB infectives against the reproduction number RT2 . It shows that making the reproduction number less than unity does not necessarily eradicate the epidemic. Using the standard linearisation of the two strain TB model to determine the local asymptotic stability of an endemic equilibrium point is boring and laborious to track mathematically. We now employ the Centre Manifold theory [11] as described in [15] (Theorem 4.1), to establish the local asymptotic stability of the endemic equilibrium. Let us make the following change of variables in oder to apply this method. P Let S = x1 , ET1 = x2 , IT1 = x3 , RT = x4 , ET2 = x5 and IT2 = x6 , so that NT = 6n=1 xn . We now use the vector notation X = (x1 , x2 , x3 , x4 , x5 , x6 )T and then, the model system (3) can be written in the dX form = F = (f1 , f2 , f3 , f4 , f5 , f6 )T as shown in Appendix A. The Jacobian matrix of dt system (97) at ET0 in Appendix A is given by  −µ  0  0  J(ET0 ) =   0  0 0

0 −(k1 + r1 + µ) k1 r1 0 0

−β1 cT p1 β1 cT (1 − p1 )β1 cT − (µ + dT1 + r2 ) qr2 (1 − q)r2 0

0 0 0 −µ 0 0

0 0 0 0 −(µ + k2 ) k2

−β2 cT 0 0 0 p2 β2 cT (1 − p2 )β2 cT − (µ + dT2 )



   .  

(22)

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C.P. Bhunu and W. Garira

Figure 2. Simulation results showing the backward bifurcation for drug resistant TB cases against RT2 . Bold and dashed lines show stable and unstable states of the equilibrium points respectively. Numerical values used are obtained from Table 1. From (22) it can be shown that RT2 =

(1 − p2 )β2 cT µ + β2 cT k2 , (µ + k2 )(µ + dT2 ) (23)

RT1

(µ + r1 )(1 − p1 )β1 cT + β1 cT k1 = . (µ + dT1 + r2 )(µ + k1 + r1 )

If β2 is taken as a bifurcation point and if we solve RT2 = 1 for β2 we obtain, β2 = β∗ =

(µ + k2 )(µ + dT2 ) . µ(1 − p2 )cT + cT k2

(24)

The linearised system of the transformed equation (97) with β2 = β∗ , has a simple zero eigenvalue. Thus the Centre Manifold theory [11] can be applied in the analysis of the

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95

dynamics of (97) near β2 = β∗ . The Jacobian of (97) at β2 = β∗ has a right eigenvector associated with the zero eigenvalue given by u = [u1 , u2 , u3 , u4 , u5 , u6 ]T where u1 = −

β1 cT u3 + β∗ cT u6 , µ

u2 =

β1 cT p1 u3 ((µ + dT1 + r2 ) − (1 − p1 )β1 cT ) u3 ⇒ RT1 = 1, u3 = u3 > 0 = k1 k1 + r1 + µ

u4 =

(µ + dT2 − (1 − p2 )β∗ cT ) u6 ((qr2 (k1 + r1 + µ) + p1 β1 cT k1 ) u3 , u5 = , µ(k1 + r1 + µ) k2

u6 =

(1 − q)r2 k2 u3 >0 (k2 + µ)(µ + dT2 ) − β∗ cT k2 − β∗ cT µ(1 − p2 )

for (k2 + µ)(µ + dT2 ) > β∗ cT k2 + β∗ cT µ(1 − p2 ).

(25) The left eigenvector associated with the zero eigenvalue at β2 = β∗ is given by w = [w1 , w2 , w3 , w4 , w5 , w6 ]T where w1 = 0, w2 = w5 =

k1 w3 w3 , w3 = w3 > 0, w4 = 0, k1 + r1 + µ

((µ + k1 + r1 )(µ + dT1 + r2 ) − β1 cT (k1 + r1 + µ − p1 (r1 + µ))) w3 >0 (k1 + µ + r1 )(1 − q)r2

for 1 > RT1 , w6 =

µ + k2 w5 . k2

(26)

Further we use Theorem 3 proven by Castillo-Chavez and Song [15]. Theorem 3. Consider the following general system of ordinary differential equations with a parameter φ dx = f (x, φ), f : Rn × R → R and f ∈ C2 (Rn × R), dt

(27)

where 0 is an equilibrium of the system that is f (0, φ) = 0 for all φ and assume   ∂fi A1: A = Dx f (0, 0) = (0, 0) is the linearisation of system (27) around the ∂xj equilibrium 0 with φ evaluated at 0. Zero is a simple eigenvalue of A and other eigenvalues of A have negative real parts; A2: Matrix A has a right eigenvector u and a left eigenvector v corresponding to the zero eigenvalue.

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C.P. Bhunu and W. Garira

Let fk be the kth component of f and

a=

n X

vk u i u j

k,i,j=1

b=

n X

k,i=1

∂ 2 fk (0, 0), ∂xi ∂xj (28)

∂ 2 fk (0, 0). vk u i ∂xi ∂φ

The local dynamics of (27) around 0 are totally governed by a and b. i. a > 0, b > 0. When φ < 0 with |φ| << 1, 0 is locally asymptotically stable, and there exists a positive unstable equilibrium; when 0 < φ << 1, 0 is unstable and there exists a negative and locally asymptotically stable equilibrium; ii. a < 0, b < 0. When φ < 0 with |φ| << 1, 0 unstable; when 0 < φ << 1, 0 is locally asymptotically stable, and there exists a positive unstable equilibrium; iii. a > 0, b < 0. When φ < 0 with |φ| << 1, 0 is unstable, and there exists a locally asymptotically stable negative equilibrium; when 0 < φ << 1, 0 is stable, and a positive unstable equilibrium appears; iv. a < 0, b > 0. When φ changes from negative to positive, 0 changes its stability from stable to unstable. Correspondingly a negative unstable equilibrium becomes positive and locally asymptotically stable. For the computations of a and b see Appendix B. Using Theorem 3 item (i) and (iv), we establish the following result. Theorem 4. If (µ + k2 )(µ + dT2 ) < β∗ cT k2 + β∗ cT µ(1 − p2 ), RT1 < 1 and there is a ϕi < 0 such that a < 0 then model system (97) has a unique endemic equilibrium ET∗1 which is locally stable for RT2 > 1 close to 1. If there is a ϕi such that a > 0 then the direction of the bifurcation at RT2 = 1 is backward.

2.2.2.

Drug Sensitive TB-only Equilibrium

If q = 1 and ET2 = IT2 = 0 then we have TB drug sensitive only model and in the next section we analyse the corresponding endemic equilibrium. This equilibrium in terms of

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the force of infection λ∗T1 is given by ET∗2 = (S ∗∗ , ET∗1 , IT∗1 , RT∗ , 0, 0) where S ∗∗ =

Λ , µ + λ∗T1

ET∗1 =  IT∗1 =  RT∗ =

z1 z2 (µ + λ∗T1 ) − z2 (p1 r1 λ∗T1

z1 z2 (µ +

λ∗T1 )

Λp1 z2 λ∗T1 , − δ1 λ∗T1 (µ + λT1 ∗)) − z1 λ∗T1 r2 (1 − p1 ) − r2 λ∗T1 (k1 p1 + δ1 λ∗T1 )

Λp1 z1 λ∗T1 (1 − p1 ) + Λλ∗T1 (k1 p1 + δ1 λ∗T1 )  − z2 (p1 r1 λ∗T1 − δ1 λ∗T1 (µ + λT1 ∗)) − z1 λ∗T1 r2 (1 − p1 ) − r2 λ∗T1 (k1 p1 + δ1 λ∗T1 )


Λλ∗T1 z2 p1 + z1 r2 (1 − p1 ) + r2 (k1 p1 + δ1 λ∗T1 )  (µ + λ∗T1 ) z1 z2 (µ + λ∗T1 ) − z2 (p1 r1 λ∗T1 − δ1 λ∗T1 (µ + λT1 ∗)) − z1 λ∗T1 r2 (1 − p1 ) − r2 λ∗T1 (k1 p1 + δ1 λ∗T1 ) (29) 

where z1 = µ + k1 + r1 , z2 = µ + dT1 + r2 . Substituting equation (29) into equation for the equilibrium value of the force of infection λ∗T1 in equation (15) we have ∗ λ∗T1 g(λ∗T1 ) = λ∗T1 (A2 λ∗2 T1 + B2 λT1 + C2 ) = 0,

(30)

where λ∗T1 = 0 corresponds to the disease free equilibrium and g(λ∗T1 ) = 0 corresponds to the existence of endemic equilibria where A2 = δ1 , B2 =, δ1 β1 cT − (µ + dT1 + r2 )(δ1 + p1 ) + (µ + r1 )(1 − p1 ) + k1

(31)

C2 = −(µ + k1 + r1 )(µ + dT1 + r2 )(1 − RT1 ). By examining the quadratic equation above we see that there is a unique endemic equilibrium if B2 < 0 and C2 = 0 or B22 − 4A2 C2 = 0, there are two if C2 > 0, B2 < 0 and B22 − 4A2 C2 > 0, and there is none otherwise. The coefficient A2 is always positive and C2 is positive or negative if RT1 is greater than or less than one respectively. We therefore rewrite these conditions in Lemma 3. Lemma 3. Model system (3) has (i) precisely one unique endemic equilibrium if C2 < 0 ⇔ RT1 < 1, (ii) precisely one unique endemic equilibrium if B2 < 0 and C2 = 0 or B22 − 4A2 C2 = 0, (iii) precisely two endemic equilibria if C2 > 0, B2 < 0 and B22 − 4A2 C2 > 0, (iv) none therwise. For this endemic equilibrium we can show that it also exists for RT1 < 1. Lemma 4. The endemic equilibrium point ET∗2 exists for RT1 < 1. p B22 − 4A2 C2 Proof. Analysing the equation = 0 we get = from 2A2 which it is clear that the disease is endemic when λ∗T1 > 0 ⇒ B22 − 4A2 C2 > B22 ⇒ 4A2 (µ + k1 + r1 )(µ + dT1 + r2 )(1 − RT2 ) > 0 ⇒ RT1 < 1. Thus the endemic equilibrium point ET∗2 also exists for RT1 < 1. g(λ∗T1 )

λ∗T1

−B2 +

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Lemma 4 suggests that making the reproduction number less than unity will not result in a decrease of the disease as the disease can even spread when reproduction number is less than unity. The stability of the equilibrium point ET∗2 it can be shown using the Centre Manifold theory manifold theory similar to the analysis of ET∗1 in the previous section but, is not shown here to avoid repetition. 2.2.3.

Co-existence of the Two TB Strains Endemic Equilibrium

This in terms of the force(s) of infection is given by ET∗3 = (S ∗∗∗ , ET∗∗∗ , IT∗∗∗ , RT∗∗∗ , ET∗∗∗ , IT∗∗∗ ) 1 1 2 2

(32)

where, S ∗∗∗ =

ET∗∗∗ = 1

IT∗∗∗ = 1

RT∗∗∗ =

IT∗∗∗ = 2 where,

Λ µ+

λ∗∗ T1

+ λ∗∗ T2

θ1 + θ2 λ∗∗ T1

θ1 + θ2 λ∗∗ T1

,

Λp1 λ∗∗ T1 (µ + dT1 + r2 ) ∗∗2 ∗∗ ∗∗ ∗∗2 , + θ3 λT1 + θ4 λ∗∗ T2 + θ5 λT1 λT2 + θ6 λT2 ∗∗2 ∗∗ ∗∗ y1 λ∗∗ T1 + y2 λT1 + y3 λT1 λT2 ∗∗ ∗∗ ∗∗ ∗∗2 + θ3 λ∗∗2 T1 + θ4 λT2 + θ5 λT1 λT2 + θ6 λT2

(33)

∗∗∗ + R∗∗∗ ) (1 − q)r2 IT∗∗1 + p2 λ∗∗ r1 ET∗∗∗ + qr2 IT∗∗∗ T2 (S T ∗∗∗ 1 1 , , E = P2 T2 δ2 λT∗∗2 + µ + k2 µ + j=1 λT∗∗j


1 ∗∗ ∗∗∗ , (1 − p2 )(S ∗∗∗ + RT∗∗∗ )λ∗∗ T2 + (k2 + δ2 λT2 )ET2 µ + dT2

θ1 = (µ + k1 + r1 )(µ + dT1 + r2 )µ, θ2 = (µ + k1 + r1 )(µ + dT1 + r2 − qr2 + p1 qr2 ) + (µ + dT1 + r2 )(δ1 µ − p1 r1 ) +k1 p1 qr2 , θ3 = δ1 (µ + dT1 + r2 − qr2 ) , θ4 = (µ + dT1 + r2 )(2µ + k1 + r1 ), θ5 = (µ + dT1 + r2 )(1 + δ1 − qr2 + p1 qr2 ), θ6 = µ + dT1 + r2 , y1 = Λ ((µ + dT1 + r2 )(1 − p1 ) + k1 p1 ) , y2 = Λδ1 , y3 = Λ(1 − p1 ),

(34) and is illustrated in Figure 3. Figure 3 shows that chemoprophylaxis and treatment of drug sensitive TB cases result in a significant decrease in the number of drug sensitive latent and active TB cases. A decrease in the number of drug sensitive TB on treatment also results in the decrease in the number of drug resistant TB cases as most cases of multi-drug resistant TB are a result of treatment failure and incomplete treatment. The recovered population starts by increasing following successful chemoprophylaxis and treatment, reaches its peak within the first twenty years and there after it shows a decrease due to a reduction in the number of drug sensitive TB cases. Stability analysis of ET∗3 can be done using the centre manifold theory similar to ET∗1 in Section 2.2.1, but is not shown here to avoid repetition.

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Figure 3. Simulation results showing the co-existence of drug sensitive TB and drug resistant TB in the presence of chemoprophylaxis and treatment of drug sensitive infectives. Numerical values used are obtained from Table 1.

We now illustrate some numerical simulations of the two strain TB model for different values of the basic reproduction number and varying initial conditions in Figure 4. The fourth order Runge-Kutta numerical scheme coded in C++ programming language and parameter values in Table 1 are used to carry out the numerical simulations. In Figure 4(a) we have RT1 > 1 > RT2 , then the drug sensitive TB strain will attain an endemic equilibrium with time while the drug resistant TB will disappear in the population. This is typical in cases where there is little or no treatment coverage of TB. This is in support of the argument that drug resistant TB is a result of incomplete treatment, implying that in the absence of treatment, there is little or no drug resistance. This is in agreement with Castillo-Chavez and Feng [12]. In Figure 4(b), both reproduction numbers are greater than unity (RT2 > RT1 > 1), and the two TB strains attain endemic states and co-exist in the population with time. In Figure 4(c), both reproduction numbers are less than

100

C.P. Bhunu and W. Garira

(a)

(b)

(c)

(d)

Figure 4. Simulations of model system (3) showing plots of individuals with drug sensitive TB infectives-only and drug resistant TB infectives-only as a function of time with various initial conditions. (a) RT1 > 1 > RT2 , (β2 = 0.1, β1 = 0.6, cT = 40), so that RT1 = 1.23, RT2 = 0.813 and RTT = 1.23. (b) RT2 > RT1 > 1, (β2 = 0.6, β1 = 0.6, cT = 40), so that RT1 = 1.23, RT2 = 4.884 and RTT = 4.884. (c) RT1 < RT2 < 1, (β2 = 0.35, β1 = 0.35, cT = 3), so that RT1 = 0.00891, RT2 = 0.213 and RTT = 0.213. (d) RT2 > 1 > RT1 , (β2 = 0.6, β1 = 0.35, cT = 10), so that RT1 = 0.0297, RT2 = 1.2209 and RTT = 1.2209. Parameter used are as in Table 1. unity (RT1 > RT2 < 1), and both TB strains will with time disappear in the population. This is typical in settings where there is effective monitoring and treatment of drug sensitive TB cases. Strong monitoring and effective treatment will imply no default and incomplete treatment of drug sensitive TB which is a major cause of drug resistance. When RT1 < 1 < RT2 , then drug sensitive TB will disappear in the population and the drug resistant strain TB will exist in the population as shown in Figure 4(d). This is likely to

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happen when the major causes of drug resistance is infection with the drug resistant strain other than inappropriate treatment regimen. Next we illustrate the effects of increasing treatment rates on the population with sensitive and resistant TB strains in Figure 5. Figure

Figure 5. Simulations of model showing plots of total number of individuals with TB. The direction of the arrow shows an increase in treatment rates from r2 = 0.4 with step size 0.1. Parameter values used are in Table 1. 5 shows that increasing treatment rates for drug sensitive TB results in decrease of total TB cases.

3.

A Two Strain HIV/AIDS Only Model

In this section we present and analyse a two strain HIV/AIDS model. The two strain HIV/AIDS model divides the population into the susceptibles S, HIV positive individuals with antiretroviral sensitive strain with no AIDS symptoms IH1 , AIDS individuals who have their health improved as result of using antiretroviral therapy AHt and this group has no AIDS-defining symptoms and do not die from AIDS, AIDS individuals with drug sensitive HIV showing AIDS symptoms AH1 , HIV positive individuals with antiretroviral resistant strain with no AIDS symptoms IH2 and AIDS individuals with antiretroviral resistant HIV showing AIDS symptoms AH2 . The population considered for this model is the sexually active population since HIV/AIDS is predominantly sexually transmitted in Sub-Saharan Africa. Individuals are recruited into the susceptible population at rate Λ and

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are infected with HIV/AIDS following sexual contact with an HIV infected individual at λAi , i = 1, 2 where i = 1 and i = 2, denotes drug sensitive and drug resistant strain respectively with

λ A1 =

βH2 cH (IH2 + ηA AH2 ) (t) βH1 cH (IH1 + ηA (AHt + AH1 )) (t) , λA2 = . NH (t) NH (t)

(35)

where βHi is the probability of getting an HIV following sexual contact with one infectious individual, cH is the number of sexual partners per unit time, ηA > 1 models the fact that HIV positive individuals in the AIDS stage have a higher viral load and therefore are more infectious than HIV positive individuals only and NH (t) is the total population size and is given by

NH (t) = S(t) + AHt (t) +

2 X

(IHi + AHi ) (t).

(36)

i=1

Susceptibles are infected with HIV at rate λAi , enters the HIV positive stage IHi . Individuals in IHi progress to the AIDS stage AHi at rate ρi . AIDS patients with antiretroviral sensitive HIV who are ill and displaying AIDS symptoms are treated at rate α, a proportion ν enters the the AIDS treated class AHt , and the complementary (1 − ν) develop drug resistance and move into AH2 . Treated AIDS move back to AH1 due to treatment failure at a rate θ. Individuals in each stage have a constant natural death rate µ. Individuals in the AIDS stage AHi , displaying AIDS symptoms have an additional disease-induced rate dAi . Parameters described in this section will assume values in Table 2 The model flow diagram is given in Figure 6.

Table 2. Model parameters and their interpretations. Parameter Recruitment rate Natural mortality rate Sexual partners per unit time Proportion of effectively treated Rate of progression to AIDS Modification parameter AIDS related death rate Probability of being infected with HIV Treatment rate for AIDS cases Drug failure rate

Symbol Λ µ cH ν (ρ1 , ρ2 ) η (dA1 , dA2 )

Value 0.029yr−1 0.02yr−1 3 yr−1 0.85 (0.028-0.19)−1 1.02yr−1 (0.333,0.4)yr−1

Source Mukandavire and Garira [33] Mukandavire and Garira [33] Mukandavire and Garira [33] Estimated Hyman et al. [27] Estimated Mukandavire and Garira [33]

(βH1 , βH2 ) α θ

0.011-0.95 0.33yr−1 0.15yr−1

Hyman et al. [27] Mukandavire and Garira [33] Mukandavire and Garira [33]

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Figure 6. Structure of model. Based on these assumptions we have the following system of equations to describe the the model. 2 X λAi S − µS, S ′ (t) = Λ − i=1

′ (t) = λ S − (ρ + µ)I , IH 1 A1 H1 1

A′Ht (t) = ναAH1 − (µ + θ)AHt ,

(37)

A′H1 (t) = ρ1 IH1 − (µ + dA1 + α)AH1 + θAHt , ′ (t) = λ S − (ρ + µ)I , IH 2 A2 H2 2

A′H2 (t) = ρ2 IH2 − (µ + dA2 )AH2 + (1 − ν)αAH1 . All parameters and state variables for model system (37) are assumed to be non-negative

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for t ≥ 0. Consider the region   Λ ΩE = (S, IH1 , AHt , AH1 , IH2 , AH2 ) : NH ≤ . µ It can be shown that all solutions of system (37) starting in ΩE remain in ΩE for all t ≥ 0. Thus, ΩE is positively invariant and it is sufficient to consider solutions in ΩE . Existence, uniqueness and continuation results for system (37) hold in this region.

3.1.

Disease-Free Equilibrium and Stability Analysis

Model system (37) has the diseases free equilibrium given by   Λ EH0 = (S, IH1 , AHt0 , AH10 , IH20 , AH20 ) = , 0, 0, 0, 0, 0 . µ

(38)

It can be shown that EH0 attracts the region ΩH0 = {(S, IH1 , AH1 , IH2 , AH2 ) ∈ ΩE : IH1 = AH1 = IH2 = AH2 = 0} . The basic reproduction number, RA is determined following [46] to obtain,    (ρ1 + µ)IH1 λ A1 S   0  (µ + θ)AHt − ναAH1     (µ + dA1 + α)AH1 − ρ1 IH1 − θAHt  0    F =  λA S  and V =  (ρ2 + µ)IH2 2     (µ + dA2 )AH2 − (1 − ν)αAH1 − ρ2 IH2  0  P2 0 i=1 λAi S + µS − Λ

(39)



   .   

(40)

The infected compartments are IH1 , AHt , AH1 , IH2 and AH2 . Thus the matrices F and V for the new infection terms and the remaining transfer terms are respectively given by,   βH1 cH βH1 cH ηA βH1 cH ηA 0 0   0 0 0 0 0    and, 0 0 0 0 0 F =    0 0 0 βH2 cH βH2 cH ηA  0 0 0 0 0 (41)   ρ1 + µ 0 0 0 0   0 µ+θ −να 0 0   .  −θ µ + dA1 + α 0 0 V =  −ρ1    0 0 0 ρ2 + µ 0 0 0 −(1 − ν)α −ρ2 µ + dA2 The dominant eigenvalues of F V −1 are RA1 =

αβH1 cH (θ + µ) + βH1 cH ηA ρ1 (θ + µ + να) + βH1 cH ((µ + θ)(µ + dA1 ) − θνα) , ((dA1 + α + µ)(θ + µ) − θνα)(µ + ρ1 )

RA2 =

βH2 cH (dA2 + µ) + βH2 cH ηA ρ2 . (dA2 + µ)(ρ2 + µ) (42)

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Thus the reproduction number for model system (37) is the spectral radius ρ(F V −1 ) = RA = max {RA1 , RA2 } ,

(43)

where RA1 and RA2 are reproduction numbers for antiretroviral sensitive HIV only and antiretroviral resistant HIV only respectively. From Theorem 2 in [46] the following holds. Theorem 5. The disease-free equilibrium point EH0 is locally asymptotically stable whenever RA < 1, and unstable otherwise. Following Castillo-Chavez [14] we write model system (37) as X ′ (t) = F (X, Y ) (44) Y

′ (t)

= G(X, Y ), G(X, 0) = 0

where X = (S) and Y = (IH1 , AHt , AH1 , IH2 , AH2 ) with X ∈ R+ (its components) denoting the number of unifected individuals and Y ∈ R5+ (its components) denoting the number of infected individuals. The disease free equilibrium is now denoted by EH0 =   Λ (X0 , 0) where X0 = . Conditions (H1) and (H2) in equation (45) below guarantee µ global asymptotic stability of EH0 if met. H1 : For X ′ (t) = F (X ∗ , 0), X ∗ is globally stable (45) b b H2 : G(X, Y ) = AY − G(X, Y ), G(X, Y ) ≥ 0 for (X, Y ) ∈ ΩE ,

where A = DY G(X ∗ , 0) is an M-matrix (the off diagonal elements of A are non-negative). If system (37) satisfies the conditions in (45)then Theorem 6 holds. Theorem 6. The fixed point EH0 is a globally asymptotically stable point of model system (37) provided RA < 1 and assumptions in (45) are satisfied. Proof. Consider F (X, 0) = [Λ − µS], 

  A=  

−(ρ1 + µ) + βH1 cH 0 ρ1 0 0 

   b and G(X, Y)=  

c1 (X, Y ) G c2 (X, Y ) G c3 (X, Y ) G c4 (X, Y )) G c5 (X, Y ) G

βH1 ηA cH −(µ + θ) θ 0 0

βH1 ηA cH 0 να 0 −(µ + dA1 + α) 0 0 −(ρ2 + µ) + βH2 cH (1 − ν)α ρ2

  S β c (I + η (A + A )) 1 − A Ht H1  H1 H H1 NH    0     0 =     S   β c (I + η A ) 1 − H2 H H2 A H2  NH 0 



0 0 0 βH2 ηA cH −(µ + dA2 ) 

    .    

(46)

b Therefore G(X, Y ) ≥ 0 for all (X, Y ) ∈ ΩA implying that EH0 is globally asymptotically stable for RA < 1.

     

106

3.2.

C.P. Bhunu and W. Garira

Endemic Equilibria

Model system (37) has three endemic equilibria, the antiretroviral resistant HIV-strain only, antiretroviral sensitive HIV-strain only and the coexistence of both strains equilibria. We start by presenting an analysis of the drug resistant only equilibrium. 3.2.1.

Anti-retroviral Resistant HIV-strain Only Equilibrium

This occurs when IH1 = AHt = AH1 = 0 and is given by ∗ ∗ EH = (S2∗ , 0, 0, 0, IH , A∗H2 ) where 1 2

S2∗ =

∗ NH

Λλ∗A2 ρ2 λ∗A2 Λ Λ ∗ ∗ I = A = , , , H2 µ + λ∗A2 H2 (µ + λ∗A2 )(ρ2 + µ) (µ + λ∗A2 )(ρ2 + µ)(µ + dA2 )


Λ (µ + ρ2 )(µ + dA2 ) + λ∗A2 (µ + dA2 + ρ2 ) , = (µ + λ∗A2 )(µ + ρ2 )(µ + dA2 )

(47) in terms of the force of infection λ∗A2 . Substituting equation (47) into the equation for the force of infection λ∗A2 we have, λ∗A2 h2 (λ∗A2 ) = λ∗A2 (C1 λ∗A2 + C2 ) = 0,

(48)

where λ∗A2 = 0 corresponds to the disease free equilibrium and h2 (λ∗A2 ) = 0 corresponds to the existence of endemic equilibrium point where C1 =

µ + d A2 + ρ 1 , C2 = 1 − RA2 . (µ + dA2 )(µ + ρ2 )

(49)

C1 is always positive and C2 is negative or positive if RA2 is greater than or less than one. ∗ exists whenever R Lemma 5. The endemic equilibrium, EH A2 > 1. 1

Proof. By examining the linear equation C1 λ∗A2 + C2 = 0 we have that λ∗A2 = −

RA2 − 1 C2 = . C1 C1

(50)

But the disease is endemic when the force of infection λ∗A2 > 0 which implies RA2 > 1. ∗ exists whenever R Therefore the endemic equilibrium EH A2 > 1. 1 We now employ the centre manifold theory [11] as described in [15] (Theorem 4.1), to establish the local asymptotic stability of the endemic equilibrium. Let us make the following change of variables in order to apply the Center Manifold theory S = x1 , IH1 = P6 x2 , AHt = x3 , AH1 = x4 , IH2 = x5 and AH2 = x6 , so that NH = n=1 xn . We now use the vector notation X = (x1 , x2 , x3 , x4 , x5 , x6 )T . Then model system (37) can be dX written in the form = F = (f1 , f2 , f3 , f4 , f5 , f6 )T , (see Appendix C). The Jacobian dt

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matrix of system (102) at EH0 in Appendix C is given by 

   J(EH0 ) =   

−µ 0 0 0 0 0

−βH1 cH βH1 cH − (ρ1 + µ) 0 ρ1 0 0

−βH1 ηA cH βH1 ηA cH −(µ + θ) θ 0 0

−βH1 ηA cH βH1 ηA cH να −(µ + dA1 + α) 0 (1 − ν)α

−βH2 cH 0 0 0 βH2 cH − (ρ2 + µ) ρ2

 −βH2 ηA cH 0   0   0  βH2 ηA cH  −(µ + dA2 ) (51)

From which it can be shown that the reproduction numbers are

RA1 =

αβH1 cH (θ + µ) + βH1 cH ηA ρ1 (θ + µ + να) + βH1 cH ((µ + θ)(µ + dA1 ) − θνα) , ((dA1 + α + µ)(ρ1 + µ) − θνα)(µ + ρ1 )

RA2 =

βH2 cH (dA2 + µ) + βH2 cH ηA ρ2 . (dA2 + µ)(ρ2 + µ)

(52) If βH2 is taken as a bifurcation point and if we consider RA2 = 1 and solve for βH2 we get

βH2 = β ∗2 =

(dA2 + µ)(ρ2 + µ) . cH (dA2 + µ + ηA ρ2 )

(53)

Note that the linearised system of the transformed equation (102) with βH2 = β ∗2 has a simple zero eigenvalue. Using the Centre Manifold theory [11] to analyze the dynamics of (102) near βH2 = β ∗2 , the Jacobian of (102) at βH2 = β ∗2 has a right eigenvector associated with the zero eigenvalue given by z = [z1 , z2 , z3 , z4 , z5 , z6 ]T where

z1 = − z2 =

βH1 cH z2 + βH1 ηA cH (z3 + z4 ) + β ∗2 cH z5 + β ∗2 ηA cH z6 , µ

θ (µ + dA1 + α)(µ + θ) βH1 cH ηA (µ + θ + να) z3 = − z3 + z3 ⇒ RA1 = 1, (ρ1 + µ − βH1 cH )να ρ1 ναρ1

z3 = z3 > 0, z4 =

µ+θ z3 , να

z5 =

β ∗2 ηA cH (1 − ν)(ν + θ) z3 , ν ((µ + dA2 )(ρ2 + µ − β ∗2 cH ) − ρ2 β ∗2 cH ηA )

z6 =

β ∗2 ηA cH (1 − ν)(ν + θ)(ρ2 + µ − β ∗2 cH ) z3 . ν ((µ + dA2 )(ρ2 + µ − β ∗2 cH ) − ρ2 β ∗2 cH ηA )

The left eigenvector of J(EH0 ) associated with the eigenvalue at βH2 =

β ∗2

(54) is given by

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C.P. Bhunu and W. Garira

y = [y1 , y2 , y3 , y4 , y5 , y6 ]T , where y1 = 0, y2 =

ρ1 (µ + θ) y3 , y3 = y3 > 0, ρ1 βH1 cH ηA + θ(ρ1 + µ − βH1 cH )

y4 =

(µ + θ)(µ + ρ1 − βH1 cH ) y3 , ρ1 βH1 cH ηA + θ(ρ1 + µ − βH1 cH )

y5 =

ρ2 (µ + dA2 ) y6 = y6 , β ∗2 ηA cH ρ2 + µ − β ∗2 cH

(ρ1 + µ − βH1 cH )[(µ + dA1 + α)(µ + θ) − ναθ] − βH1 ηA cH ρ1 (µ + θ + να) y3 . (1 − ν)(ρ1 βH1 ηA cH + θ(ρ1 + µ − βH1 cH )) (55) For the computations of a and b see Appendix D. Using Theorem 3 item (iv), we establish the following result. y6 =

∗ is locally asymptotically stable Theorem 7. If RA1 < 1, the endemic equilibrium point EH 1 for RA2 > 1 but close to 1.

3.2.2.

Antiretroviral Sensitive HIV-strain Only Equilibrium

This occurs when IH2 = AH2 = 0, ν = 1 and is given by ∗ ∗ EH = (S1∗ , IH , A∗Ht , A∗H1 , 0, 0) where 2 1

S1∗

Λλ∗A1 Λ ∗ , , I = = µ + λ∗A1 H1 (µ + λ∗A1 )(ρ1 + µ)

A∗Ht =

A∗H1

αρ1 Λλ∗A1 , (µ + λ∗A1 )(ρ1 + µ) [(µ + θ)(µ + dA1 ) + µα]

(56)

(µ + θ)ρ1 Λλ∗A1 = , (µ + λ∗A1 )(ρ1 + µ) [(µ + θ)(µ + dA1 ) + µα]

in terms of of the force of infection λ∗A1 . Substituting (56) into the equation for the force of infection λ∗A1 we have λ∗A1 h1 (λA1 ) = λ∗A1 (B1 λ∗A1 + B2 ) = 0,

(57)

where λ∗A1 = 0 corresponds to the disease free equilibrium and h1 (λ∗A1 ) = 0 corresponds to the existence of endemic equilibrium point where, B1 = (µ + θ)(µ + dA1 + ρ1 ) + α(µ + ρ1 ), B2 = (ρ1 + µ)[(µ + θ)(µ + dA1 ) + αµ](1 − RA1 ). (58) B1 is always positive and B2 is negative or positive if RA1 is greater than or less than one. ∗ exists whenever R Lemma 6. The endemic equilibrium EH A1 > 1. 2

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Proof. By examining the linear equation B1 λ∗A1 + B2 = 0 we have that λ∗A1 = −

B2 (ρ1 + µ)[(µ + θ)(µ + dA1 ) + αµ](RA1 − 1) = . B1 B1

(59)

But the disease is endemic when the force of infection λ∗A1 > 0 ⇒ RA1 > 1. Therefore the ∗ exists whenever R endemic equilibrium EH A1 > 1. 2 ∗ we use the Centre Manifold theory similar to the of E ∗ in the For stability analysis of EH H1 2 previous section.

3.2.3.

Co-existence of Both HIV Strains Endemic Equilibrium

When both HIV strains co-exist the endemic equilibrium in terms of the force(s) of infection ∗ = (S ∗∗ , I ∗∗ , A∗∗ , A∗∗ , I ∗∗ , A∗∗ ) where is given by EH 2 H1 Ht H1 H2 H2 3 S2∗∗ =

∗∗ IH = 1

A∗∗ Ht = A∗∗ H1 = ∗∗ IH = 2

A∗∗ H2

Λ µ+

∗∗ λA 1

(µ +

+ λ∗∗ A2

,

Λλ∗∗ A1 , + λ∗∗ A2 )(ρ1 + µ)

λ∗∗ A1

(µ + λ∗∗ A1

Λνρ1 αλ∗∗ A1 , + λ∗∗ A2 )(ρ1 + µ) ((µ + dA1 + α)(µ + θ) − νθα)

(µ + λ∗∗ A1

Λρ1 λ∗∗ A1 , + λ∗∗ )(ρ + µ) ((µ + dA1 + α)(µ + θ) − νθα) 1 A2

(60)

λ∗∗ A2 Λ ∗∗ , (µ + ρ2 )(µ + λ∗∗ A1 + λ A2 ) ∗∗ ρ Λ λA 2 2 ∗∗ + λ∗∗ ) (µ + ρ2 )(µ + λA A2 1

1 = µ + dA2



+

(1 − ν)αΛρ1 λ∗∗ A1 + λ∗∗ )(ρ + µ) ((µ + d + α)(µ + θ) − νθα) 1 A 1 A2

(µ + λ∗∗ A1



,

in terms of the equilibrium value of the forces of infection λ∗∗ Ai . ∗ exists whenever R > 1. Lemma 7. The endemic equilibrium EH A 3 ∗ . Since both strains Proof. We have to prove that RA1 > 1 and RA2 > 1 at the point EH 3 exist (S, IH1 , AHt , AH1 , IH2 , AH2 ) > 0. In terms of IH2 , AH1 we have

AH2 =

ρ2 IH2 (1 − ν)αAH2 ρ2 IH2 + , then AH2 ≥ . µ + d A2 µ + d A2 µ + d A2

(61)

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C.P. Bhunu and W. Garira

′ (t) in (37) at E ∗ , we have Substituting equation (61) into the equation for IH H3 2 ∗∗ + η A∗∗ )S ∗∗ βH2 cH (IH A H2 3 ∗∗ 2 − (ρ2 + µ)IH = 0, ∗∗ 2 NH ∗∗ + η A∗∗ ) − (ρ + µ)I ∗∗ > 0, but βH2 cH (IH 2 A H2 H2 2

βH2 cH (µ + dA2 + ηA ρ2 implying that IH2 − (ρ2 + µ)IH2 > 0, µ + d A2 giving

(62)

βH2 cH (µ + dA2 + ηA ρ2 ) > 1, hence RA2 > 1. (ρ2 + µ)(µ + dA2 )

In terms of IH1 , AHt we have AH1 =

ρ1 IH1 + θAHt µ+θ AHt and AH1 = να µ + d A1 + α (63)

implying that IH1

(µ + dA1 + α)(µ + θ) − νθα = AHt . ρ1 να

′ in (37)at E ∗ we have, Substituting (63) into the equation for IH H3 1 ∗∗ ∗∗ βH1 cH (IH + ηA (A∗∗ Ht + AH1 )) 1

S3∗∗ ∗∗ ∗∗ − (ρ1 + µ)IH1 = 0, NH

∗∗ + η (A∗∗ + A∗∗ )) − (ρ + µ)I ∗∗ > 0, giving βH1 cH (IH 1 A H1 Ht H1 1

then, βH1 cH A∗∗ Ht >



(µ + dA1 + α)(µ + θ) − ναθ + ρ1 ηA (µ + θ + να) ρ1 να



(ρ1 + µ) ((µ + dA1 + α)(µ + θ) − ναθ) ∗∗ AHt , which gives ρ1 αν

βH1 cH ((µ + dA1 + α)(µ + θ) − να) + βH1 ηA cH ρ1 (µ + θ + να) > 1 ⇒ RA1 > 1. (ρ1 + µ) ((µ + dA1 + α)(µ + θ) − ναθ) (64) ∗ exists whenever R > 1, since R Thus EH > 1 and R > 1 ⇔ R > 1. A A1 A2 A 3 ∗ can also be shown using the centre manifold Stability analysis of the equilibrium point EH 3 ∗ theory similar to the analysis of EH1 in the previous section but is not shown here to avoid repetition.

3.3.

Numerical Simulations

We now illustrate some numerical simulations of the two strain HIV model for different values of the basic reproduction number and varying initial conditions in Figure 7. The fourth

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order Runge-Kutta numerical scheme coded in C++ programming language and parameter values in Table 2 are used in carrying out the numerical simulation.

(a)

(b)

(c)

(d)

Figure 7. Simulations of model (37) showing plots of individuals with sensitive HIV strain only and resistant HIV strain only as a function of time with various initial conditions. (a) RA2 > 1 > RA1 (βH1 = 0.012, βH2 = 0.6), so that RA1 = 0.125, RA2 = 6.448 and RA = 6.448. (b) RA1 > 1 > RA2 (βH1 = 0.6, βH2 = 0.012), so that RA1 = 0.625, R2 = 0.128 and RA = 0.625. (c) RA1 < RA2 < 1 (βH1 = 0.012, βH2 = 0.012), so that RA1 = 0.125, RA2 = 0.128 and RA = 0.128. (d) RA1 > RA2 > 1 (βH1 = 0.2, βH2 = 0.2), so that RA1 = 2.111, RA2 = 1.912 and RA = 2.111. Parameters values used are as in Table 2. When RA2 > 1 > RA1 , resistant HIV strain will exist in the population as shown in Figure 7 (a) and when RA1 > 1 > RA2 , sensitive HIV strain will exist in the population as shown in Figure 7 (b). From the simulations in Figure 7 (a) we notice that when the reproduction number of a strain is less than unity and the other greater than unity, then the later strain

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C.P. Bhunu and W. Garira

exists as shown in Figure 7 (a) and (b). If the reproduction numbers of both strains are less than unity ( RA1 < RA2 < 1) then both strains will disappear in the population with time as shown in Figure 7 (c). Simulations for the case where the both reproduction numbers are greater than unity (RA1 > RA2 > 1) show that the two strains can co-exist as shown in Figure 7 (d). Next, we illustrate the effects of increasing antiretroviral treatment rates on the population with sensitive and resistant HIV strains in Figure 8.

(a)

(b)

(c) Figure 8. Simulations of model (37) showing plots of individuals with sensitive HIV strain only, resistant HIV strain only and AIDS patients as a function of time with various treatment rates. (a) population with sensitive HIV strain. (b) population with resistant HIV strain. (c) AIDS patients. The direction of the arrow shows an increase in treatment rates starting from θ = 0.3 with step size 0.1. Parameters values used are as in Table 2. Simulations in Figure 8 illustrates the effects of increasing treatment with amelioration as a single-strategy approach in a community. Figures 8 (a) and (b) show that treatment alone as an intervention strategy results in an increase in the number of infective individuals

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(asymptomatic that is with sensitive and resistant HIV strains) and the arrow show the direction of increase of the infected classes with increase in treatment rates. This suggests that treatment with amelioration intended to lengthen the lives of AIDS patients results in more HIV infections and thus may not benefit the community. This is in agreement with Mukandavire and Garira [33] and Hsu Schmitz [25, 26] that treatment significantly prolongs the incubation period but does not reduce infectiousness. The only way in which such treatment can help the community is when the efficacy of treatment drugs intended for this practice is 100 % [33]. Figure 8 (c) shows treatment alone results in a decrease in number of AIDS patients and the arrow shows the direction of decrease of the AIDS classes with increase in treatment rates. Thus, of course treatment with amelioration as a single-strategy approach enlarges the epidemic but will help the community by reducing HIV/AIDS morbidity/mortality and hence reduces the number of orphans in affected communities. In the next section we give an analysis of the effects of treatment and drug resistance in TB individuals co-infected with HIV/AIDS.

4.

Effects of Treatment and Drug Resistance in TB Individuals Co-infected with HIV/AIDS

In this section, we combine our earlier models (model systems (3) and (37)) to take into account possible HIV/AIDS and TB co-infections. We add the following additional classes EHi Tj (HIV positive exposed to TB), EAi Tj (AIDS individuals exposed to TB), IHi Tj (HIV positive individuals with TB), EAt Tj (treated AIDS cases exposed to TB), AAt Tj (treated AIDS cases with TB), and AAi Tj (AIDS cases with TB) with i = 1 (denoting drug sensitive HIV strain), i = 2 (drug resistant HIV strain), j = 1 (denoting drug sensitive TB) and j = 2 (denoting drug resistant TB). It is assumed that susceptible humans are recruited into the population at per capita rate Λ. Susceptible individuals acquire HIV infection following sexual contact with HIV infected individuls at a rate λHi , λH1

  2 2 X X βH1 cH  IH1 + IH1 Tj  + EH1 Tj + = N j=1 j=1

  2 2 X X βH1 cH ηA AH1 + AHt + ηHT (θHT (EA1 Tj + EAt Tj )) , (AA1 Tj + AAt Tj ) + N j=1 j=1 λH2

   2 2 2 2 X X X X βH2 cH  = (IH2 + IH2 Tj ) + ηA AH2 + ηHT (θHT EA2 Tj ) . EH2 Tj + AA2 Tj + N j=1 j=1 j=1 j=1 (65)

In (65), βHi , cH , are as defined for equation (35), the modification parameter ηHT ≥ 1 accounts for the relative infectiousness of individuals with HIV and in the AIDS stage dually infected with TB (EAi Tj , EAt Tj , AAt Tj and AAi Tj ), in comparison to those solely infected with HIV (IHi ). Further, θHT ≥ 1 models the fact that dually infected people in the AIDS stage of the disease displaying symptoms of TB (AAt Tj , AAi Tj ) are more infectious than the corresponding individuals in the AIDS stage who are only exposed to TB,

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C.P. Bhunu and W. Garira

(EAt Tj , EAi Tj ). Finally the parameter ηA > 1 captures the fact that individuals who are in the AIDS stage of infection are more infectious than HIV-infected individuals with no AIDS symptoms. This is so because people in the AIDS stage have a higher viral load compared to other HIV infected individuals with no symptoms and there is a positive correlation between viral load and infectiousness. Susceptibles acquire TB infection following contact with an infectious individual at a rate λTj with, λTj =

β1 cT (IT1 + AAt Tj +

P2

i=1 (AAi Tj

+ IHi Tj ))

N

(66)

,

where βj and cT are as defined for equation (1). The total population size is given by, N (t) = S(t) +

2 X

ETj (t) +

j=1

+ +

2 X

i=1 2 X

AHi +

2 2 X X

2 X

2 X

IHi + AHt

i=1

j=1

EHi Tj

(67)

i=1 j=1

(EAt Tj + AAt Tj ) +

j=1

ITj (t) + RT (t) +

2 X 2 X i=1 j=1

EAi Tj +

2 X 2 X i=1 j=1

IHi Tj +

2 X 2 X

AAi Tj .

i=1 j=1

It is assumed that individuals suffering from drug sensitive TB may recover and enter the recovered class (RT ) at rates r1 and qr2 following chemoprophylaxis for the exposed with drug sensitive Mtb and treatment for the drug sensitive TB infectives. All individuals in different human subgroups suffer from natural death (at a rate µ). Susceptibles infected with Mtb enter the exposed class at rate pj λTj and then progress to active TB at rate kj and the complementary proportion (1 − pj ) develop fast TB. Individuals infected with TB can acquire HIV at rates λHi and δj λHi for the exposed and infectives respectively. Individuals with TB suffer disease-induced death at rates dT1 and dT2 for drug sensitive TB and drug resistant TB, respectively. Individuals infected with HIV only (with no symptoms) is generated following infection at rate λHi . This further reduces following progression to AIDS at rate ρi and through being infected with Mtb to enter the EHi Tj class. Individuals infected with HIV exposed to TB (EHi Tj ) develop (i) active TB at rates ψ2 , ψ4 , ζ2 and ζ4 and (ii) AIDS at rates ψ1 , ψ3 , ζ1 and ζ3 . Individuals in IHi Tj class die due to TB at rate dTj and progress to AIDS at rates nl , l = 1, 2, 3, 4. The population of individuals with AIDS alone is generated following progression to AIDS of people infected with HIV only at rate ρi . Individuals in the AIDS stage with the drug sensitive HIV are treated at a rate α and a proportion ν move into the treated AIDS and the complementary proportion (1 − ν) develop drug resistance and move to the AIDS class with drug resistance. Treated AIDS cases move back to the AIDS class due to waning of the vaccine at rate θ. The population of individuals with AIDS and displaying TB symptoms is generated following progression to AIDS of people dually infected with no AIDS symptoms displaying TB symptoms and/or progression to TB of people with AIDS exposed to TB at rates nl and ml , respectively. Individuals with AIDS die at a constant rate dAj . AIDS individuals displaying symptoms of TB die at an increased death rate υj dAj with υj ≥ 1. The assumptions result in the following differential equations that describe the interaction of HIV and TB.

A Theoretical Assessment of the Effects of Chemoprophylaxis...

2 2 X X dS λHi S − λTj S − µS, =Λ− dt j=1 i=1 2

X dET1 λHi ET1 , = p1 λT1 (S + RT ) − (k1 + r1 + µ)ET1 − (δ1 λT1 + λT2 )ET1 − dt i=1 2

X dIT1 = (1 − p1 )λT1 (S + RT ) + k1 ET1 + δ1 λT1 ET1 − (µ + dT1 + r2 )IT1 − λHi IT1 , dt i=1 2

2

X X dRT λHi RT − λTj RT , = r1 ET1 + qr2 IT1 − µRT − dt j=1 i=1 2

X dET2 λHi ET2 − (µ + k2 )ET2 , = (1 − q)r2 IT1 + p2 λT2 (S + RT ) + λT2 ET1 − δ2 λT2 ET2 − dt i=1 2

X dIT2 = k2 ET2 + δ2 λT2 ET2 + (1 − p2 )λT2 (S + RT ) − (µ + dT2 )IT2 − λHi IT2 , dt i=1 2

X dIH1 λTi IH1 + qr2 IH1 T1 + r1 EH1 T1 , = λH1 (S + RT ) − (ρ1 + µ)IH1 − dt i=1 2

X dAHt = ναAH1 − (µ + θ)AHt − λTj AHt + r1 EAt T1 dt j=1 2

X dAH1 σj λTj AH1 − (µ + dA1 + α)AH1 + θAHt + r1 EA1 T1 + qr2 AA1 T1 , = ρ1 IH1 − dt j=1 2

X dIH2 = λH2 (S + RT ) − (ρ2 + µ)IH2 − λTj IH2 + qr2 IH2 T1 + r1 EH2 T1 , dt j=1 2

X dAH2 δj λTj AH2 − (µ + dA2 )AH2 + (1 − ν)αAH1 + r1 EA2 T1 + qr2 AA2 T1 , = ρ2 IH2 − dt j=1 dEH1 T1 = p1 λT1 IH1 + λH1 ET1 − (µ + r1 + ψ1 + ψ2 )EH1 T1 , dt dEH1 T2 = p2 λT2 IH1 + λH1 ET2 − (µ + ψ3 + ψ4 )EH1 T2 + (1 − q)r2 IH1 T1 , dt dEH2 T1 = p1 λT1 IH2 + λH2 ET1 − (µ + ζ1 + ζ2 + r1 )EH2 T1 , dt dEH2 T2 = p2 λT2 IH2 + λH2 ET2 − (µ + ζ3 + ζ4 )EH2 T2 + (1 − q)r2 IH2 T1 , dt dEA1 T1 = ψ1 EH1 T1 + p1 σ1 λT1 AH1 − (µ + dA1 + r1 + α + m1 )EA1 T1 + θEAt T1 , dt dEA1 T2 = ψ3 EH1 T2 + p2 σ2 λT2 AH1 − (µ + dA1 + α + m2 )EA1 T2 + (1 − q)r2 AA1 T1 + θEAt T2 , dt dEA2 T1 = ζ1 EH2 T1 + p1 δ1 λT1 AH2 − (µ + dA2 + r1 + m3 )EA2 T1 + (1 − ν)αEA1 T1 , dt dEA2 T2 = ζ3 EH2 T2 + p2 δ2 λT2 AH2 − (µ + dA2 + m4 )EA2 T2 + (1 − q)r2 AA2 T1 + (1 − ν)αEA1 T2 , dt

115

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C.P. Bhunu and W. Garira

dEAt T1 dt

= p1 λT1 AHt + ναEH1 T1 − (µ + θ + r1 + γ1 )EAt T1 ,

dEAt T2 = p2 λT2 AHt + ναEH1 T2 − (µ + θ + γ2 )EAt T2 , dt AA t T 1 = (1 − p1 )λT1 AHt + γ1 EAt T1 − (µ + dT1 + r2 + θ)AAt T1 + ναAA1 T1 , dt AA t T 2 = (1 − p2 )λT2 AHt + γ2 EAt T2 − (µ + dT2 + θ)AHt T2 + ναAA1 T2 , dt dIH1 T1 = (1 − p1 )λT1 IH1 + ψ2 EH1 T1 + λH1 IT1 − (µ + dT1 + n1 + r2 )IH1 T1 , dt dIH1 T2 = (1 − p2 )λT2 IH1 + ψ4 EH1 T2 + λH1 IT2 − (µ + dT2 + n2 )IH1 T2 , dt dIH2 T1 = (1 − p1 )λT1 IH2 + ζ2 EH2 T1 + λH2 IT1 − (µ + dT1 + r2 + n3 )IH2 T1 , dt dIH2 T2 = (1 − p2 )λT2 IH2 + ζ4 EH2 T2 + λH2 IT2 − (µ + dT2 + n4 )IH2 T2 , dt dAA1 T1 = (1 − p1 )σ1 λT1 AH1 + θAAt T1 + m1 EA1 T1 + n1 IH1 T1 − (µ + υ1 dA1 + dT1 + r2 + α)AA1 T1 , dt dAA1 T2 = (1 − p2 )σ2 λT2 AH1 + m2 EA1 T2 + n2 IH1 T2 − (µ + υ1 dA1 + dT2 + α)AA1 T2 , dt dAA2 T1 = (1 − p1 )δ1 λT1 AH2 + m3 EA2 T1 + n3 IH2 T1 − (µ + υ2 dA2 + dT1 + r2 )AA2 T1 + (1 − ν)αAA1 T1 , dt dAA2 T2 = (1 − p2 )δ2 λT1 AH2 + m4 EA2 T2 + n4 IH2 T2 − (µ + υ2 dA2 + dT2 )AA2 T2 + (1 − ν)αAA1 T2 . dt

(68)

Based on biological considerations model system (68) will be studied in the following region, D=



Λ µ (69)

(S, ETj , ITj , RT , IHi , AHt , AHi , EHi Tj , EAi Tj , EAt Tj , IHi Tj , AAt Tj , AAi Tj ) ∈ R31 + : N (t) ≤

It can be shown that all solutions of model system (68) starting D remain D for all t ≥ 0. Thus, D is positively invariant and it is sufficient to consider solutions in D. Existence, uniqueness and continuation results for system (68) hold in this region.

4.1.

Disease-Free Equilibrium and Stability Analysis

Model system (68) has a disease-free equilibrium given by 

 Λ G0 = , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 . µ (70) It can be shown that G0 attracts the region  D0 = (S, ETj , ITo j , RT , IHi , AHi , AHt , EHi Tj , EAi Tj , EAt Tj , IHi Tj , AAt Tj , AAi Tj ) ∈D:S=

Λ µ

,

(71)



.

A Theoretical Assessment of the Effects of Chemoprophylaxis...

117

and the rest are zeros. The linear stability of G0 is governed by the reproduction number RAT where RAT = max {RA , RTT }

(72)

with RA and RTT as defined in equations (43) and (15) respectively. We now state Theorem 8 whose proof follows from [46] (Theorem 2). Theorem 8. The disease free equilibrium G0 , is locally asymptotically stable for RAT < 1 and unstable otherwise.

4.2.

Endemic Equilibria

For model system (68) there are fifteen possible endemic equilibria states that is (i) drug sensitive TB only, (ii) drug resistant TB only, (iii) coexistence of drug sensitive and resistant TB only, (iv) anti-retroviral sensitive HIV only, (v) anti-retroviral resistant HIV only, (vi) coexistence of anti-retroviral sensitive and resistant HIV only, (vii) coexistence of drug sensive TB and anti-retroviral sensitive HIV, (viii) coexistence of drug sensive TB and anti-retroviral drug HIV, (ix) coexistence of drug sensive TB, anti-retroviral sensitive HIV and anti-retroviral resistant HIV, (x) coexistence of drug resistant TB and anti-retroviral sensitive HIV, (xi) coexistence of drug resistant TB and anti-retroviral resistant HIV, (xii) coexistence of drug resistant TB, anti-retroviral sensitive HIV and anti-retroviral resistant HIV, (xiii) coexistence of drug sensitive TB, drug resistant TB, and anti-retroviral sensitive HIV, (xiv) coexistence of drug sensitive TB, drug resistant TB, and anti-retroviral resistant HIV, (xv) coexistence of drug sensitive TB, drug resistant TB, anti-retroviral sensitive HIV, and anti-troviral resistant HIV endemic equilibria. (i) TB drug sensitive only equilibrium is given by G1 with G1 = (ST∗1 , ET∗1 , IT∗1 , RT∗ , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0). (73) The equilibrium point G1 is simply the equilibrium ET∗2 with 0 added to all compartments of drug resistant TB forms, HIV-infected and doubly-infected individuals. (ii) TB drug resistant only is given by G2 with G2 = (ST∗2 , 0, 0, 0, ET∗2 , IT∗2 , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0). (74) The equilibrium point G2 is simply the equilibrium ET∗1 with 0 added to all compartments of drug sensitive TB forms, HIV-infected and doubly-infected individuals. (iii) Coexistence of drug sensitive and resistant TB only is given by G3 with,

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C.P. Bhunu and W. Garira

G3 = (S ∗∗∗ , ET∗∗∗ , IT∗∗∗ , RT∗∗∗ , ET∗∗∗ , IT∗∗∗ , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 1 2 2 0, 0, 0, 0, 0, 0, 0). (75) The equilibrium point G3 is simply the equilibrium ET∗3 with 0 added to all compartments of HIV-infected, and those dually infected with HIV and TB. (iii) Anti-retroviral resistant HIV-strain only equilibrium is given by G4 which is ∗ with 0 added to all compartments of TB-infected, those simply the equilibrium EH 1 infected with anti-retroviral sensitive HIV-strain and those doubly-infected and is given by ∗ , 0, 0, 0, 0, 0, 0, 0, 0, 0, I ∗ , A∗ , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0). G4 = (SH H2 H2 2 (76) (iv) Anti-retroviral sensitive HIV-strain only equilibrium is given by G5 which is simply the ∗ with 0 added to all compartments of TB-infected and those doubly-infected equilibrium EH 2 and is given by ∗ ∗ , 0, 0, 0, 0, 0, 0, IH , A∗Ht , A∗H1 , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0). G5 = (SH 1 1 (77)

(iv) Coexistence of anti-retroviral sensitive and resistant HIV only is given by G6 which is ∗ with 0 added to the all compartments of TB-infected and those simply the equilibrium EH 3 dually-infected with HIV and TB and is given as ∗∗ ∗∗ ∗∗ ∗∗ G6 = (S ∗∗ , 0, 0, 0, 0, 0, 0, IH , A∗∗ Ht , AH1 , IH2 , AH1 , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0). 1 (78)

Instead of writing all the other remaining endemic states let us just consider the coexistence of antiretroviral sensitive HIV, antiretroviral resistant HIV, drug sensitive and resistant TB endemic equilibrium which is quite involving to express explicitly in terms of the force(s) of infection, so we just generalise this equilibrium point to be G15 = (S, ETj , ITj , RT , IHi , AHi , AHt , EHi Tj , EAi Tj , EAt Tj , IHi Tj , AAt Tj , AAi Tj ), j = 1, 2 and i = 1, 2 such that S, ETj , ITj , RT , IHi , AHi , AHt , EHi Tj , EAi Tj , EAt Tj , IHi Tj , AAt Tj , AAi Tj > 0. (79)

We now state Theorem 9 for the local asymptotic stability G15 whose proof can be shown using the Centre Manifold theory done in the Sections 2 and 3, but is not shown here to avoid repetition. Theorem 9. The endemic equilibrium point G15 is locally asymptotically stable for RAT > 1.

4.3.

Analysis of the Reproduction Number RAT RAT = max{RA , RTT }.

From (80) we have the following scenarios,

(80)

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119

Case 1: No intervention for all TB and HIV/AIDS infections In the absence of any intervention strategy we have lim

(rj ,θ,α)→(0,0,0)

RAT = max



(1 − pj )βj cT µ + βj cT kj βHi cH (µ + dAi + ηA ρi ) , (µ + kj )(µ + dTj ) (µ + ρi )(µ + dAi )



= R0 ,

which is the pre-treatment reproduction number for model system (68) i = 1, 2; j = 1, 2. (81) Case 2: Only drug sensitive latent and active forms of TB cases are treated In this case α = θ = 0 thus n o lim RAT = lim max {RA1 , RA2 , RTT } = max RAN1 , RA2 , RT (α,θ)→(0,0)

(α,θ)→(0,0)

(82)

where RAN1

βH1 cH (µ + dA1 + ηA ρ1 ) = , (µ + dA1 )(µ + ρ1 )

is the reproduction number for drug sensitive HIV/AIDS in the absence of antiretroviral treatment. In this case the HIV/AIDS epidemic is allowed to grow. Rewrite RA1 as RA1 = H1 RAN1 with H1 ∈ (0, 1) where H1 =

(µ + dA1 ) ((µ + dA1 + α)(µ + θ) − θνα + ηA ρ1 (µ + θ + να)) <1 (µ + dA1 + ηA ρ1 ) ((µ + dA1 + α)(µ + θ) − θνα)

(83)

for 0 < (µ + θ)(1 − ν) − νdA1 . Thus if H1 < 1, then antiretroviral therapy slows the spread of AIDS individuals if adopted in a community. If RAN1 < 1, AIDS cannot develop into an epidemic and antiretroviral therapy may not be necessary and for RAN1 > 1, we want to determine conditions necessary for slowing down the AIDS epidemic. Following Hsu Schmitz [25, 26] we have ∆1 = RAN1 (1 − H1 ) for which ∆1 > 0 is expected if antiretroviral therapy is to reduce the epidemic. Differentiating RA1 partially with respect to α we have, (µ + dA1 )(µ + θ) ((µ + θ)(ν − 1) + νdA1 ) ∂RA1 = . ∂α (µ + dA1 + ηA ρ1 ) ((µ + dA1 + α)(µ + θ) − θνα)2

(84)

∂RA1 < 0 and these ∂α = 1 and solving for the critical

The conditions for slowing the HIV/AIDS epidemic are ∆1 > 0 and occur when 0 < (µ + θ)(1 − ν) − νdA1 . Setting RA1 antiretroviral rate we have, αc =

(µ + dA1 + ηA ρ1 )(µ + θ)(µ + dA1 )(RAN1 − 1)

(µ + θ(1 − ν))(µ + dA1 )(1 − RAN1 ) + ηA ρ1 (µ + θ(1 − ν) − (µ + dA1 )νRAN1 ). (85) Thus antiretroviral therapy will have a positive impact on HIV/AIDS individuals with TB for α > αc . When antiretroviral therapy is administered in a population with both strains of HIV, the long term outcome depends on the reproduction numbers of both strains. Solving

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RA1 = 1, we have, RA2 (µ + dA1 + ηA ρ1 )(µ + dA1 )(µ + θ)(RAN1 − RA2 )

∗

α c1 =

(µ + dA1 + ηA ρ1 )(µ + θ(1 − ν))RA2 − (µ + dA1 )(µ + θ(1 − ν) + ηρ1 ν)RAN1

which exists if

RAN1 RA2

>

,

(µ + dA1 + ηA ρ1 )(µ + θ(1 − ν)) ≥ 1. (µ + dA1 )(µ + θ(1 − ν) + ηρ1 ν)

(86) ∗ If the antiretroviral rate α < αc1 , then all HIV/AIDS cases remain sensitive to antiretrovirals provided there is no primary transmission of anti-retroviral resistant HIV strain to ∗ the susceptibles. If α > αc1 some AIDS cases will become antiretroviral resistant in the long run. If antiretroviral therapy is implemented in a population with (a) drug sensitive TB without any intervention, (b) drug sensitive TB with chemoprophylaxis and treatment, (c) drug resistant TB, we obtain the following critical antiretroviral therapy rates (µ + dA1 + ηA ρ1 )(µ + dA1 )(µ + θ)(RAN1 − RTN1 )

∗

(a) αc2 =

(µ + dA1 + ηA ρ1 )(µ + θ(1 − ν))RTN1 − (µ + dA1 )(µ + θ(1 − ν) + ηρ1 ν)RAN1

which exists if

RAN1 RTN1

(µ + dA1 + ηA ρ1 )(µ + θ(1 − ν))RT1 − (µ + dA1 )(µ + θ(1 − ν) + ηρ1 ν)RAN1

which exists if

RAN1 RT1

>

(µ + dA1 + ηA ρ1 )(µ + θ(1 − ν))RT2 − (µ + dA1 )(µ + θ(1 − ν) + ηρ1 ν)RAN1

which exists if

RAN1 RT2

>

,

(µ + dA1 + ηA ρ1 )(µ + θ(1 − ν)) ≥ 1, (µ + dA1 )(µ + θ(1 − ν) + ηρ1 ν)

(µ + dA1 + ηA ρ1 )(µ + dA1 )(µ + θ)(RAN1 − RT2 )

∗

(c) αc4 =

(µ + dA1 + ηA ρ1 )(µ + θ(1 − ν)) ≥ 1, (µ + dA1 )(µ + θ(1 − ν) + ηρ1 ν)

(µ + dA1 + ηA ρ1 )(µ + dA1 )(µ + θ)(RAN1 − RT1 )

∗

(b) αc3 =

>

(µ + dA1 + ηA ρ1 )(µ + θ(1 − ν)) ≥ 1. (µ + dA1 )(µ + θ(1 − ν) + ηρ1 ν)

(87)

Case 3: Only drug sensitive AIDS cases are treated In this case rj = 0, (j = 1, 2) thus n o lim RAT = lim max {RA , RT1 , RT2 } = max RA , RTN1 , RT2 ,

rj →0

rj →0

where RTN1

(1 − p1 )β1 cT µ + β1 cT k1 , = (µ + dT1 )(µ + k1 )

(88)

is the reproduction number for the drug sensitive only TB model in the absence of any intervention. In this case the TB epidemic is allowed to grow unchecked. Rewrite RT1 as

,

,

A Theoretical Assessment of the Effects of Chemoprophylaxis...

121

RT1 = H2 RTN1 with H2 ∈ (0, 1) where H2 =

((1 − p1 )(µ + r1 ) + k1 ) (µ + dT1 )(µ + k1 ) < 1, (µ + dT1 + r2 )(µ + k1 + r1 ) ((1 − p1 )µ + k1 )

(89)

is a factor by which chemoprophylaxis of latent and treatment of active forms of drug sensitive TB reduce the number of secondary TB cases in a community. If RTN1 < 1, then TB may not develop into an epidemic and intervention strategies may not be necessary and for RTN1 > 1 we need to determine conditions for slowing down the TB epidemic. We have that ∆2 = RTN1 (1 − H2 ) > 0 for chemoprophylaxis and treatment to reduce the epidemic and is satisfied for rj ∈ (0, 1), j = 1, 2. Differentiating RT1 partially with respect to r1 and r2 we have, k1 p1 (µ + dA1 )(µ + k1 )RTN1 ∂RT1 =− < 0, ∂r1 (µ + dA1 + r2 )((1 − p1 )µ + k1 )(µ + k1 + r1 )2

(90)

((1 − p1 )(µ + r1 ) + k1 )(µ + dA1 )(µ + k1 )RTN1 ∂RT1 =− <0 ∂r2 (µ + k1 + r1 )((1 − p1 )µ + k1 )(µ + dA1 + r2 )2 Conditions in equation (90) and ∆2 > 0 are necessary for slowing down the TB epidemic and these are satisfied for rj ∈ (0, 1). Setting RT1 = 1 when r2 = 0 that is when chemoprophylaxis is the only intervention strategy, we have the critical chemoprophylaxis rate as, ((1 − p1 )µ + k1 )(µ + k1 )(RTN1 − 1) , r1c = (1 − p1 )(1 − RTN1 )µ + k1 (1 − (1 − p1 )RTN1 ) (91) (1 − p1 )µ + k1 . which exists if 1 < RTN1 < (1 − p1 )(µ + k1 )

When r1 > r1c , chemoprophylaxis will be able to eradicate the TB epidemic in the community but for r1 < r1c chemoprophylaxis alone will reduce the TB epidemic but not eradicate it and some intervention strategies like treatment, of infectives may be necessary. If chemoprophylaxis is implemented in the presence of the drug resistant TB strain, the long term RT1 = 1 when outcome depends on the reproduction numbers of both strains. Solving for RT2 r2 = 0 for the critical treatment rate we obtain c∗

r11 =

((1 − p1 )µ + k1 )(µ + k1 )(RTN1 − RT2 )

(1 − p1 )(RT2 − RTN1 )µ + k1 (RT2 − (1 − p1 )RTN1 )

which exists for 1 < c∗

RTN1 RT2

<

, (92)

(1 − p1 )µ + k1 . (1 − p1 )(µ + k1 )

If the chemoprophylaxis rate r1 > r11 , then chemoprophylaxis will be able to eradicate TB in communities where drug sensitive and drug resistant TB strains coexist. If chemoprophylaxis is implemented in a population with (a) drug sensitive HIV strains without AIDS

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treatment, (b) drug sensitive HIV strains with AIDS treatment and (c) drug resistant HIV strains without AIDS treatment, the following critical chemoprophylaxis rates are obtained, ((1 − p1 )µ + k1 )(µ + k1 )(RTN1 − RAN1 )

c∗

(a) r12 =

(1 − p1 )(RAN1 − RTN1 )µ + k1 (RAN1 − (1 − p1 )RTN1 )

which exists if 1 <

<

(1 − p1 )µ + k1 , (1 − p1 )(µ + k1 )

(1 − p1 )(RA1 − RTN1 )µ + k1 (RA1 − (1 − p1 )RTN1 )

which exists if 1 <

RTN1 RA1

(1 − p1 )(RA1 − RTN1 )µ + k1 (RA2 − (1 − p1 )RTN1 )

which exists if 1 <

RTN1 RA2

<

, (93)

(1 − p1 )µ + k1 < , (1 − p1 )(µ + k1 )

((1 − p1 )µ + k1 )(µ + k1 )(RTN1 − RA2 )

c∗

(c) r14 =

RAN1

((1 − p1 )µ + k1 )(µ + k1 )(RTN1 − RA1 )

c∗

(b) (a) r13 =

RTN1

,

,

(1 − p1 )µ + k1 , (1 − p1 )(µ + k1 )

Setting RT1 = 1 when r1 = 0, that is when treatment of infectives is the only intervention strategy, we have the critical treatment rate as,   (94) r2c = (µ + dT1 ) RTN1 − 1 , and exists if RTN1 > 1.

When r2 > r2c , treatment of infectives will be able to eradicate the TB epidemic in the community, but for r2 < r2c , treatment of infectives alone will not be able to control the TB epidemic and some intervention strategies like chemoprophylaxis of the latently infected may be necessary. When TB treatment is implemented in a population in the presence of resistant TB, then the outcome depends on the reproduction numbers of the sensitive and RT1 = 1 when r1 = 0 for the critical drug resistant TB infections. Solving the equation RT2 c∗ treatment rate r21 , we obtain, c∗ r21

= (µ + dT1 ) c∗



RTN1 RT2

 − 1 and exists if RTN1 > RT2 .

(95)

If treatment rate r2 < r21 , all TB cases remain drug sensitive provided there is no primary c∗ transmission of drug resistant TB to the susceptibles and if r2 > r21 all TB cases will RT1 become drug resistant in the long run. The ratio is a decreasing function of r2 and RT2 this means that resistance spreads faster in areas with better access to drugs. If treatment of TB infectives is implemented in a population with (a) antiretroviral sensitive HIV without

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AIDS treatment, (b) antiretroviral sensitive HIV with AIDS treatment and (c) antiretroviral resistant HIV to obtain the following critical treatment rates ! RTN1 c∗2 − 1 which exists if RTN1 > RAN1 , (a) r2 = (µ + dT1 ) RAN1 (b)

c∗ r23

(c)

c∗ r24

= (µ + dT1 )



RTN1

 − 1 which exists if RTN1 > RA1 ,

= (µ + dT1 )



RTN1

 − 1 which exists if RTN1 > RA1 .

RA1

RA2

(96)

Figure 9 is graphical representation showing critical treatment rates in 5 different settings

Figure 9. Graphs of the critical treatment rates against RTN1 . Parameter values are obtained from Table 2. Series 1 to 5 denote critical treatment when (a) treatment of TB infectives is implemented in a population with TB resistance, (b) treatment of TB infectives is implemented in a population with antiretroviral sensitive HIV but without treatment, (c) TB treatment is implemented in a population with antiretroviral sensitive HIV/AIDS with treatment, (d) TB treatment is implemented in a population with antiretroviral resistant HIV/AIDS and (e) TB treatment is implemented in a population without HIV/AIDS and TB resistance, respectively. against the no intervention reproduction number RTN1 . This shows that c∗

c∗

c∗

c∗

r2c < r22 < r24 < r23 < r21 for all values of RTN1 > 1 suggesting that treatment of TB infectives is more effective in controlling TB in the absence of anti-TB drug resistance and c∗ c∗ c∗ HIV/AIDS. We note that graphs for r2c , r22 , r24 , r23 are close to one another suggesting that treatment is able to control TB almost equally in individuals co-infected with HIV and c∗ c∗ c∗ those with drug sensitive TB only. The fact that r22 < r24 < r23 suggests that antiretroviral

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therapy have a negative impact in individuals on TB treatment implying that treatment of TB infectives will do better in AIDS individuals who are not on antiretroviral therapy. This suggests that a different treatment regimen for those coinfected with HIV/AIDS if available c∗ will be more appropriate. The fact that r21 is very large shows that in the presence of drug resistant TB, it is difficult to manage TB using first line anti-TB drugs.

5.

Summary and Concluding Remarks

Mathematical models have been presented and studied to assess the the impact of chemoprophylaxis, treatment of drug sensitive TB infectives, drug resistance and antiretroviral therapy on TB cases in areas with high HIV/AIDS prevalence. A two strain TB model incorporating chemoprophylaxis and treatment of infectives for drug sensitive TB was first presented. This was followed by presentation and analysis of a two strain HIV/AIDS model and a full HIV/AIDS-TB co-infection model incorporating drug resistance of both infections. We computed and compared the reproduction numbers of the submodels to assess the effectiveness of chemoprophylaxis and treatment of infectives in the control of TB and antiretroviral therapy in controlling HIV/AIDS. Chemoprophylaxis to prevent progression from latent TB to active TB tuberculosis among those with HIV infection (EHi Tj ) will reduce the local burden of TB disease for several years. This is more pronounced when coverage levels of chemoprophylaxis are high. Effective HIV/AIDS control lessens the impact of HIV on TB epidemics. It is noted from the study that a decrease in number of drug sensitive TB infectives results in a decrease of drug resistant TB cases as most cases of TB drug resistance are due to improper anti-TB drug use. So, a decrease in the number of individuals using TB drugs results in creation of very low rates of drug resistance. Using the same argument, an increase in the number of TB infectives on TB treatment results in an increase of drug resistant TB. The obtained reproduction numbers RTN1 > RT1 and RAN1 > RA1 suggests that chemoprophylaxis and treatment of infectives is effective in controlling TB and that antiretroviral therapy is effective in improving the lives of HIV/AIDS individuc∗ c∗ c∗ c∗ als. The obtained critical treatment rates r2c < r22 < r24 < r23 < r21 for all values of RTN1 > 1 suggests that TB treatment using first line drugs is more effective in controlling TB in settings without or with low levels of drug resistant TB and HIV/AIDS. From c∗ c∗ c∗ the analysis of the critical treatment rates r2c , r22 , r24 , r23 , we conclude that treatment of TB infectives are equally effective in controlling TB in individuals with drug sensitive TB c∗ c∗ c∗ co-infected with HIV/AIDS. We also note from this study that r22 < r24 < r23 implying that treatment of drug sensitive forms of TB in individuals co-infected with HIV/AIDS on antiretrovirals is not as effective as those co-infected with HIV/AIDS and not on antiretrovirals, suggesting that antiretroviral therapy have a negative impact in individuals on TB treatment. However, the scenario is different when there is TB drug resistance as chemoprophylaxis and treatment of infectives fail to contain the epidemic. We note from the study that chemoprophylaxis and treatment of TB with first line drugs alone is not enough in the fight against HIV/AIDS-TB co-infection with resistance, but in the absence of other measures it is important to use the current regimen to control drug sensitive TB in individuals co-infected with HIV/AIDS. To achieve maximum benefit from HIV/AIDS-TB co-infection treatment, it is vital to have a constant reliable supply of drugs as erratic supply results in

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drug resistance of both infections which is quite difficult to manage. Combined HIV/AIDSTB treatment in individuals co-infected results in significant reduction of the TB pandemic and not the same for HIV/AIDS as AIDS treatment results in an increase in the number of AIDS individuals due to a reduction of AIDS related deaths. Thus the succesful control of TB in high-burden HIV settings may require policies that simultenously target the risk of progression when infected by Mtb and the risk of transmission when one has TB.

Acknowledgements This work was made possible through a fellowship received by C. P. Bhunu from the Forgarty International Centre through the International Clinical, Operational and Health Services and Training Award (ICOHRTA).

Appendix A Model system (3) can be written in the form

x′1 (t)

= f1 = Λ −

P2

j=1 βj cT x3j x1 P6 n=1 xn

− µx1 ,

p1 β1 cx3 δ1 β1 cT x3 β2 cT x6 x′2 (t) = f2 = P6 (x1 + x4 ) − (k1 + r1 + µ)x2 − P6 x2 − P6 x2 , n=1 xn n=1 xn n=1 xn x′3 (t) = f3 =

x′4 (t)

(1 − p1 )β1 cT x3 δ1 β1 cT x3 (x1 + x4 ) + k1 x2 + P6 x2 − (µ + dT1 + r2 )x3 , P6 n=1 xn n=1 xn

= f4 = r1 x2 + qr2 x3 − µx4 −

P2

j=1 βj cT x3j x4 , P6 n=1 xn

δ2 βj cT x6 β2 cT x6 p2 β2 cT x6 (x1 + x4 ) + P6 x2 − P6 x5 − (µ + k2 )x5 , x′5 (t) = f5 = (1 − q)r2 x3 + P6 x x x n n n n=1 n=1 n=1

(1 − p2 )β2 cT x6 δ2 β2 cT x6 x′6 (t) = f6 = k2 x5 + P6 x5 + (x1 + x4 ) − (µ + dT2 )x6 . P6 n=1 xn n=1 xn

(97)

Appendix B Computations of a and b: For system (97), the associated non-zero partial derivatives of F at the disease free equilib-

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rium are given by ∂ 2 f2 p1 β1 cT µ + δ1 β1 cT µ ∂ 2 f2 ∂ 2 f2 β∗ cT µ ∂ 2 f2 , , = =− = =− ∂x2 ∂x3 ∂x3 ∂x2 Λ ∂x6 ∂x2 ∂x2 ∂x6 Λ ∂ 2 f2 ∂ 2 f2 ∂ 2 f2 ∂ 2 f2 p1 β1 cT µ = = = =− , ∂x3 ∂x5 ∂x5 ∂x3 ∂x3 ∂x6 ∂x6 ∂x3 Λ 2(1 − p1 )β1 cT µ ∂ 2 f3 δ1 β1 cT µ − (1 − p1 )β1 cT µ ∂ 2 f3 ∂ 2 f3 =− = = , , 2 ∂x2 ∂x3 ∂x3 ∂x2 Λ Λ ∂x3 ∂ 2 f3 ∂ 2 f3 ∂ 2 f3 ∂ 2 f3 (1 − p1 )β1 cT µ = = = =− , ∂x3 ∂x5 ∂x5 ∂x3 ∂x3 ∂x6 ∂x6 ∂x3 Λ ∂ 2 f4 β1 cT µ ∂ 2 f4 ∂ 2 f4 β∗ cT µ ∂ 2 f4 = =− , = =− , ∂x3 ∂x4 ∂x4 ∂x3 Λ ∂x4 ∂x6 ∂x6 ∂x4 Λ ∂ 2 f5 ∂ 2 f5 β∗ cT µ − p2 β∗ cT µ ∂ 2 f5 ∂ 2 f5 p2 β∗ cT µ = = , = =− , ∂x2 ∂x6 ∂x6 ∂x2 Λ ∂x3 ∂x6 ∂x6 ∂x3 Λ 2p2 β∗ cT µ ∂ 2 f6 ∂ 2 f6 ∂ 2 f6 ∂ 2 f6 (1 − p2 )β∗ cT µ ∂ 2 f5 =− , = = = =− , 2 Λ ∂x2 ∂x6 ∂x6 ∂x2 ∂x3 ∂x6 ∂x6 ∂x3 Λ ∂x6 ∂ 2 f6 δ2 β∗ cT µ − (1 − p2 )β∗ cT µ ∂ 2 f6 ∂ 2 f6 2(1 − p2 )β∗ cT µ = = , =− 2 ∂x5 ∂x6 ∂x6 ∂x5 Λ Λ ∂x6

(98)

It follows from (98) that a = ϕ1 + ϕ2 + ϕ3 + ϕ4 , 



k2 + µ + dT2 − (1 − p2 )β∗ cT k2

ϕ1 = −

2w2 Λ

ϕ2 = −


2w3 u2 u3 β1 cT µ(1 − p1 − δ1 ) + u23 (1 − p1 )β1 cT µ + u3 u6 p2 β∗ cT µ , Λ

ϕ3 = − ϕ4 = −

u2 u3 β1 cT µ(p1 + δ1 ) + u2 u6 β∗ cT µ + u3 u6 p1 β1 cT µ


2w5 β∗ cT µ(u3 u6 p2 + u26 − u2 u6 (1 − p2 )) , Λ


2w6 β∗ cT µ (u2 u6 + u3 u6 )(1 − p2 ) + u26 (1 − p2 ) − u5 u6 (δ2 + (1 − p2 )) . Λ

(99)

Thus a < 0 for some ϕi < 0, i = 1, 2, 3, 4 otherwise a > 0. For the sign of b, it is associated with the following non-vanishing partial derivatives of F , ∂ 2 f1 ∂ 2 f5 ∂ 2 f6 = −cT , = p 2 cT , = (1 − p2 )cT . ∂x6 ∂β∗ ∂x6 ∂β∗ ∂x6 ∂β∗

(100)



,

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It follows from (100) that, b=

u 6 w5 c T (k2 + µ(1 − p2 )) > 0 for (k2 + µ)(µ + dT2 ) > β∗ cT k2 + k2

(101)

β∗ cT µ(1 − p2 ), 1 > RT1 .

Appendix C Model system (37) can be written in the form

x′1 (t) = f1 = Λ −

βH1 cH (x2 + ηA (x3 + x4 )) + βH2 cH (x5 + ηA x6 ) x1 − µx1 P6 n=1 xn

βH cH x′2 (t) = f2 = P6 1 (x2 + ηA (x3 + x4 ))x1 − (ρ1 + µ)x2 , n=1 xn x′3 (t) = f3 = ναx4 − (µ + θ)x3 ,

(102)

x′4 (t) = f4 = ρ1 x2 − (µ + dA1 + α)x4 + θx3 , βH cH x′5 (t) = f5 = P6 2 (x5 + ηA x6 )x1 − (ρ2 + µ)x5 , n=1 xn

x′6 (t) = f6 = ρ2 x5 − (µ + dA2 )x6 + (1 − ν)αx4 .

Appendix D Computations of a and b: For the sign of b, it is associated with the following non-vanishing partial derivatives of F , ∂ 2 f1 ∂ 2 f1 ∂ 2 f5 ∂ 2 f5 = −c , = −η c , = c , = ηA cH . H A H H ∂x5 ∂β ∗2 ∂x6 ∂β ∗2 ∂x4 ∂β ∗2 ∂x6 ∂β ∗2

(103)

From (103) it follows that b = y5 cH (z5 + z6 ηA ) > 0 for RA1 < 1.

(104)

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For the sign of a it is associated with the following non-vanishing partial derivatives of F 2βH1 cH µ ∂ 2 f2 ∂ 2 f2 βH cH (1 + ηA )µ ∂ 2 f2 = − , , = =− 1 2 Λ ∂x2 ∂x3 ∂x3 ∂x2 Λ ∂x2 ∂ 2 f2 ∂ 2 f2 βH cH (1 + ηA )µ ∂ 2 f2 ∂ 2 f2 βH cH µ = =− 1 , = =− 1 , ∂x2 ∂x4 ∂x4 ∂x2 Λ ∂x2 ∂x5 ∂x2 ∂x6 Λ ∂ 2 f2 2βH1 cH ηA µ ∂ 2 f2 ∂ 2 f2 βH cH ηA µ ∂ 2 f2 = = − , = =− 1 , 2 ∂x3 ∂x4 Λ ∂x3 ∂x5 ∂x3 ∂x6 Λ ∂x3 ∂ 2 f2 2βH1 cH ηA µ ∂ 2 f2 ∂ 2 f2 βH cH ηA µ = − , = =− 1 , 2 Λ ∂x4 ∂x5 ∂x4 ∂x6 Λ ∂x4 ∂ 2 f5 ∂ 2 f5 β ∗2 cH µ ∂ 2 f5 = = =− , ∂x5 ∂x2 ∂x5 ∂x3 ∂x5 ∂x4 Λ ∂ 2 f5 ∂ 2 f5 β ∗2 cH ηA µ ∂ 2 f5 = = =− ∂x6 ∂x2 ∂x6 ∂x3 ∂x6 ∂x4 Λ ∂ 2 f5 2β ∗2 cH µ ∂ 2 f5 2β ∗2 cH ηA µ β ∗2 cH (1 + ηA )µ ∂ 2 f5 = − = − =− , , . ∂x5 ∂x6 Λ Λ Λ ∂x25 ∂x26

(105)

From (105) it follows that
2µ (z2 + z3 + z4 + z5 + z6 ) y2 βH1 cH (z2 + ηA (z3 + z4 )) + y5 β ∗2 cH (z5 + ηA z6 ) . Λ (106) Thus a < 0 and b > 0. a=−

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In: Infectious Disease Modelling Research Progress ISBN 978-1-60741-347-9 c 2009 Nova Science Publishers, Inc. Editors: J.M. Tchuenche, C. Chiyaka, pp. 133-150

Chapter 4

W HEN Z OMBIES ATTACK !: M ATHEMATICAL M ODELLING OF AN O UTBREAK OF Z OMBIE I NFECTION Philip Munz1,∗, Ioan Hudea1,†, Joe Imad2,‡ and Robert J. Smith?3,§ 1 School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, ON K1S 5B6, Canada 2 Department of Mathematics, The University of Ottawa, 585 King Edward Ave, Ottawa ON K1N 6N5, Canada 3 Department of Mathematics and Faculty of Medicine, The University of Ottawa, 585 King Edward Ave, Ottawa ON K1N 6N5, Canada

Abstract Zombies are a popular figure in pop culture/entertainment and they are usually portrayed as being brought about through an outbreak or epidemic. Consequently, we model a zombie attack, using biological assumptions based on popular zombie movies. We introduce a basic model for zombie infection, determine equilibria and their stability, and illustrate the outcome with numerical solutions. We then refine the model to introduce a latent period of zombification, whereby humans are infected, but not infectious, before becoming undead. We then modify the model to include the effects of possible quarantine or a cure. Finally, we examine the impact of regular, impulsive reductions in the number of zombies and derive conditions under which eradication can occur. We show that only quick, aggressive attacks can stave off the doomsday scenario: the collapse of society as zombies overtake us all.

1.

Introduction

A zombie is a reanimated human corpse that feeds on living human flesh [1]. Stories about zombies originated in the Afro-Caribbean spiritual belief system of Vodou (anglicised ∗

E-mail address: E-mail address: ‡ E-mail address: § E-mail address: †

[email protected] [email protected] [email protected] [email protected]. Corresponding author.

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Philip Munz, Ioan Hudea, Joe Imad et al.

voodoo). These stories described people as being controlled by a powerful sorcerer. The walking dead became popular in the modern horror fiction mainly because of the success of George A. Romero’s 1968 film, Night of the Living Dead [2]. There are several possible etymologies of the word zombie. One of the possible origins is jumbie, which comes from the Carribean term for ghost. Another possible origin is the word nzambi which in Kongo means ‘spirit of a dead person’. According to the Merriam-Webster dictionary, the word zombie originates from the word zonbi, used in the Louisiana Creole or the Haitian Creole. According to the Creole culture, a zonbi represents a person who died and was then brought to life without speech or free will. The followers of Vodou believe that a dead person can be revived by a sorcerer [3]. After being revived, the zombies remain under the control of the sorcerer because they have no will of their own. Zombi is also another name for a Voodoo snake god. It is said that the sorcerer uses a ‘zombie powder’ for the zombification. This powder contains an extremely powerful neurotoxin that temporarily paralyzes the human nervous system and it creates a state of hibernation. The main organs, such as the heart and lungs, and all of the bodily functions, operate at minimal levels during this state of hibernation. What turns these human beings into zombies is the lack of oxygen to the brain. As a result of this, they suffer from brain damage. A popular belief in the Middle Ages was that the souls of the dead could return to earth one day and haunt the living [4]. In France, during the Middle Ages, they believed that the dead would usually awaken to avenge some sort of crime committed against them during their life. These awakened dead took the form of an emaciated corpse and they wandered around graveyards at night. The idea of the zombie also appears in several other cultures, such as China, Japan, the Pacific, India, Persia, the Arabs and the Americas. Modern zombies (the ones illustrated in books, films and games [1, 5]) are very different from the voodoo and the folklore zombies. Modern zombies follow a standard, as set in the movie Night of the Living Dead [2]. The ghouls are portrayed as being mindless monsters who do not feel pain and who have an immense appetite for human flesh. Their aim is to kill, eat or infect people. The ‘undead’ move in small, irregular steps, and show signs of physical decomposition such as rotting flesh, discoloured eyes and open wounds. Modern zombies are often related to an apocalypse, where civilization could collapse due to a plague of the undead. The background stories behind zombie movies, video games etc, are purposefully vague and inconsistent in explaining how the zombies came about in the first place. Some ideas include radiation (Night of the Living Dead [2]), exposure to airborne viruses (Resident Evil [6]), mutated diseases carried by various vectors (Dead Rising [7] claimed it was from bee stings of genetically altered bees). Shaun of the Dead [8] made fun of this by not allowing the viewer to determine what actually happened. When a susceptible individual is bitten by a zombie, it leaves an open wound. The wound created by the zombie has the zombie’s saliva in and around it. This bodily fluid mixes with the blood, thus infecting the (previously susceptible) individual. The zombie that we chose to model was characterised best by the popular-culture zombie. The basic assumptions help to form some guidelines as to the specific type of zombie we seek to model (which will be presented in the next section). The model zombie is of the classical pop-culture zombie: slow moving, cannibalistic and undead. There are other ‘types’ of zombies, characterised by some movies like 28 Days Later [9] and the 2004 re-

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make of Dawn of the Dead [10]. These ‘zombies’ can move faster, are more independent and much smarter than their classical counterparts. While we are trying to be as broad as possible in modelling zombies – especially since there are many varieties – we have decided not to consider these individuals.

2.

The Basic Model

For the basic model, we consider three basic classes: • Susceptible (S). • Zombie (Z). • Removed (R). Susceptibles can become deceased through ‘natural’ causes, i.e., non-zombie-related death (parameter δ). The removed class consists of individuals who have died, either through attack or natural causes. Humans in the removed class can resurrect and become a zombie (parameter ζ). Susceptibles can become zombies through transmission via an encounter with a zombie (transmission parameter β). Only humans can become infected through contact with zombies, and zombies only have a craving for human flesh so we do not consider any other life forms in the model. New zombies can only come from two sources: • The resurrected from the newly deceased (removed group). • Susceptibles who have ‘lost’ an encounter with a zombie. In addition, we assume the birth rate is a constant, Π. Zombies move to the removed class upon being ‘defeated’. This can be done by removing the head or destroying the brain of the zombie (parameter α). We also assume that zombies do not attack/defeat other zombies. Thus, the basic model is given by S ′ = Π − βSZ − δS

Z ′ = βSZ + ζR − αSZ

R′ = δS + αSZ − ζR .

This model is illustrated in Figure 1. This model is slightly more complicated than the basic SIR models that usually characterise infectious diseases [11], because this model has two mass-action transmissions, which leads to having more than one nonlinear term in the model. Mass-action incidence specifies that an average member of the population makes contact sufficient to transmit infection with βN others per unit time, where N is the total population without infection. In this case, the infection is zombification. The probability that a random contact by a zombie is made with a susceptible is S/N ; thus, the number of new zombies through this transmission process in unit time per zombie is: (βN )(S/N )Z = βSZ .

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Philip Munz, Ioan Hudea, Joe Imad et al.

Figure 1. The basic model. We assume that a susceptible can avoid zombification through an altercation with a zombie by defeating the zombie during their contact, and each susceptible is capable of resisting infection (becoming a zombie) at a rate α. So, using the same idea as above with the probability Z/N of random contact of a susceptible with a zombie (not the probability of a zombie attacking a susceptible), the number of zombies destroyed through this process per unit time per susceptible is: (αN )(Z/N )S = αSZ . The ODEs satisfy S ′ + Z ′ + R′ = Π and hence S+Z +R → ∞ as t → ∞, if Π 6= 0. Clearly S 6→ ∞, so this results in a ‘doomsday’ scenario: an outbreak of zombies will lead to the collapse of civilisation, as large numbers of people are either zombified or dead. If we assume that the outbreak happens over a short timescale, then we can ignore birth and background death rates. Thus, we set Π = δ = 0. Setting the differential equations equal to 0 gives −βSZ = 0

βSZ + ζR − αSZ = 0

αSZ − ζR = 0 .

From the first equation, we have either S = 0 or Z = 0. Thus, it follows from S = 0 that we get the ‘doomsday’ equilibrium ¯ Z, ¯ R) ¯ = (0, Z, ¯ 0) . (S, When Z = 0, we have the disease-free equilibrium ¯ Z, ¯ R) ¯ = (N, 0, 0) . (S,

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These equilibrium points show that, regardless of their stability, human-zombie coexistence is impossible. The Jacobian is then   −βZ −βS 0 J =  βZ − αZ βS − αS ζ  . αZ αS −ζ The Jacobian at the disease-free equilibrium is  0 −βN J(N, 0, 0) =  0 βN − αN 0 αN

We have

 0 ζ . −ζ

det(J − λI) = −λ{λ2 + [ζ − (β − α)N ]λ − βζN } . It follows that the characteristic equation always has a root with positive real part. Hence, the disease-free equilibrium is always unstable. Next, we have   −β Z¯ 0 0 ¯ 0) =  β Z¯ − αZ¯ 0 ζ  . J(0, Z, αZ¯ 0 −ζ

Thus,

det(J − λI) = −λ(−β Z¯ − λ)(−ζ − λ) . Since all eigenvalues of the doomsday equilibrium are negative, it is asymptotically stable. It follows that, in a short outbreak, zombies will likely infect everyone. In the following figures, the curves show the interaction between susceptibles and zombies over a period of time. We used Euler’s method to solve the ODE’s. While Euler’s method is not the most stable numerical solution for ODE’s, it is the easiest and least timeconsuming. See Figures 2 and 3 for these results. The MATLAB code is given at the end of this chapter. Values used in Figure 3 were α = 0.005, β = 0.0095, ζ = 0.0001 and δ = 0.0001.

3.

The Model with Latent Infection

We now revise the model to include a latent class of infected individuals. As discussed in Brooks [1], there is a period of time (approximately 24 hours) after the human susceptible gets bitten before they succumb to their wound and become a zombie. We thus extend the basic model to include the (more ‘realistic’) possibility that a susceptible individual becomes infected before succumbing to zombification. This is what is seen quite often in pop-culture representations of zombies ([2, 6, 8]). Changes to the basic model include:

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Figure 2. The case of no zombies. However, this equilibrium is unstable. • Susceptibles first move to an infected class once infected and remain there for some period of time. • Infected individuals can still die a ‘natural’ death before becoming a zombie; otherwise, they become a zombie. We shall refer to this as the SIZR model. The model is given by S ′ = Π − βSZ − δS

I ′ = βSZ − ρI − δI

Z ′ = ρI + ζR − αSZ

R′ = δS + δI + αSZ − ζR The SIZR model is illustrated in Figure 4 As before, if Π 6= 0, then the infection overwhelms the population. Consequently, we shall again assume a short timescale and hence Π = δ = 0. Thus, when we set the above equations to 0, we get either S = 0 or Z = 0 from the first equation. It follows again from our basic model analysis that we get the equilibria: Z=0

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Thus, coexistence between humans and zombies/infected is again not possible. In this case, the Jacobian is   −βZ 0 −βS 0  βZ −ρ βS 0   J =   −αZ ρ −αS ζ  . αZ 0 αS −ζ

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Figure 4. The SIZR model flowchart: the basic model with latent infection.

First, we have  −λ 0 −βN 0   0 −ρ − λ βN 0  det(J(N, 0, 0, 0) − λI) = det    0 ρ −αN − λ ζ 0 0 αN −ζ − λ   −ρ − λ βN 0  ρ −αN − λ ζ = −λ det  0 αN −ζ − λ  3 2 = −λ −λ − (ρ + ζ + αN )λ − (ραN + ρζ − ρβN )λ 

+ρζβN ] .

Since ρζβN > 0, it follows that det(J(N, 0, 0, 0) − λI) has an eigenvalue with positive real part. Hence, the disease-free equilibrium is unstable.

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¯ −ρ, −ζ. Since all eigenvalues are nonpositive, it The eigenvalues are thus λ = 0, −β Z, follows that the doomsday equilibrium is stable. Thus, even with a latent period of infection, zombies will again take over the population. We plotted numerical results from the data again using Euler’s method for solving the ODEs in the model. The parameters are the same as in the basic model, with ρ = 0.005. See Figure 5. In this case, zombies still take over, but it takes approximately twice as long. ;<=>5?. @,1!5>!5A5&5B*3C5
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4. The Model with Quarantine In order to contain the outbreak, we decided to model the effects of partial quarantine of zombies. In this model, we assume that quarantined individuals are removed from the population and cannot infect new individuals while they remain quarantined. Thus, the changes to the previous model include: • The quarantined area only contains members of the infected or zombie populations (entering at rates κ and σ, respectively). • There is a chance some members will try to escape, but any that tried to would be killed before finding their ‘freedom’ (parameter γ). • These killed individuals enter the removed class and may later become reanimated as ‘free’ zombies.

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The model equations are: S ′ = Π − βSZ − δS

I ′ = βSZ − ρI − δI − κI

Z ′ = ρI + ζR − αSZ − σZ

R′ = δS + δI + αSZ − ζR + γQ

Q′ = κI + σZ − γQ . The model is illustrated in Figure 6.

Figure 6. Model flow diagram for the Quarantine model. For a short outbreak (Π = δ = 0), we have two equilibria, ¯ I, ¯ Z, ¯ R, ¯ Q) ¯ = (N, 0, 0, 0, 0), (0, 0, Z, ¯ R, ¯ Q) ¯ . (S, In this case, in order to analyse stability, we determined the basic reproductive ratio, R0 [12] using the next-generation method [13]. The basic reproductive ratio has the property that if R0 > 1, then the outbreak will persist, whereas if R0 < 1, then the outbreak will be eradicated. If we were to determine the Jacobian and evaluate it at the disease-free equilibrium, we would have to evaluate a nontrivial 5 by 5 system and a characteristic polynomial of degree of at least 3. With the next-generation method, we only need to consider the infective differential equations I ′ , Z ′ and Q′ . Here, F is the matrix of new infections and V is the matrix of transfers between compartments, evaluated at the disease-free equilibrium.     ρ+κ 0 0 0 βN 0 F =  0 0 0  , V =  −ρ αN + σ 0  −κ −σ γ 0 0 0 V −1 =

F V −1 =

 γ(αN + σ) 0 0 1   ργ γ(ρ + κ) 0 γ(ρ + κ)(αN + σ) ρσ + κ(αN + σ) σ(ρ + κ) (ρ + κ)(αN + σ)   βN ργ βN γ(ρ + κ) 0 1  0 0 0 . γ(ρ + κ)(αN + σ) 0 0 0 

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Philip Munz, Ioan Hudea, Joe Imad et al.

This gives us R0 =

βN ρ . (ρ + κ)(αN + σ)

It follows that the disease-free equilibrium is only stable if R0 < 1. This can be achieved by increasing κ or σ, the rates of quarantining infected and zombified individuals, respectively. If the population is large, then R0 ≈

βρ . (ρ + κ)α

If β > α (zombies infect humans faster than humans can kill them, which we expect), then eradication depends critically on quarantining those in the primary stages of infection. This may be particularly difficult to do, if identifying such individuals is not obvious [8]. However, we expect that quarantining a large percentage of infected individuals is unrealistic, due to infrastructure limitations. Thus, we do not expect large values of κ or σ, in practice. Consequently, we expect R0 > 1. As before, we illustrate using Euler’s method. The parameters were the same as those used in the previous models. We varied κ, σ, γ to satisfy R0 > 1. The results are illustrated in Figure 7. In this case, the effect of quarantine is to slightly delay the time to eradication of humans. =>?@A7B0 C.3 !7@!7D7"7E,5 F7>G7H7IJK :94- .7L432.97M0 N714N4- .5.N9729.C7,671N.L,0 297M,O2N.< 7

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143

to exist, and there must still be zombies in that class. The zombies measured by the curve in the figure are considered the ‘free’ zombies: the ones in the Z class and not in Q.

5.

A Model with Treatment

Suppose we are able to quickly produce a cure for ‘zombie-ism’. Our treatment would be able to allow the zombie individual to return to their human form again. Once human, however, the new human would again be susceptible to becoming a zombie; thus, our cure does not provide immunity. Those zombies who resurrected from the dead and who were given the cure were also able to return to life and live again as they did before entering the R class. Things that need to be considered now include: • Since we have treatment, we no longer need the quarantine. • The cure will allow zombies to return to their original human form regardless of how they became zombies in the first place. • Any cured zombies become susceptible again; the cure does not provide immunity. Thus, the model with treatment is given by S ′ = Π − βSZ − δS + cZ I ′ = βSZ − ρI − δI

Z ′ = ρI + ζR − αSZ − cZ

R′ = δS + δI + αSZ − ζR . The model is illustrated in Figure 8.

Figure 8. Model flowchart for the SIZR model with cure. As in all other models, if Π 6= 0, then S + I + Z + R → ∞, so we set Π = δ = 0. When Z = 0, we get our usual disease-free equilibrium, ¯ I, ¯ Z, ¯ R) ¯ = (N, 0, 0, 0) . (S,

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Philip Munz, Ioan Hudea, Joe Imad et al.

However, because of the cZ term in the first equation, we now have the possibility of an ¯ I, ¯ Z, ¯ R) ¯ satisfying endemic equilibrium (S, −β S¯Z¯ + cZ¯ = 0 β S¯Z¯ − ρI¯ = 0 ¯ − αS¯Z¯ − cZ¯ = 0 ρI¯ + ζ R

¯ = 0. αS¯Z¯ − ζ R

Thus, the equilibrium is ¯ I, ¯ Z, ¯ R) ¯ = (S,



c c ¯ ¯ αc ¯ , Z, Z, Z β ρ ζβ



.

The Jacobian is

J

We thus have

 βZ 0 −βS + c 0  βZ −ρ βS 0   =   −αZ ρ −αS − c ζ  . αZ 0 αS −ζ 

β Z¯ 0 0 0  β Z¯ −ρ c 0 ¯ I, ¯ Z, ¯ R) ¯ − λI) = det  det(J(S,  −αZ¯ ρ − αc ζ β −c αc ¯ αZ 0 −ζ β  −ρ c αc ¯  ρ − = −(β Z − λ) det β −c αc 0 β    αc = −(β Z¯ − λ) −λ λ2 + ρ + β  ζαc ραc + + ρζ + cζ . + β β 

   

 0 ζ  −ζ



+c+ζ λ

Since the quadratic expression has only positive coefficients, it follows that there are no positive eigenvalues. Hence, the coexistence equilibrium is stable. The results are illustrated in Figure 9. In this case, humans are not eradicated, but only exist in low numbers.

6.

Impulsive Eradication

Finally, we attempted to control the zombie population by strategically destroying them at such times that our resources permit (as suggested in [14]). It was assumed that it would be difficult to have the resources and coordination, so we would need to attack more than once, and with each attack, try and destroy more zombies. This results in an impulsive effect [15, 16, 17, 18].

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= = = =

Π − βSZ − δS βSZ + ζR − αSZ δS + αSZ − ζR −knZ

t t t t

6= tn 6= tn 6= tn = tn ,

where k ∈ (0, 1] is the kill ratio and n denotes the number of attacks required until kn > 1. The results are illustrated in Figure 10. Eradication with increasing kill ratios 1000 900 800

Number of Zombies

700 600 500 400 300 200 100 0

0

2

4

6

8

10

Time

Figure 10. Zombie eradication using impulsive attacks. In Figure 10, we used k = 0.25 and the values of the remaining parameters were

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Philip Munz, Ioan Hudea, Joe Imad et al.

(α, β, ζ, δ) = (0.0075, 0.0055, 0.09, 0.0001). Thus, after 2.5 days, 25% of zombies are destroyed; after 5 days, 50% of zombies are destroyed; after 7.5 days, 75% of remaining zombies are destroyed; after 10 days, 100% of zombies are destroyed.

7.

Discussion

An outbreak of zombies infecting humans is likely to be disastrous, unless extremely aggressive tactics are employed against the undead. While aggressive quarantine may eradicate the infection, this is unlikely to happen in practice. A cure would only result in some humans surviving the outbreak, although they will still coexist with zombies. Only sufficiently frequent attacks, with increasing force, will result in eradication, assuming the available resources can be mustered in time. Furthermore, these results assumed that the timescale of the outbreak was short, so that the natural birth and death rates could be ignored. If the timescale of the outbreak increases, then the result is the doomsday scenario: an outbreak of zombies will result in the collapse of civilisation, with every human infected, or dead. This is because human births and deaths will provide the undead with a limitless supply of new bodies to infect, resurrect and convert. Thus, if zombies arrive, we must act quickly and decisively to eradicate them before they eradicate us. The key difference between the models presented here and other models of infectious disease is that the dead can come back to life. Clearly, this is an unlikely scenario if taken literally, but possible real-life applications may include allegiance to political parties, or diseases with a dormant infection. This is, perhaps unsurprisingly, the first mathematical analysis of an outbreak of zombie infection. While the scenarios considered are obviously not realistic, it is nevertheless instructive to develop mathematical models for an unusual outbreak. This demonstrates the flexibility of mathematical modelling and shows how modelling can respond to a wide variety of challenges in ‘biology’. In summary, a zombie outbreak is likely to lead to the collapse of civilisation, unless it is dealt with quickly. While aggressive quarantine may contain the epidemic, or a cure may lead to coexistence of humans and zombies, the most effective way to contain the rise of the undead is to hit hard and hit often. As seen in the movies, it is imperative that zombies are dealt with quickly, or else we are all in a great deal of trouble.

Acknowledgements We thank Shoshana Magnet, Andy Foster and Shannon Sullivan for useful discussions. RJS? is supported by an NSERC Discovery grant, an Ontario Early Researcher Award and funding from MITACS. function [ ] = zombies(a,b,ze,d,T,dt) % This function will solve the system of ODE’s for the basic model used in % the Zombie Dynamics project for MAT 5187. It will then plot the curve of % the zombie population based on time. % Function Inputs: a - alpha value in model: "zombie destruction" rate % b - beta value in model: "new zombie" rate

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% ze - zeta value in model: zombie resurrection rate % d - delta value in model: background death rate % T - Stopping time % dt - time step for numerical solutions % Created by Philip Munz, November 12, 2008 %Initial set up of solution vectors and an initial condition N = 500; %N is the population n = T/dt; t = zeros(1,n+1); s = zeros(1,n+1); z = zeros(1,n+1); r = zeros(1,n+1); s(1) = N; z(1) = 0; r(1) = 0; t = 0:dt:T; % Define the ODE’s of the model and solve numerically by Euler’s method: for i = 1:n s(i+1) = s(i) + dt*(-b*s(i)*z(i)); %here we assume birth rate = background deathrate, so only term is -b term z(i+1) = z(i) + dt*(b*s(i)*z(i) -a*s(i)*z(i) +ze*r(i)); r(i+1) = r(i) + dt*(a*s(i)*z(i) +d*s(i) - ze*r(i)); if s(i)<0 || s(i) >N break end if z(i) > N || z(i) < 0 break end if r(i) <0 || r(i) >N break end end hold on plot(t,s,’b’); plot(t,z,’r’); legend(’Suscepties’,’Zombies’) -----------function [z] = eradode(a,b,ze,d,Ti,dt,s1,z1,r1) % This function will take as inputs, the initial value of the 3 classes. % It will then apply Eulers method to the problem and churn out a vector of % solutions over a predetermined period of time (the other input). % Function Inputs: s1, z1, r1 - initial value of each ODE, either the % actual initial value or the value after the % impulse. % Ti - Amount of time between inpulses and dt is time step % Created by Philip Munz, November 21, 2008 k = Ti/dt; %s = zeros(1,n+1); %z = zeros(1,n+1); %r = zeros(1,n+1);

148 %t = 0:dt:Ti; s(1) = s1; z(1) = z1; r(1) = r1; for i=1:k s(i+1) = s(i) = background z(i+1) = z(i) r(i+1) = r(i) end

Philip Munz, Ioan Hudea, Joe Imad et al.

+ dt*(-b*s(i)*z(i)); %here we assume birth rate deathrate, so only term is -b term + dt*(b*s(i)*z(i) -a*s(i)*z(i) +ze*r(i)); + dt*(a*s(i)*z(i) +d*s(i) - ze*r(i));

%plot(t,z)

-----------function [] = erad(a,b,ze,d,k,T,dt) % This is the main function in our numerical impulse analysis, used in % conjunction with eradode.m, which will simulate the eradication of % zombies. The impulses represent a coordinated attack against zombiekind % at specified times. % Function Inputs: a - alpha value in model: "zombie destruction" rate % b - beta value in model: "new zombie" rate % ze - zeta value in model: zombie resurrection rate % d - delta value in model: background death rate % k - "kill" rate, used in the impulse % T - Stopping time % dt - time step for numerical solutions % Created by Philip Munz, November 21, 2008 N = 1000; Ti = T/4; %We plan to break the solution into 4 parts with 4 impulses n = Ti/dt; m = T/dt; s = zeros(1,n+1); z = zeros(1,n+1); r = zeros(1,n+1); sol = zeros(1,m+1); %The solution vector for all zombie impulses and such t = zeros(1,m+1); s1 = N; z1 = 0; r1 = 0; %i=0; %i is the intensity factor for the current impulse %for j=1:n:T/dt % i = i+1; % t(j:j+n) = Ti*(i-1):dt:i*Ti; % sol(j:j+n) = eradode(a,b,ze,d,Ti,dt,s1,z1,r1); % sol(j+n) = sol(j+n)-i*k*sol(j+n); % s1 = N-sol(j+n); % z1 = sol(j+n+1); % r1 = 0; %end sol1 = eradode(a,b,ze,d,Ti,dt,s1,z1,r1); sol1(n+1) = sol1(n+1)-1*k*sol1(n+1); %347.7975;

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s1 = N-sol1(n+1); z1 = sol1(n+1); r1 = 0; sol2 = eradode(a,b,ze,d,Ti,dt,s1,z1,r1); sol2(n+1) = sol2(n+1)-2*k*sol2(n+1); s1 = N-sol2(n+1); z1 = sol2(n+1); r1 = 0; sol3 = eradode(a,b,ze,d,Ti,dt,s1,z1,r1); sol3(n+1) = sol3(n+1)-3*k*sol3(n+1); s1 = N-sol3(n+1); z1 = sol3(n+1); r1 = 0; sol4 = eradode(a,b,ze,d,Ti,dt,s1,z1,r1); sol4(n+1) = sol4(n+1)-4*k*sol4(n+1); s1 = N-sol4(n+1); z1 = sol4(n+1); r1 = 0; sol=[sol1(1:n),sol2(1:n),sol3(1:n),sol4]; t = 0:dt:T; t1 = 0:dt:Ti; t2 = Ti:dt:2*Ti; t3 = 2*Ti:dt:3*Ti; t4 = 3*Ti:dt:4*Ti; %plot(t,sol) hold on plot(t1(1:n),sol1(1:n),’k’) plot(t2(1:n),sol2(1:n),’k’) plot(t3(1:n),sol3(1:n),’k’) plot(t4,sol4,’k’) hold off

References [1] Brooks, Max, 2003 The Zombie Survival Guide - Complete Protection from the Living Dead, Three Rivers Press, pp. 2-23. [2] Romero, George A. (writer, director), 1968 Night of the Living Dead. [3] Davis, Wade, 1988 Passage of Darkness - The Ethnobiology of the Haitian Zombie, Simon and Schuster pp. 14, 60-62. [4] Davis, Wade, 1985 The Serpent and the Rainbow, Simon and Schuster pp. 17-20, 24, 32. [5] Williams, Tony, 2003 Knight of the Living Dead - The Cinema of George A. Romero, Wallflower Press pp.12-14. [6] Capcom, Shinji Mikami (creator), 1996-2007 Resident Evil. [7] Capcom, Keiji Inafune (creator), 2006 Dead Rising.

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[8] Pegg, Simon (writer, creator, actor), 2002 Shaun of the Dead. [9] Boyle, Danny (director), 2003 28 Days Later. [10] Snyder, Zack (director), 2004 Dawn of the Dead. [11] Brauer, F. Compartmental Models in Epidemiology. In: Brauer, F., van den Driessche, P., Wu, J. (eds). Mathematical Epidemiology. Springer Berlin 2008. [12] Heffernan, J.M., Smith?, R.J., Wahl, L.M. (2005). Perspectives on the Basic Reproductive Ratio. J R Soc Interface 2(4), 281-293. [13] van den Driessche, P., Watmough, J. (2002) Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180, 29-48. [14] Brooks, Max, 2006 World War Z - An Oral History of the Zombie War, Three Rivers Press. [15] Bainov, D.D. & Simeonov, P.S. Systems with Impulsive Effect. Ellis Horwood Ltd, Chichester (1989). [16] Bainov, D.D. & Simeonov, P.S. Impulsive differential equations: periodic solutions and applications. Longman Scientific and Technical, Burnt Mill (1993). [17] Bainov, D.D. & Simeonov, P.S. Impulsive Differential Equations: Asymptotic Properties of the Solutions. World Scientific, Singapore (1995). [18] Lakshmikantham, V., Bainov, D.D. & Simeonov, P.S. Theory of Impulsive Differential Equations. World Scientific, Singapore (1989).

In: Infectious Disease Modelling Research Progress ISBN 978-1-60741-347-9 c 2009 Nova Science Publishers, Inc. Editors: J.M. Tchuenche, C. Chiyaka, pp. 151-176

Chapter 5

A R EVIEW OF M ATHEMATICAL M ODELLING OF THE E PIDEMIOLOGY OF M ALARIA C. Chiyaka1∗, Z. Mukandavire1 , S. Dube2 , G. Musuka3 and J.M. Tchuenche4 1 Department of Applied Mathematics, National University of Science and Technology, P. O. Box AC 939 Ascot, Bulawayo, Zimbabwe 2 Department of Applied Biology/Biochemistry, National University of Science and Technology, P. O. Box AC 939 Ascot, Bulawayo, Zimbabwe 3 African Comprehensive HIV/AIDS Partnerships, Private Bag X033 Gaborone, Botswana 4 Mathematics Department, University of Dar es Salaam, Box 35062, Dar es Salaam, Tanzania

Abstract Malaria is one of the most prevalent and debilitating diseases afflicting humans. It is widespread in tropical regions especially Sub-Saharan Africa in which most areas including uplands report malaria transmission throughout the year, though it increases during and soon after the rainy season. Immunologically vulnerable populations are generally afflicted by malaria epidemics and people of all age groups remain susceptible to the full range of its clinical effects. Its spread in a community poses unique intervention strategies. Mathematical models have played a significant role in understanding the epidemiology of malaria. The purpose of this review is to highlight the challenges faced in combating malaria, its control strategies and to discuss the mathematical models that have been developed and analyzed to better understanding the disease dynamics.

Keywords: malaria, mathematical modelling, epidemiology, intervention strategies. ∗

E-mail address: [email protected]; [email protected]. Corresponding author.

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Introduction

Malaria is the world’s most prevalent vector borne disease and it still remains among the most devastating diseases occurring in the world. It represents 10% of Africa’s overall disease burden (World Malaria Report, 2005). Hundreds of millions of African children and adults are chronically infected with malaria. This situation is exacerbated by generally poor access to health care, under-funding of public health services, and deficient training of health workers, who are inadequately motivated, equipped and supported (Van Lerberghe et al., 1997). In 2000, malaria was the principal cause of around 18% of deaths of children under 5 years of age in Sub-Saharan Africa (Rowe et al., 2006). In endemic African countries, malaria accounts for 25-35% of all outpatient visits, 20-45% of hospital admissions and 15-35% of hospital deaths, imposing a great burden on already fragile health-care systems (World malaria report, 2005). In a review done in Kinshasa, Democratic Republic of Congo, an unprecedented increase in the prevalence of malaria in school age children from 1980 to 2000 was shown (Kazadi et al., 2004). In Africa alone, the 28 million reported cases of malaria are believed to represent only 5-10% of the total malaria incidence on the continent (Hamoudi and Sachs, 1999). While malaria remains under control in most developed and stable areas, the situation is deteriorating in all frontier areas of economic development, that is, in areas where the exploitation of natural resources or illegal trade occurs, in jungle areas or areas burdened with problems of civil war and other conflicts, and where mass movements of refugees exist. Studies have shown that in areas where sprinkler irrigation scheme is practiced there is potential transmission of malaria (Chimbari et al., 2004). The association of irrigation and vectors that transmit malaria is well documented (Keiser et al., 2004; Mutero et al., 2000; Jobin, 1999; Hunter et al., 1993). In the past three decades, malaria has, however, encroached upon areas where it had formerly been eradicated or had successfully been controlled (Baird, 2000) by use of chemicals, therapeutic drugs and insecticides, thus offsetting the gains attained in the latter half of the past century. The prevalence of malaria has an enormous impact on a country’s economy, it dramatically inhibits economic growth (by restricting individual worker productivity, tourism and transportation). In Africa, malaria slows economic growth by up to 1.3% each year (WHO, 2000). The symptoms and epidemiological manifestations from this single micro-organism are highly variable and geographically determined by a balanced interplay of the parasite with the human host and the vector. The explosiveness of malaria epidemics always strains the capacity of health facilities, causing case fatality rates to increase five-fold or more during outbreaks. People of all ages remain susceptible to the full range of clinical effects. Malaria epidemics display the full explosive power of vector-borne infections, erupting with a suddenness and intensity that can overwhelm vulnerable communities (for example, in Burundi 2000-2001, (Kiszewski and Teklehaimanot, 2004) and the 1998 epidemic in Rwanda (Hammerlich et al., 2002)). The expected rise in global temperature and climate change might also herald an increase in the geographical distribution of malaria transmission (Patz et al., 1996) due to the role of temperature and rainfall in the population dynamics of its mosquito vector (Zhou et al., 2004; Lindsay and Martens, 1998). Malaria cases are also being exacerbated by the high levels of HIV infection, that weakens the immune system rendering people with HIV more

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susceptible to contracting the disease (Bush et al., 2001). It also enhances mortality in advanced HIV patients by a factor of about 25% in nonstable malaria areas (Grimwade et al., 2004). In endemic areas the heaviest toll of morbidity and mortality falls on young children. This is due to a slow acquisition of immunity by children in endemic areas. Plasmodium infection only confers labile and partial immunity which only reduces the incidence of clinical malaria attacks without preventing infection (Gilles and Warrell, 1993; Miller, 1958). Although campaigns against malaria were initially successful in some areas, the emergence of resistance of the parasite to drugs and of the mosquito vector to insecticides, combined with the difficulties in implementing and maintaining effective control (vector control and epidemic prevention and control) schemes have led to a resurgence of the disease in many parts of the world (Roush, 1993; Schapira et al., 1993; Wernsdorfer, 1991). An effective vaccine would constitute a powerful addition to malaria control, but up to date there is no licensed vaccine for malaria. Different parasite-specific features and characteristics of the interaction of human immune system and the parasites, might explain the difficulties in developing effective immuno-therapeutic intervention strategies. In addition to infectioninduced mortality, malaria is also associated with public health problems resulting from the impairment of immune response. Although this immunosuppression may have evolved as a result of mechanisms by which the parasite can prevent immune mediated clearance (Ho et al., 1986; Theander et al., 1986) it leaves malaria infected individuals more susceptible to secondary infections such as herpes zooster virus (Cook, 1985) and hepatitis B virus (Thursz, 1995). There are no simple solutions to the world’s malaria problem and it is unlikely that a single strategy for control will be applicable to all countries and all epidemiological situations. Global changes in weather (Greenwood et al., 2005; Tanser et al., 2003; Dobson and Carter, 1992) accompanied by changes in demographic structures, increasing resistance to prevalent drugs (Parola et al., 2007), a vaccine long promised but still due (Chauhan, 2007) and outright public exhaustion are some of the factors compounding the scientific challenge of outsmarting an organism fully equipped with nature’s remarkable set of survival tools. Therefore, expansion of our understanding of the biology, epidemiology, pathogenesis and clinical manifestations of this complex, heterogeneous disease will be critical to the development of additional strategies for control.

2.

Malaria

There are several ways in which malaria can be transmitted in humans and these are: (a) The bite of an infected female Anopheles mosquito. (b) Malaria can be transmitted through blood transfusion. Among people living in malarious areas, partial (semi) immunity to malaria allows donors to have parasitaemia without any fever or other clinical manifestations. (c) Organ transplantation may transmit malaria. (d) Transplacental malaria (that is, congenital malaria) can be significant in populations who are partially immune to malaria. The mother may have placental parasitaemia,

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The incidence of direct human to human transmission is significantly lower than that from mosquitoes. There are about 3500 species of mosquitoes worldwide, of which 430 are in the genus Anopheles (Baker, 1966). Of the approximately 430, only 30-40 species of Anopheles are capable of transmitting malaria (they are suitable for the development of malaria). The susceptibility of these mosquitoes to the malaria parasite is known to be variable with some mosquitoes more permissive to infection than others (Hume et al., 2007). The Anopheles mosquitoes are easy to recognize because of their characteristic stance. They rest with their head down and their abdomen sticking out, while other mosquitoes rest with their bodies parallel to the resting surface (Parry and Pharms, 2007). These vectors also differ in their biology and ecology, varied epidemiological patterns and capability to develop resistance to insecticides. Both the female and male mosquitoes feed on nectar, the females however require blood meals for the protein necessary for egg development (Klowden, 1995). Some species of the mosquito prefer human blood (anthropophilic) while others prefer animal blood (zoophilic). Some prefer to bite indoors (endophagic) and others prefer outdoors (exophagic). Similarly some rest indoors (endophilic), while others rest outdoors (exophilic). Once a good blood meal has been taken, the vector searches for a convenient breeding site. The Anopheles mosquito, like several other insect vectors, goes through several separate and distinct stages of development. The eggs are laid on water and after about 2-3 days, they hatch into larva. This process is temperature-dependent and can take up to 2-3 weeks in cold weather. In about 4-10 days, the larva changes into pupa. The pupa then changes into the adult mosquito in about 2-4 days. The duration of the whole cycle (from egg laying to an adult mosquito) varies between 7 and 20 days, depending on the ambient temperature and the mosquito species (Giles and Warrel, 1993). The four species that cause malaria in humans are the protozoan parasites of the genus Plasmodium: Plasmodium vivax, Plasmodium malariae, Plasmodium ovale and Plasmodium falciparum. These protozoan parasites are transmitted by the bite of an Anopheles mosquito. These parasites can also infect some non-human primates (Garnham, 1966). P. vivax is a major cause of clinical malaria but is rarely fatal. This species is not found in tropical Africa because of the lack of red cell surface Duffy antigen in African populations that P. vivax requires for cell invasion. P. malariae and P. ovale are an infrequent cause of clinical malaria, often persisting as low grade parasitaemia with other species and relapse many years after apparent cure. P. falciparum accounts for the most severe and often potentially lethal forms of malaria. Chronic infections persisting over 2-3 years do occur but relapses are uncommon, since no dormant stages, viz. hypnozoites exist in the liver. Although there are four species of Plasmodium that infect humans, only two (P. vivax and P. falciparum) cause significant disease, with nearly all deaths being caused by P. falciparum (Aide et al., 2007). An appreciation of the life cycle and transmission of Plasmodia and the pathophysiology of infection is the key to understanding the disease. The biology of the four species of malaria parasites is generally similar and consists of two discrete phases - sexual and asexual. A mosquito can systematically target and identify human beings (McCall and Kelly, 2002) through various stimuli, which include higher concentrations of carbon dioxide, certain body odours, warmth and movement. Once attracted, it may seek and bite humans

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as many times as possible until it takes a blood meal and depending on the disease status of both organisms, will either infect or be infected. In the process of this interaction, the mosquito may loose its life. When a mosquito alights on the skin, it attempts to pierce a small blood vessel with its proboscis in order to suck blood. The mosquito first injects some saliva together with an anticoagulant (anti-clotting chemical) to prevent blood from clotting and closing the wound so the mosquito can drink it. If the mosquito is infected with malaria parasites then, sporozoites are introduced into the human body together with saliva during the bite . The sporozoites, leave the site of the bite fairly rapidly, penetrate capillary and travel through the bloodstream to gain access to hepatocytes. About 30 minutes after being introduced into the bloodstream, these sporozoites enter the hepatocytes and form schizonts (McQueen and McKenzie, 2004; Rouzine and McKenzie, 2003; Gravenor et al., 2002; Bailey, 1982). Schizonts undergo a process of maturation and multiplication known as pre-erythrocytic or hepatic schizogony. In P. vivax and P. ovale infections, some sporozoites convert to dormant forms called hypnozoites, which can cause disease after months or years. They further continue to differentiate and divide into exo-erythrocytic schizonts, each containing thousands of infectious merozoites. The liver stage is also called the preerythrocytic stage. The sporozoite stage and the liver stage that follows subsequently are clinically silent stages of infection and parasite clearance or reductions in the parasite burden at these stages can markedly attenuate disease. Six to sixteen days after infection (contingent on the species), the schizont ruptures releasing merozoites into the blood stream. The number of merozoites released is estimated as 2,000 for P. malariae, 10,000 for P. vivax/P. ovale, and up to 30,000 for P. falciparum (Garnham, 1966). These merozoites released into the blood stream invade neighbouring or circulating erythrocytes and become first trophozoites and then erythrocytic schizonts (by a phase of asexual reproduction). After about 48-72 hours (Diebner et al., 2000; Hoshen et al., 2000), depending on the Plasmodium species, the infected erythrocyte bursts releasing 16-32 merozoites and multiple antigenic and pyrogenic substances into the bloodstream. These merozoites again quickly infect fresh erythrocytes incessantly propagating the parasite cycle. This cycle also known as the erythrocytic stage, causes the pathological hallmarks of a malaria infection (Gravenor et al., 2002). Merozoites represent one of the developmental stages in which parasites are extracellular and thus theoretically readily accessible to antibodies during repeated cycles of merozoites released from rupturing affected erythrocytes. Some merozoites, responding to diverse triggers (Bilker et al., 1997), follow an alternative developmental pathway that yields the sexual form called the gametocytes (Diebner et al., 2000). These gametocytes develop through morphologically distinct stages, designated I to V within the host red blood cell (Piper et al., 1999). Mature (stage V ) gametocytes enclosed within the red blood cell membrane, circulate in the host’s blood, awaiting uptake by the mosquito vector during a blood meal. Once inside the stomach (of the mosquito), erythrocytes and asexual blood stage parasites perish while the gametocytes responding to a drop in temperature (Sinden and Croll, 1975) and other mosquito associated factors (Nijhout and Carter, 1978; Carter and Nijhout, 1977) undergo rapid transformation to yield male and female gametes. The male gametocyte nucleus divides into eight sperm-like flagellated micro-gametes each of which also leaves the erythrocyte, reaches the midgut and actively moves to fertilize a macrogamete within 60 minutes of ingestion of blood (Simonetti, 1996). The result of the fertilization

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process is the zygote, which develops into a motile ookinete within 10-25 hours from the blood meal. The ookinete invade the midgut wall of the mosquito where they develop into oocysts. The oocysts grow, rupture and release sporozoites, which make their way to the mosquito’s salivary glands over a period of 7-12 days. Here, they stay for possibly very long periods until injected into another vertebrate host when the next blood meal is taken (Carter and Graves, 1988; Sinden, 1984). Time required for the entire sexual development of the parasite in the body of the mosquito depends critically on the prevailing temperature and for P. falciparum it is about 12 days within the temperature range of 25-27◦ C (Anderson and May, 1991). Inoculation of sporozoites into a new human host perpetuates the Plasmodium life cycle. The cycle of blood meal and oviposition continues throughout the life of a female mosquito and requires repeated contacts with the vertebrate host, allowing for ingestion of malaria parasites, their multiplication and maturation, and transmission to other individual hosts during subsequent feedings.

2.1.

Control Strategies

In 1955, the Eighth World Health Assembly, singled out malaria as the first priority and set up the most grandiose health project ever undertaken with the World Health Organization (WHO) responsible for overall coordination and for the technical approval of individual projects (Brown, 2002). The Global Malaria Eradication Campaign successfully eliminated or controlled the disease in countries with temperate climates and in some countries where malaria transmission was low or moderate. One of the projects on the study of epidemiology and control of malaria was conducted by WHO and the government of Nigeria in Garki, Nigeria, from 1969 to 1976 (Nedelman, 1988). The results of the study have been reported by the WHO in a monograph (Molineaux and Gramiccia, 1980). The emergence of drug and insecticide resistance, coupled with concerns about the feasibility and sustainability of tackling malaria in areas with weak infrastructure and high transmission, brought an end to the eradication era, as well as to the bulk of international funding for malaria control and investment in malaria research. Some of the factors which account for limited successes in eradicating malaria are: (i) lack of political will and commitment, (ii) poor awareness of the magnitude of the malaria burden, (iii) poor health practices by individuals and communities and (iv) resistance to drugs by malaria parasites and resistance to insecticides by mosquitoes. A new approach to the war against malaria was then put up in 1998, by WHO, in conjunction with the United Nations Children’s Fund (UNICEF), the United Nations Development Programme (UNDP), and the World Bank. These organizations launched the Roll Back Malaria Global Partnership (RBM), with the goal of halving the worldwide burden of malaria by 2010 (WHO, 1999; Nabarro and Tayler, 1998). The RBM relies on the following strategies: • curative measures based on research and evidence,

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• early diagnosis, • prompt and appropriate treatment, • multiple prevention methods, • dynamic global movement against the problem and • broad-based coordinated actions backed by focused research. The malaria pathogen can be combated either in its human host or mosquito vector and both strategies have received enormous attention over the years. Of the numerous antimalarial activities and research efforts supported by RBM and others, we describe some of the control strategies here which have also been documented previously (Munday, 2007; Sharp et al., 2007; Walker and Lynch, 2007; WHO, 2006; Chitnis, 2005; Prudhomme O’Meara et al., 2005; Guyatt et al., 2004; Killeen et al., 2002; Margos et al., 1998; Alonso et al., 1993; Smith? and Hove-Musekwa, 2008; Chaves et al., 2008). • Larval control: This strategy includes methods such as the destruction of mosquito breeding sites, to reduce the number of mosquitoes. • Indoor residual spraying: Spraying reduces mosquito longevity. This strategy is also likely to kill mosquitoes that rest indoors after feeding so it increases the chances of killing infected mosquitoes. • Insecticide-treated bed nets (ITNs): RBM has been promoting the use of insecticide treated bed nets in many countries including regions of Africa to reduce the transmission of malaria; and has succeeded in doing so in many regions. As some recent studies have shown (Igwe, 2007), ITNs have had a significant impact on disease prevalence and mortality, but their ownership is generally low. • Intermittent prophylactic treatment: This is a new area of research that involves administering antimalarial drugs at regular intervals, even to those who are not sick, to reduce parasitaemia load. This is essentially similar to the treatment taken by travelers from malaria-free regions when visiting malaria-endemic countries. This form of control would most likely be applied in areas of high transmission where almost everyone has some Plasmodia in their blood. Intermittent prophylactic treatment is also carried out for pregnant women and for infants. Initial studies have started in this area and have shown significant effects in reducing infant mortality. • Prompt and effective case management: This strategy involves the quick identification and treatment of malaria cases. Although it may seem obvious, it is not always possible in many places because of poor health infrastructure and a lack of resources. Quick treatment is doubly effective because it directly reduces the suffering and lack of productivity due to malaria, and it reduces the transmission of infection to mosquitoes. • Transgenically modified mosquitoes: As there are some species of Anopheles mosquitoes that have an immune response to kill the Plasmodium parasites, there

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C. Chiyaka, Z. Mukandavire, S. Dube et al. is hope that genetically modified mosquitoes could be introduced into the wild that would be incapable of transmitting malaria. This is a promising area of research, although still in its early stages. There would be need for strict controls to ensure that the new mosquitoes created are not accidentally given the capability of transmitting other diseases such as influenza or AIDS. As these mosquitoes would be immune to malaria, having a population of only transgenically modified mosquitoes would eliminate the transmission of malaria. However, we would expect some wild-type mosquitoes to persist in the population. Li (2005, 2004) proposed some population models for the introduction of transgenic mosquitoes. • Vaccination: Several approaches to malaria vaccine development target different stages of the parasite’s life cycle in humans and mosquitoes. Pre-erythrocytic vaccines are designed to prevent the parasite’s infective sporozoite stage from entering or developing within the liver cells of an individual bitten by an infected mosquito. Erythrocytic (asexual blood stage) vaccine prevents the malaria parasite from entering or developing in red blood cells. Transmission blocking (sexual stage) vaccine are directed against the sexual stage antigens of the life cycle to prevent fertilisation.

Treatment of malaria is one of the earliest methods of control. Vaccination is another new prevention method for malaria, but is still under research. We briefly discuss each of these strategies and the associated problems. Treatment of malaria depends on the infecting plasmodia species, the geographic area of acquisition (which affects the likelihood of drug resistance) and the severity of infection (Suh et al., 2004; White, 1996). Falciparum malaria in the non immune person is a medical emergency and requires rapid initiation of antimalarial therapy. If the species cannot be immediately identified, the patient should be assumed to have drug-resistant falciparum malaria until proven otherwise. Hospital admission is advised for those with falciparum malaria or in whom the infecting species cannot be identified, and for those who are severely ill. Malaria is treated with drugs that block the growth of the Plasmodium parasite but do not harm the patient. Antimalarials have varying effects on the different stages of the malaria parasite’s life cycle (Terzian, 1970). Some drugs interfere with the parasites metabolism of food, while others prevent the parasite from reproducing. The antifolates, quinine and mefloquine appear to affect only the schizonts (actively dividing forms of the Plasmodium species) (Gutteridge and Trigg, 1971; McGregor and Smith, 1952; Jones et al., 1948). Parasites treated with antifolates will continue to mature, cytoadhere and develop into gametocytes following treatment. Chloroquine, artemesinin and other drugs act on the early ring stages (ter Kuile et al., 1993; 1992; Landau et al., 1992; Geary et al., 1989) and will enhance clearance of parasites shortly after administration, potentially preventing further development of susceptible parasites and worsening of clinical illness (Enosse et al., 2000; White, 1994). Historically, chloroquine has been the drug of choice for the treatment of non-severe or uncomplicated malaria and for chemoprophylaxis, although drug resistance has dramatically reduced its usefulness. Treatment of malaria has become more difficult with the spread of drug resistant strains of the parasite and the emergence of drug-resistant strains has become a significant health problem. Drug resistance involves mutations in the drug target so that the drug does not bind or inhibit the target as well. Fansidar resistance is correlated with specific mutations in the enzymes targeted by sulfadoxine

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and pyrimethamine (dihydropterote sythetase and dihydrofolate reductase, respectively). Chloroquine resistance has been correlated with mutations in a transporter found on the food vacuole membrane (chloroquine resistance transporter). The mechanisms involved in the resistance of malaria parasites are well documented elsewhere (Gregson and Plowe, 2005; Hastings, 2004; Hastings, 2003; Wellems and Plowe, 2001; Nzila et al., 2000). The best long-term hope for control of malaria may be a vaccine. There is an urgent need to accelerate the development of promising malaria vaccines and ensuring their availability and accessibility in the developing world and in fitting them into other prevention strategies. An effective vaccine against malaria would be a critical component to aid in the control of this disease since the parasite is fast growing resistance to drugs and the vector is developing resistance to insecticides. There are, however, many factors that make malaria vaccine development difficult and challenging. These factors are: (i) Due to the size and genetic complexity of the Plasmodium genus, each infection presents a myriad of antigens which vary throughout the different stages of its life cycle, and against which sequential consecutive immune responses are required (Aide et al., 2007). Understanding which of these antigens can be a useful target for vaccine development has been complicated, and up to date at least 40 promising antigens have been identified. (ii) The parasite changes through several life stages even while in the human host, presenting different subset of molecules for the immune system to combat at each stage. (iii) The parasite has evolved a series of strategies that allow it to confuse, hide, and misdirect the human immune system. (iv) It is possible to have multiple malaria infections of not only different species but also different strains at the same time. Pioneering work towards a malaria vaccine in the 1960’s showed that sterile immunity in humans can be attained by immunization with irradiated sporozoites (Clyde et al., 1973; Nussenzweig et al., 1967). This approach was however considered unfeasible for mass vaccination, therefore considerable efforts were directed towards identifying sporozoite components targeted by protective immune responses (Egan et al., 1993; Herrington et al., 1987). Several malaria vaccine candidates have been tested in clinical trials and many more are in various stages of development. The trials of a vaccine candidate RTS,S/AS02A, carried out in Gambia and Mozambique marked an important landmark in the history of malaria vaccine development (Bojang et al., 2001). The initial Phase IIb trials were conducted in semi immune Gambian adults, who were immunised with three doses of RTS,S/AS02A during a period of low transmission and followed up on occurrence of new infections during 16 weeks of active malaria transmission. The estimated efficacy during the first 9 weeks of follow up was 71% and zero thereafter. In another trial carried out in Mozambique in children aged 1-4 years, RTS,S/AS02A imparted 30% reduction in the incidence of clinical malaria, a 45% delayed time to first infection and a reduced incidence of clinical malaria by 58% at a 6 month follow up (Alonso et al., 2004). Ross (1916) pointed out the important differences between the dependent happenings in infectious diseases, where incidence depends on the prevalence and the independent events in non infectious diseases. Because of this dependence of events in malaria, vaccination programs that suppress transmission without eliminating it will alter existing host-parasite balances, causing complex changes in the pattern of both the infection and the disease (Halloran and Struchiner, 1992). The distinction between the three main biological characteristics, infection blocking, disease modification and transmission blocking in malaria, correspond roughly to three distinct stages of the parasite. A malaria vaccine can have

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different potential effects which include reducing mortality due to malaria, increasing recovery rate, increasing acquired immunity rate or reducing the transmission probability. An ideal vaccine for the pre-erythrocytic stage would induce high titers of functional antibodies against sporozoites to prevent all parasites from entering the liver stage, and induce potent cytotoxic T-lymphocytes immunogenicity against the liver stage to kill infected hepatocytes (Moorthy et al., 2004). Merozoite surface protein-1 (MSP-1) is the most well characterized antigen involved in invasion of red blood cells by merozoites, and is therefore the basis of several malaria blood stage vaccines. Some of the blood stage vaccines, glutamate rich protein (GLURP) and MSP-3, have been clinically assessed (Oeuvray et al., 2000; 1994). For a sexual-stage vaccine (transmission blocking vaccine), induction of antibodies to gametocyte antigens can prevent gametocyte fertilisation in the mosquito. A candidate vaccine for this stage is Pfs25, a recombinant protein (Moorthy et al., 2004).

3.

Malaria Immunoepidemiology

Immunoepidemiology is the study of the distribution of immune responses and infection in populations, and of the factors influencing this distribution (Woolhouse and Hagan, 1999). It enables the understanding of processes affecting immune responses and the influence of immunity on infection (Quinnell et al., 2004). Further, it examines how inter-individual differences in immune responses affect the population dynamics of micro- and macro-parasites to produce the epidemiological patterns of infection observed in heterogeneous host populations (Hellriegel, 2001) and it is an important part of the study of infectious diseases. By combining immunology with epidemiology, we study not only the distribution and frequency of infection and disease but we also uncover how types and levels of immunity vary over time in relation to exposure to infection or clinical presentation. At the individual level, this can reveal how factors such as frequency of infection or genetic variation affect levels of immunity and, at the community level, can reveal how immune responses influence the prevalence of infection and the burden of disease. Undoubtedly there is a close link between the within-host progression of an infection leading to certain associated immunity and the development of the disease at epidemiological level. Immune status of the hosts, however, their level of innate, naturally and/or artificially acquired immunity, has a significant impact on the spread of an infectious disease in a population. Incorporating immunological concepts in epidemic models is not new, it is often accomplished through incorporation of more classes, which sometimes increases the complexity of the model (Martcheva and Pilyugin, 2006). To capture a complex immune structure of the disease-affected individuals, some models incorporate n recovered classes (Thieme and van den Driessche, 1999; Hethcote et al., 1981). Another approach in immunoepidemiology was employed by Gilchrist and Sasaki (2002) where the epidemiology of the disease was modeled with age-since-infection SIR model while the immunology of the individual level is described by a simple predatorprey type model for the pathogen and the immune response cells. A physiological model that structures the individuals by their immune status was proposed by Martcheva and Pilyugin (2006). The results of the immunological battle between host and parasite determine the ability of the parasite to spread. In persons infected by malaria, for example, the course of infection depends heavily on the immunological status of the host (Dushoff, 1996). The variability of the host parasite specific immunity is also incorporated in epidemic models

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by subdividing the entire population of hosts into various subclasses corresponding to different levels of immune protection, some of which are; naive or completely susceptible, completely or partially immune, vaccinated and treated (Martcheva and Pilyugin, 2006). In this section, we review some of the literature that has contributed into the epidemiology of malaria due to the existence of different levels of immune protection. The different levels of immune protection for malaria infection are brought up by the immune responses elicited by the dynamic interaction between the blood stages of the parasite with the host cells. To understand the progression of the malaria parasite from host to host, we first need to discuss the models of the in host dynamics of the malaria parasite. Considerable work on intra-host models of malaria has been done by several authors. These describe the dynamics of blood stages of the parasite and their interaction with the host cells, in particular red blood cells and immune effectors (Molineaux and Dietz, 1999). This interaction is the one that gives rise to clinical manifestation of the disease and would in turn determine its spread from one host to the other. The asexual blood stages of this protozoan parasite cause the symptoms and pathology of malaria. One of the leading intra-host models is that of Anderson et al. (1989). The purpose of their modeling was to compare the effectiveness of immune responses against merozoites and parasitized red blood cells and they concluded that the immune response against merozoites is less effective than the immune response against infected red blood cells. The same model was used (Gravenor et al., 1995), without immunity to evaluate to what extent the consumption of red blood cells by P. falciparum can explain the control of the parasite density and the anaemia observed during malaria therapy. The conclusion made by Gravenor et al. (1995) was that the basic interaction of parasite and erythrocyte is insufficient to explain intra-host parasite population regulation but that the role of erythrocyte destruction by the parasite is a more important determinant of anaemia than is generally appreciated. Hetzel and Anderson (1996) assumed that super-infecting merozoites are wasted on the original model (Anderson et al., 1989) and found that the change has negligible effects on the dynamics. Some of the intra-host models of malaria have been reviewed (Molineaux and Dietz, 1999). Tumwiine et al., (2008) recently analyzed an age-structured mathematical model for the within-host dynamics of malaria and the immune system, and their results reveal that the density of the merozoites and gametocytes released by the infected red blood cells rise and then fall via damped oscillations to a stable steady state. Hoshen et al. (1998) modelled the regression of parasitaemia by P. falciparum, its subsequent elimination from the body, or recrudescence, for population of cells treated with chloroquine. The model for chemotherapy of malaria by chloroquine predicts that at doses lower than those needed for full cure, recrudescence will occur. Hoshen et al. (2000) proposed a mathematical model for the in-host asexual erythrocytic development of P. falciparum malaria by introducing a delay parameter τ as life span of infected red blood cells, equal to 48 hours in falciparum malaria and linearising the original model of Anderson et al. (1989). They also show how the level of immunity affects development of disease by simulating the effects of an induced host immune response. A mathematical model to investigate the dynamics of parasite phenotypes in a malaria-infected host, with respect to critical interactions between their immune-mediated competition, relative drug sensitivities and persistent superinfection was developed (Gurarie and McKenzie, 2006). Their conclusions support the hypothesis that immune mediated interactions can shape the spread of drug resistance, even if the phenotypic traits are not linked

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genetically. This model builds on earlier models of the within-host dynamics (Gurarie et al., 2006; Mason and McKenzie, 1999; Hertzel and Anderson, 1996). The development of acquired clinical immunity is an important feature of P. falciparum malaria because it reduces the frequency of symptomatic episodes and thus disease-associated morbidity and mortality. Development of acquired clinical immunity to P. falciparum was considered in a mathematical model (Gatton and Cheng, 2004) and it was found that time to develop clinical immunity was proportional to parasite diversity and inversely proportional to transmission intensity. The effects of immune effectors on merozoite invasion of erythrocytes and suppression of parasite production were considered (Chiyaka et al., 2008a). There is also vast literature on in-host modeling and notable work in this has been published (Dietz et al., 2006; McQueen and McKenzie, 2004; Haydon et al., 2003; Simpson et al., 2002; Gatton et al., 2001; Anderson, 1998; Gravenor and Kwiatkowski, 1998; Koella and Antia, 1995; Kwiatkowski and Nowak, 1991). Several authors have put forward mathematical models of malaria that describe the transmission of the parasite from one host to the other in completely susceptible populations (Ngwa, 2004; Freeman et al., 1999; Anderson and May, 1990; Aron, 1983, Bailey, 1982; McDonald, 1957; Tumwiine et al., 2007). Some of these models take into account the time lapse between being infected to being infectious (Freeman et al., 1999; Anderson and May, 1991; Dietz et al., 1974; Macdonald, 1957). The first to model malaria transmission including the latent period was Macdonald (1957) whose work was refined by Anderson and May (1991). The model of Anderson and May (1991) was modified (Freeman et al., 1999) by allowing people in the latent state to die or recover during their period of latency. For simplicity some models considered constant population sizes (Anderson and May, 1991; Freeman et al., 1999; Macdonald, 1957). On the other hand, Smith and McKenzie (2004) analyze the statics and dynamics of malaria infection in Anopheles mosquitoes. Some of the authors considered the dynamics of mosquitoes as one time dependent variable as vectorial capacity (Dietz et al., 1974). The effect of partial host immunity to malaria parasites is crucial to malaria epidemiology. Partial anti-parasite immunity develops only after several years of endemic exposure (Chauhan, 2007; Taylor-Robinson, 2002). Evidence suggests that this inefficient induction of immunity is partly a result of antigenic polymorphism, poor immunogenicity of individual antigens, the ability of the parasite to interfere with the development of immune responses and to cause apoptosis of effector and memory T and B cells, and the interaction of maternal and neonatal immunity. Without recurrent infection, partial immunity is relatively short-lived. The distinction between infection and disease is particularly important in malaria, since infection with the parasite does not necessarily result in disease (Oaks, 1991). Many infected people in areas where malaria is endemic are asymptomatic: they may harbor large numbers of parasites yet exhibit no outward signs and symptoms of the disease. Asymptomatic individuals are major contributors to the transmission of malaria parasites. Immunity to malaria can be subdivided into immunity against infection (mild and severe), disease and transmission. While immunity does not eradicate the disease from the body, or stop new infections, it usually prevents symptoms. Immunity to malaria is characterized by reduction in parasite burden, clinical symptoms, and prevalence of severe disease in individuals residing in an endemic area. Children are at increased risk since they have not acquired immunity and pregnant women transiently lose some of their ac-

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quired immunity due to the relative immunosuppression of pregnancy. In fact, malaria is a cause of pregnancy loss, low birth weight, and neonatal mortality (Breman et al., 2006). Individuals living in endemic areas may develop partial immunity to disease following repeated infections. These individuals are not immune to infection per se. Following a bite by an infected mosquito, they will still develop parasitaemia, but the severity of clinical symptoms is typically limited. P. falciparum infection confers only labile and partial immunity (Cancr´ e et al., 2000). This immunity only reduces the incidence of clinical malaria attacks without preventing infection (Garnham, 1949; Christophers, 1924). Antimalarial antibodies have been detected in high titers in such patients (Plebanski and Hill, 2000). The Muench (1959) catalytic model divides the population into three groups: susceptible, infected and partially immune individuals. Aron and May (1982) modified the model to an SIRS type by adding a return path from the partially immune back to the susceptible. Aron (1988) further modified the model by adding an immediate return path following the concept that partial immunity is not immediately acquired, where the effect of boosting is incorporated into the rate at which immune individuals revert to being susceptible. In the model (Aron, 1988), the rate of reversion is chosen so that the average duration of immunity corresponds to the assumption that immunity lasts until the occurrence of a gap of τ years without exposure. The conclusions made show that different formulations of boosting capture qualitatively different epidemiological features. The effect of partial host immunity on the transmission of malaria parasites have been discussed (Buckling and Read, 2001). Yang (2000) presented a transmission model of malaria for different levels of acquired immunity and acquiring immunity was treated as something that has a finite duration and requires boosting and show that the types of immunity loss rates do not appear in the basic reproductive number. Ngwa and Shu (2000) introduced in their model a class of persons who are partially immune to malaria but who may be infectious. They assumed that these immune individuals lose their immunity at a certain rate. Their model shows that if more and more humans become asymptomatic immune carriers with enough parasite load to transmit the infection to mosquitoes, then the value of the reproductive number gets larger and it then becomes difficult to control the infection. The same conclusion was also made (Chiyaka et al., 2007a) after the analysis of a model which considered delays in disease latency and partial immunity. History shows that vaccines are most easily developed for those organisms that induce natural immunity after a single infection. For malaria, vaccine strategies that are likely to be ultimately successful are those that combine many antigens to induce maximal response to protective determinants that might not be normally recognized following normal infection of naive individuals. A large body of literature exists on the search for a malaria vaccine and a number of researchers have looked at the potential impact of a vaccine on the malaria epidemic (Aide et al., 2007; Druilhe and Barnwell, 2007; Smith et al., 2006; Pouniotis, 2004; Tsuji and Zavala, 2001; de Zoysa, 1990; Anderson et al., 1989; Halloran et al., 1989). Anderson et al. (1989) assumed that the pre-erythrocytic vaccine completely blocks infection so none of the vaccinated humans become infected as long as the vaccine does not wear off. Struchner et al. (1989) built upon the model of Dietz et al. (1974) and considered natural immunity that can wane with time but its duration and effectiveness is prolonged by boosting from natural infection. The effects of stage specific vaccines were independently considered (Halloran et al., 1989). Similar to the results of Anderson et al.

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(1989), Halloran et al. (1989) showed that the total prevalence in humans is reduced more with lifelong sporozoite vaccine than with the life long gametic vaccine and the prevalence of infectives and transmission to mosquitoes is the same with both vaccines. It was shown that vaccination of less than half the population achieved over a 60% reduction in peak incidence, whereas vaccinating the entire population resulted in a reduction of 94% in peak incidence. The effects of combining all the malaria vaccine subunits were also discussed (Chiyaka et al., 2007b).

The epidemiological impact of treatment and the spread of anti malarial drug resistance has been analysed using a number of mathematical modelling studies (Baca¨ er and Sokhna, 2005; Koella and Antia, 2003; Aneke, 2002). The model proposed by Baca¨ er and Sokhna (2005) begins with resistant parasites that have been introduced in an area by migrating humans but then considers the area as closed to migration and focuses on the diffusion of resistance due to the mobility of mosquitoes. Koella and Antia (2003) used their model to investigate under which conditions drug resistance will spread in the population of hosts as well as to predict the rate of spread of resistance and concluded that resistance does not spread if the fraction of infected individuals treated is less than a threshold value, if drug treatment exceeds this threshold resistance will eventually become fixed in a population. Aneke (2002) considered resistance in his model which was developed from the basic model (Bailey, 1982). The work done so far in modelling treatment and spread of resistance either did not consider the infectiousness of the treated and the effects of partially immune humans in a population where treatment is available as a control strategy, or did not consider the treated humans to recover with or without acquiring immunity or assumed that the disease induced death rate is negligible. These factors were considered elsewhere (Chiyaka et al., 2009) and it was deduced that for treatment to effectively reduce the number of infections, a certain condition should be met if the treated do not immediately become un-infectious.

Combination of multiple interventions may be more effective than using single interventions, specifically individual interventions that may not be fully effective alone may be completely effective, either by additive or synergistic effects, when combined with other interventions. Multiple interventions may prevent complications that might arise if one of the interventions were used alone. An example includes loss of natural immunity after vector control measures are employed could be prevented by vaccines that restore natural immunity, or, spread of drug resistance parasites could be slowed by adding anti-gametocydal agents. Mathematical models have been used to assess the impact of using a single intervention strategy such as vaccination (Halloran et al., 1989) treatment and the spread of resistance (Baca¨ er and Sokhna, 2005; Koella and Antia, 2003; Aneke, 2002). Chiyaka et al., (2008b) analysed the effects of combining multiple interventions which are personal protection against mosquito bites, treatment with vaccination in the epidemiology of malaria. From these studies it was concluded that eradication of malaria using only personal protection measures from mosquito bites may be difficult because the compliance needed to achieve this may be very suggesting that multiple control strategies are to be adopted for reducing the infection in a community.

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Conclusion

Malaria is the world’s most prevalent vector borne disease and it still remains among the most devastating diseases occurring in the world, especially in Sub-Saharan Africa. It is a leading cause of morbidity and mortality (if not properly managed) in both outpatient attendance and inpatient admissions accounting for around 10% of Africa’s overall disease burden. It is caused by four species of parasites that are transmitted by Anopheles mosquitoes, with Plasmodium falciparum being the most virulent of the parasites. The most severe cases are typically limited to those who have impaired immune function or who have developed little or no immunity to malaria through previous exposure. The population group most at risk, therefore, are children under five years of age. Also, at particular risk are pregnant women, who are vulnerable because of their reduced natural immunity. Pregnant women are four times more likely to suffer from complications of malaria than non-pregnant women (Breman et al., 2006). Malaria continues to pose a high burden in both social and economic terms worldwide, ranging from school absenteeism to low productivity at workplaces. This affects agricultural production (which is generally the only source of income) and outputs from other economic sectors in the developing world. Despite the geographical variations in altitude, seasonality, and humidity, malaria is endemic in tropical regions, with most areas being holoendemic. In this review, we have provided an overview of the malaria epidemic and the dynamic mathematical models for predicting its spread, the effects of intervention strategies and some of the challenges faced in combating the disease.

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In: Infectious Disease Modelling Research Progress ISBN 978-1-60741-347-9 c 2009 Nova Science Publishers, Inc. Editors: J.M. Tchuenche, C. Chiyaka, pp. 177-196

Chapter 6

A M ATHEMATICAL M ODEL OF THE W ITHIN V ECTOR DYNAMICS OF THE Plasmodium Falciparum P ROTOZOAN PARASITE Miranda I. Teboh-Ewungkem∗, Thomas Yuster and Nathaniel H. Newman Department of Mathematics, Lafayette College, Easton, PA, 18042, USA

Abstract Based upon experimental data, a mathematical model is developed that simulates the within-vector dynamics of Plasmodium falciparum in an Anopheles mosquito. The model takes as input a mosquito blood meal and the final output is the salivary gland sporozoite load, a probable measure of mosquito infectivity. Sensitivity analysis of the model parameters suggests that reduction of gametocyte density in the blood meal most significantly lowers mosquito infectivity, and is thus an attractive target for malaria control. Model extensions and other model implications for malaria control are also discussed.

Keywords: Gametocytes, Sporogony and Sporozoite load, Infectivity, Oocysts, Transmission, Mathematical modeling within Anopheles mosquito.

1.

Introduction

Malaria, a long time human malady, continues to this day to affect huge numbers of people across the globe, but the problem is most severe in Africa. In 2006, an estimated 3.3 billion people were at risk of malaria of whom 2.1 billion were at low risk (< 1 reported case per 1000 population), and 97% of whom were living in regions other than Africa. The other 1.2 billion were at high risk (≥ 1 case per 1000 population), 49% of whom were living in the WHO African region and 37% in WHO South-East Asia regions [39]. In addition, there were an estimated 881,000 malaria deaths in 2006, of which 91% were in Africa and 85% of which were children below 5 years of age [39]. Of the four common protozoan species that transmit the disease in humans, Plasmodium falciparum (P. falciparum) is the deadliest type [29, 38, 39]. ∗

E-mail address: [email protected]. Corresponding author.

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The need to continue and expand the fight against malaria is clear, and new research tools and ideas are needed to enhance old control schemes and to develop new control schemes to better contain the disease. A thorough understanding of the P. falciparum biology and life cycle will likely prove very important in that regard, and it is to this understanding that we hope to contribute. Mathematical models to study the malaria parasite and its transmission have been and continue to be constructed by researchers. However, most of the focus have been on population models and within-host dynamics of the asexual blood stages of its life cycle (see for example [7, 16, 17, 31 and 32]), with very little discussion and focus on the within-vector dynamics of the parasite (see for example [10]). An important aspect of malaria transmission is the parasite’s ability to infect and proliferate successfully within a vector (mosquito), and to subsequently infect a non-infected human. Transmission of parasite from mosquito to human occurs when sporozoites are successfully conveyed from a mosquito’s salivary glands to the circulating blood of a human during the mosquito’s blood meal. The number of sporozoites present in the mosquito’s salivary glands appears to be a determining factor in a mosquito’s ability to infect a human. In fact, there is evidence that for Plasmodium yoelii, infectiousness is a function of sporozoite loads [25]. Though our model concerns P. falciparum, we will assume that sporozoite load is correlated with infectivity in that case as well. We now give a brief outline of the P. falciparum lifec ycle. The P. falciparum protozoan parasite has a complex, multi-stage life cycle that involves both a vertebrate, human (Host), and an invertebrate, female Anopheles mosquito (Vector). During its life cycle, it undergoes asexual reproduction in the human which will lead to the sexual component of the parasite and later undergoes sexual reproduction in the mosquito midgut. P. falciparum enters a human from the saliva of an infected female Anopheles mosquito when the mosquito draws blood during its meal. When a mosquito is infective, sporozoites, the forms of the parasite that are conveyed from the mosquito to the human, are present in the mosquito’s salivary glands and can be transmitted into the human blood stream when the mosquito draws blood. Some of these sporozoites eventually invade hepatocytes in the human liver. Upon invasion of a hepatocyte, a sporozoite transforms into a trophozoite, which undergoes asexual reproduction, eventually producing numerous merozoites. These merozoites cause the host hepatocyte to burst, which releases the merozoites into the circulating blood. These merozoites invade erythrocytes in the blood stream. After infecting an erythrocyte, a merozoite transforms into a trophozoite, undergoes another round of asexual reproduction, and produces several more merozoites that again cause the host cell to rupture. A merozoite generated from within an erythrocyte has one of two potential fates after bursting its original host cell and infecting a new erythrocyte host cell: (1) it may become a trophozoite and repeat the cycle of merozoite production, or (2) after transforming into a trophozoite, the parasite may alternatively undergo gametocytogenesis, a process in which either a male or female gametocyte is generated within the host erythrocyte. This process represents the initiation of the sexual phase of the parasite’s life cycle. The gametocyte is the only form of the parasite that is infectious to the mosquito [29]. When mature male gametocytes (microgametocytes) and mature female gametocytes (macrogametocytes), the forms that must be transmitted to a mosquito for the parasite to continue its life cycle are present within human erythrocytes, the human is said to be infective. When a female mosquito feeds, by drawing blood from an infected human, it may pick

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up these matured male and female gametocytes that are present within some of the ingested human erythrocytes in the blood meal. After ingestion, within the mosquito gut the male and female gametocytes undergo gametogenesis, losing the structures associated with the erythrocyte host cell and respectively transforming into male and female gametes. Each female gametocyte emerges completely giving rise to a single non-motile female gamete, whereas the male gametocyte frequently is initially trapped within the host erythrocyte plasma membrane, but eventually undergoes the process of exflagellation producing up to eight multiple flagellated male gametes [3, 6, 35]. After gametogenesis, a male gamete finds and fertilizes a female gamete, generating a zygote within the mosquito gut. The zygote undergoes further development into an ookinete, which invades the mosquito midgut lining and ultimately develops into an oocyst. The process of sporogony then occurs, which results in the generation of multiple sporozoites within the oocyst. The sporozoites cause the oocyst to burst, and they are released into the mosquito’s circulation. Approximately 25% [3, 4], of these sporozoites successfully reach their intended destination, the mosquito salivary glands, where they can be transmitted through saliva to a new human host during future blood meals, beginning a new parasitic life cycle, Figure 1.

Sporozoites (in salivary glands) SPOROGONY

Oocysts

Mosquito to Human Transmission

Ookinete

Zygotes

FERTILIZATION

Gametes (male and female) GAMETOGENESIS

Gametocytes (male and female)

Human to Mosquito Transmission

Figure 1. Anopheles mosquito stages of the life cycle of the Plasmodium falciparum parasite. Our model will be of the within-vector dynamics of P. falciparum, taking as input the blood meal and with the final output being the sporozoite load in the mosquito salivary glands. This is not the only factor that can affect mosquito infectivity [5], but it is likely an important measure of infectivity and our model will focus on the sporozoite load to see how best to reduce it and hence the mosquito infectivity. This chapter is divided up as follows: The basic mathematical model is developed in Section 2, and in Section 3 model parameter values are estimated and initial conditions are

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Miranda I. Teboh-Ewungkem, Thomas Yuster and Nathaniel H. Newman Table 1. Dependent variables and their description

Variables M F Z O S

Description Number of Male gametes produced in the mosquito’s midgut from one feeding Number of Female gametes produced in the mosquito’s midgut from one feeding Number of Zygotes produced in the mosquito’s midgut from one feeding Number of Oocyst in the mosquito midgut epithelium as a result of one feeding Number of Sporozoites in mosquito salivary glands

Unit Number Number Number Number Number

discussed, all based on empirical data. In Section 4, the model is solved numerically and the results are presented. Section 5 contains a sensitivity analysis of important parameters and initial conditions and Section 6 discusses some possible extensions of the basic model. Section 7 presents our conclusions and discusses implications for prevention and control strategies.

2.

The Basic Mathematical Model

The basic model we will develop will take the form of a nonlinear deterministic continuoustime system of differential equations. This is not the only reasonable approach. We will attempt to model the stages of P. falciparum parasite development in the mosquito, but we will make many simplifying assumptions. We assume that the ambient temperature is about 26◦ C and that the initial conditions are the result of a single blood meal. We will discuss those initial conditions in the next section of the paper. The independent variable is t, which is the time in days since the ingestion of the blood meal. The dependent variables are M , the number of male gametes (called microgametes), F , the number of female gametes (called macrogametes), Z, the number of zygotes, O, the number of oocysts, and S, the number of sporozoites in the mosquito salivary glands. All these variables are described in Table 1. When a mosquito takes a blood meal from an infectious human, it picks up G0 number of gametocytes from the human, of which a fraction m are males and the rest (1 − m) are females. Within minutes, gametogenesis begins, [3, 27] and see Figure 2, which results in the formation of the respective male and female gametes. The male gametes, M , either die or fertilize female gametes, and female gametes, F , either die or are fertilized by males. We assume a constant per microgamete death rate a (per day), and a constant per macrogamete death rate b, (per day). We also assume that the fertilization rate will be proportional to both the number of male gametes and the number of female gametes, and we call this constant

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Figure 2. Figure showing the transitional stages of the Plasmodium parasite and the time frame from gametocyte ingestion to spotozoite formation. This diagram was taken from Baton et al., [3]. of proportionality r (per number of parasites per day). Thus: dM = −aM − rM F, dt dF = −bM − rM F. dt

(1) (2)

Notice that starting with an initial number of male and female gametes, which will be denoted as M0 for the male gamete population and by F0 for the female gamete population, each gamete population decays down to zero. Upon fertilization of a female gamete by a male gamete (which occurs, fairly quickly, within an hour of gamete formation [3]), a zygote, Z, is formed. This zygote will transform to an ookinete, then to an oocyst, and finally burst to release sporozoites, or will fail/die somewhere in the chain. There are various levels of detail a model of this chain might have. The simplest model would have a single equation involving sporozoite formation based only on Z. But, the number of oocysts has been an important measure of mosquito infectivity and is relatively easy to measure experimentally [1, 4, 13, 14, 26, 33], so including O, the number of oocysts, in the model seems reasonable. Thus, a zygote will either eventually transform into an oocyst or die, and we assume constant per zygote rates, σz = 1/(transformation time) (per day), and µz = 1/(life time of zygotes) (per day). Thus: dZ = rM F − µz Z − σz Z. dt

(3)

An oocyst will either undergo mitosis and sporoblast formation to eventually burst to release sporozoites or die/fail. We assume that the per oocyst failure rate, µo (per day), is constant throughout the process. The bursting term is more complex. Oocysts take several days to mature and only then are they able to burst and release sporozoites. We will model this with

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the variable coefficient k(t), where k (t) =



0

if 0 < t < t0 . d, if t ≥ t0

(4)

and where d (per day) being the rate of transformation from a mature oocyst to sporozoites that will be released upon bursting of the oocyst, and t0 being the time (in days) since gametocyte ingestion for a mature oocysts (required for sporoblast formation) to be established. We will discuss the basis for this particular form for k(t) in the next section. Therefore, the equation that models the oocyst population is dO = σz Z − µo O − k (t) O. (5) dt It is worth noting that once oocysts have been established, they tend to survive the entire sporogony period, [3, 4], although mortality is an occasional outcome. Hence µo is expected to be small. Once the oocysts mature, sporozoite production commences immediately and sporozoites are formed within the oocysts. When the oocysts burst, n sporozoites are released per mature oocyst, of which a fraction p will successfully migrate and invade the salivary glands. Evidence suggests that almost all of sporozoites that make it to the salivary glands will persist for the remainder of the adult mosquito life [4, 12]. Therefore, the equation that models the sporozoite population is dS = npk (t) O. (6) dt Notice that the equation that models the sporozoite population in the salivary glands can be easily obtained once O (t) is known. Combining all our equations together yields the system dM = −aM − rM F, dt dF = −bM − rM F dt dZ = rM F − µz Z − σz Z (7) dt dO = σz Z − µo O − k (t) O dt dS = npk (t) O. dt All the parameters to be used in our model and those shown in the model equation (7) are non-negative and are compiled and shown on Table 2.

3.

Initial Conditions and Parameters

The first goal of this chapter is to validate the model (7), using empirical data to compare the results obtained against biologically observed results. To do so, a set of initial conditions must be specified and values for model parameters determined, using empirical data.

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Table 2. The parameters for the model and their description Parameters, Initial conditions and their Description a, Death/failure rate of male gametes (microgamete) b, Death/failure rate of female gametes (macrogametes) r, Fertilization rate of male and female gametes µz , Death rate of zygotes µo , Death rate of oocysts σz , Transformation rate of zygotes k (t) , Rate of sporozoite production n, Number of sporozoites produced per oocysts p, Fraction of sporozoites that successfully migrate and invade salivary glands m, Proportion of gametocytes that are microgametocytes (male gametocytes) α, Fraction of male gametocytes that successfully undergo gametogenesis β, Fraction of female gametocytes that successfully undergo gametogenesis υ, Number of female gametes produced per female gametocyte ρ, Number of male gametes produced per male gametocyte G0 , Number of gametocytes picked up in a blood meal M0 , Initial Number of male gametes F0 , Initial Number of female gametes

Unit day−1 day−1 Number−1 ×day−1 day−1 day−1 day−1 day−1 Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Number Number Number

At the onset of the plasmodium life cycle in the mosquito, a number of gametocytes are picked up in a blood meal, where blood meal sizes range from 2 to 10 µL [11, 24, 37]. From this number of gametocytes, G0 , the number of male, M0 , and female, F0 , gametes produced via gametogenesis can be estimated. To estimate M0 and F0 , we must estimate the proportion of ingested gametocytes that are microgametocytes (male gametocyte) and macrogametocytes (female gametocytes), the number of male and female gametes produced per the corresponding sex gametocyte, and the fraction of male and female gametocytes that successfully undergo gametogenesis. From this we can estimate that the initial values, M0 , and F0 , are M0 = mG0 ρα; F0 = (1 − m) G0 υβ, (8) where m is the proportion of ingested gametocytes that are male gametocytes, ρ is the number of male gametes produced per male gametocyte, α is the fraction of male gametocytes that successfully undergo gametogenesis, υ is the number of female gametes produced per female gametocyte and β is the fraction of female gametocytes that successfully undergo gametogenesis. These parameters are all non-negative and will be estimated from empirical data. They appear in Table 2. The number of P. falciparum gametocytes, G0 , picked up in a blood meal is known to vary considerably [3, 9, 22, 24, 30, 37], even for mosquitoes that fed on the same humans [24]. This number also varied depending on the age of the human, the season, the infectiousness level of the human and other factors [22, 37]. Values ranging from 4 × 100 to 1.8 × 105 gametocytes per µL of blood have been observed in patients [3, 9, 22, 24, 30, 37], which when multiplied by the size of a blood meal will give ranges from 8 × 100 to 1.8 × 106 . However, most of the mean number of gametocytes in a blood meal observed and quoted fall within a range of 2 × 101 − 1 × 103 , which we will take as our possible range for G0 . The ratio of male gametocytes to female gametocytes ingested in the blood meal, m,

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also varies [28, 29, 36, 37] depending on a host of conditions, some of which are the endemic malaria location, and whether there is multiple infection from other plasmodium species, i.e., heterogeneity of infection. Some of the values observed are 0.22 in Cameroon [28, 36], 0.346 in Senegal [28], 0.18 in Papua New Guinea [36]. Averages of 0.076 have also been reported [28], but typically observed is 1 male to 3 to 4 females [29]. Thus 0.076 ≤ m ≤ 0.35 is a possible range of values for m, but a value of 0.25 will be used for the basic model. The value of α, the fraction (efficiency ) of male gametocytes that successfully undergo gametogenesis to become viable gametes is assumed to be at most 0.4 because of multiple developmental abnormalities that occur during microgametogenesis. There is evidence that as many as 60% of male gametes that develop are anucleate (no nucleus) [6, 27], with others having 2 or more nuclei and still others failing to detach from the residual body and/or failure of plasmalemma [6], making them non-functional. Hence, a suitable range for α is 0 < α < 0.4. On the other hand, the fraction of female gametocytes that successfully undergo macrogametogenesis is very high, hence higher efficiency, and so a value of β close to one is reasonable. This is based on the evidence that female gametogenesis (macrogametogenesis) is relatively simple and involves relatively little of the complexities and abnormalities involved in male gametogenesis [6]. Once the process of gametogenesis is successful, a male gametocyte will give rise to ρ male gametes where from empirical data, ρ falls in the range 4-8 as observed and quoted in [3, 6, 35, 36]. However, a female gametocyte will give rise to only one female gamete [3, 6], and so the value of υ, the number of female gametes produced per female gametocyte, will be taken to be one. For the basic model, the value of ρ will be taken to be six and the value of υ taken to be 1. The initial values M0 and F0 can now be computed from an estimate of G0 and from these estimates of m, ρ, α, υ and β. We next estimate the parameters a and b. It takes between 10-30 minutes for male gametes to arise via gametogenesis after a mosquito has taken a blood meal while female gametes arise fairly quickly, within the first 5 minutes [3]. Male gametes are short-lived and this is so because they have no mitochondrion [3, 6]. Once the male and female gametes emerge, fertilization by fusion of the male gametes to the female gametes occurs within the next one hour [3, 4, 6]. Any gametes that are not involved in fertilization will be broken down by proteases, which are enzymes secreted from the mosquito’s midgut [3]. Hence, the lifespan of a male gamete can be estimated to be between 40 to 50 minutes and that for a female gamete between 55 to 60 minutes. From these values, the parameters a = 1/(lifespan of male gamete) and b = 1/(lifespan of the female gamete) can be estimated. There is no direct empirical evidence useful for estimating the rate constant r, the fertilization rate, associated with fusion of the male and female gametes to form a zygotes. However, r will be chosen such that the number of oocysts generated is within biologically observed numbers, which in Africa is typically between 1-5 oocysts per infected mosquito [4, 14, 26, 33], with even much higher numbers if blood factors are considered [14]. Gouagna et al., [14] observed a range of 1-105 in their field study of P. falciparum malaria disease manifestations in humans and transmission to Anopheles gambiae in Western Kenya, but such high end range values are not typically observed. In the simulation of the basic model, the parameter, r, will be chosen such that the maximum number of oocysts obtained is at most 5 (see Section 4.). We believe that a feasible range for r is 0-0.22. A

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sensitivity analysis on this parameter will be carried out in Section 5.. The zygote transformation commences, through meiosis, about an hour after the blood meal [3], and has a time frame of 24 to 30 hours [4, 12]. It ends with an establishment of an oocyst. Therefore, σz , the transformation rate of zygote to oocyst is between 24/29 to 24/23 per day. Within two days, zygotes that do not transform are ingested by proteases [3], and so the lifespan of a zygote can be taken to be between 1 to 2 days, and thus the death rate of a zygote µz = 1/(lifespan of the zygote) can be estimated. With the establishment of the oocysts by day two, maturation of the oocysts takes place, in which the oocysts stretch the basal lamina and increase in thickness, reaching maturation and a maximum thickness by days 5-7 as quoted by [3] or days 6-9 as quoted by [5]. At the end of this maturation process, sporoblast formation begins via mitosis to produce sporozoites. It can take between 1-2 weeks for sporozoites to be released. In fact, Abraham [1] and Beier [4], both state that sporozoite release occurs at least 10 days after the blood meal. Hence, once an oocyst matures and sporoblast formation commences, we can estimate a range for the rate of conversion from mature oocyst to sporozoites to be  0 if 0 < t < 10 . (9) k(t) = 1/9 − 1/7 if t ≥ 10 Note that once oocysts have been established, they will almost always survive the entire sporogony period [3], so we will take µo , the death rate of oocysts, to be zero. The number of sporozoites produced per successful oocyst, n, is about 103 − 104 [3, 4, 5], a number that is extremely variable within each mosquito and between mosquitoes. Some of the released sporozoites will find their way to invade the salivary glands. From empirical evidence, Beier [4, 5], and Baton et al., [3], reported that about a quarter or less of sporozoites produced by oocysts successfully invade the salivary glands. Hillyer et al., [15], also quoted a range of 10 − 20%. Hence, the fraction, p, can be taken to be in the range 0.1-0.25. The parameters and their possible values or range of values are summarized in Table 3.

4.

Solution and Results from the Basic Model

The basic model equations (7) were solved numerically using the initial conditions and parameter values from Table 3, together with M (0) = M0 , F (0) = F0 , Z (0) = 0, O (0) = 0 and S (0) = 0. A fourth order Runge-Kutta method was used with 1000 steps per day, resulting in a step size of slightly less than 1.5 minutes. When analyzing the results of the model, it is useful to think of the model not as describing the within-vector process in a single mosquito, but as describing an average of processes occurring in many identical mosquitoes. Figure 3 shows the solution curves for the male and female gamete parasite populations during the first day of the within-vector process. From this graph, it can be seen that there is a sharp decrease in the male and female gamete populations with time, with the male population dropping to slightly less than 20% of its initial value within 0.05 days = 1.2 hours. The female gamete population declines only slightly more slowly, dropping to slightly more than 20% of its initial value in that same time. This means that the fertil-

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Miranda I. Teboh-Ewungkem, Thomas Yuster and Nathaniel H. Newman Table 3. Parameters, their description, their values/range of values and the parameters used in the numerical simulation

Parameters m, Proportion of gametocytes that are microgametocytes (male gametocytes) α, Fraction of male gametocytes that successfully undergo gametogenesis β, Fraction of female gametocytes that successfully undergo gametogenesis υ, Number of female gametes produced per female gametocyte ρ, Number of male gametes produced per male gametocyte a, Death/failure rate of male gametes (microgametes) b, Death/failure rate of female gametes (macrogametes) r, Fertilization rate of male and female gametes µz , Death rate of zygotes σz , Transformation rate of zygotes µo , Natural death rate of oocysts k (t) , Rate oocysts transform to sporozoites n, Number of sporozoites produced per oocyst p, Percentage of sporozoites that successfully migrate and invade salivary glands G0 , Number of gametocytes ingested in a blood meal

Values/range of values

Values used in Simulation

0 < m ≤ 0.36 (dimensionless)

0.25

0 < α < 0.4 (dimensionless)

0.39

0.9 < β ≤ 1 (dimensionless)

0.96

1 (dimensionless)

1

4 − 8 (dimensionless)

6

24 (50/60)

−

24 (40/60)

(day−1 )

24 (45/60)

24 (60/60)

−

24 (55/60)

(day−1 )

24 (55/60)

0 − 0.22 (Number−1 ×day−1 ) 1 − 1 (day−1 ) 2 24 − 24 (day−1 ) 29 23 0  0, if 0 < t < 10 (day−1 ) 1 − 71 , if t ≥ 10 9 103 − 104 (dimensionless)

0.011 0.5 0.85 0  0, if 0 < t < 10 1 , if t ≥ 10 7 4 10

0.1 − 0.25 (dimensionless)

0.22

2 × 101 − 1 × 103 (Number)

325

ization term, rM F , has dropped to about 4% of its initial value. Thus, virtually all zygote formation occurs within the first 1.2 hours. Figure 4 shows the solution curves of the zygote parasite population superimposed on the male and female gametes curves during the first two days. Recall that in the basic model, there is no ookinete equation, so Z represents both zygotes and (later) ookinetes. By day 2, Z is almost 0, which is consistent with the biological phenomenon. Notice that there is a huge drop in the number of parasites from 325 gametocytes, to a maximum of a little under 8 (zygotes) parasites, an over 40 fold (multiplicative) reduction. This drastic reduction has been observed and quoted by others [12]. Figure 5 shows the oocyst solution curve together with the zygote solution curve during the entire within-vector process. Note that by day 2, almost all of the oocysts have been established. Sporozoite release begins 10 days after the blood meal, hence the decreasing oocyst value after day 10, with maturation occurring between days 2 and 10. Figure 6 shows the sporozoite abundance (numbers) in the mosquito salivary glands. Notice that the populating of the salivary glands by sporozoites commences at day 10 and increases to about 104 , a number that is within experimentally observed ranges [3, 4]. The results above simulate P. falciparum dynamics within an infected mosquito. The

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Figure 3. The male and female gamete solution curves during the first hour of the withinmosquito dynamics of the P. falciparum parasite life cycle. - - - - represents the male gamete solution curve and —— represents the female gamete solution curve.

Figure 4. The zygote, male gamete and female gamete solution curves during the first 2 days of the parasite life cycle in an infected mosquito. · · ·· represesents the zygote solution curve, - - - - represents the male gamete solution curve and —— represents the female gamete solution curve. graphs follow what is biologically expected as described earlier in this chapter. The sporozoite load increases to about 104 , of which about 100 or less will be transmitted to a human during a blood meal, for about 95% of mosquitoes [4]. An important question to ask is whether high sporozoite load in the salivary glands can be used as a measure of mosquito infectivity. In his paper [4], Beier stated that the biological basis of sporozoite infectivity is unknown and that there were no biochemical or molecular markers known (at that time) for assessing sporozoite infectivity. Currently, we do not know of any in existence, hence

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Figure 5. The oocyst and zygote solution curves for the Plasmodium falciparum parasite within the mosquito. - - - - represesents the zygote solution curve and —— represents the oocyst solution curve.

Figure 6. Sporozoite profile in the mosquito salivary glands during the life of an infected mosquito. it is impossible to assess the infectivity of sporozoites in mosquitoes with great confidence. However, a long-standing assumption has been to consider that sporozoites in the salivary glands of wild mosquitoes are infectious [4]. Therefore, we think it is reasonable to consider the sporozoite load in the salivary glands, which is our model’s final output, as a measure of mosquito infectivity.

5.

Sensitivity Analysis of the Parameters

To better understand the relative importance of the various parameters and initial conditions on the final sporozoite load within the mosquito salivary glands, we carry out a sensitivity analysis of the model equations (7). Different feasible values of r (the fertilization rate), G0 (the number of gametocytes ingested in a single blood meal) and of a and b (the death/failure rates of the male and female gametes) are used to solve the model equations and their impact on reducing the sporozoite load in the salivary glands is analyzed at the end of the

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mosquito’s life, which is taken to be 24 days [31]. All other parameters will be held fixed and are as shown on Table 3. Figure 7 shows the sporozoite load for different values of the number of ingested gametocytes in the blood meal. As the number of gametocytes increases, the number of sporozoites in the salivary glands increases, and this increase is more than linear in G0 . This indicates that raising the gametocyte density is extremely dangerous, as it leads to rapidly increasing sporozoite numbers in the salivary glands, and hence, higher mosquito infectivity. Conversely, lowering gametocyte density should have a significant effect on reducing mosquito infectivity.

Figure 7. The number of sporozoites in the mosquito’s salivary glands as a function of the number of gametocytes ingested in a blood meal. Figure 8 shows the sporozoite load for different fertilization rates. The higher the fertilization rate, the higher the sporozoite load. However, this increase is somewhat less than linear, which means that the model is not nearly as sensitive to changes in r as it is to changes in G0 . Figure 9 shows the sporozoite load for different male and female gamete death/failure rates, with all other parameters as shown on Table 3. From the curves, it can be seen that changes in the death rates of the male and female gametes become significant in reducing sporozoite abundance relatively slowly, but can be quite significant when the male and female gametes are killed off fairly quickly after they have been formed. For example, if they are killed off within the first 5 minutes of their formation (equivalent to a = b = 288), the final load drops from over 10, 000 to under 2, 000. The above results suggest that the parameter with the greatest significance in reducing sporozoite load for the basic model is the initial number of gametocytes that are ingested in a blood meal. The fertilization rate and the gamete death rates are also important but less so. However, a drastic increase in the death rate of the male and female gametes, or a drastic decrease in r, also yielded significant salivary gland sporozoite reduction.

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Figure 8. The number of sporozoites in the mosquito’s salivary glands as a function of the fertilization rate, r.

Figure 9. The number of sporozoites in the mosquito’s salivary glands as a function of the male and female gamete death/failure rates. The curve - - - - represents the change with a, holding b constant at 24/ (55/60) per day while the curve —— represents the case when the change is with b, holding a constant at 24/ (45/60) per day.

6.

Model Extension

The primary advantage of the basic model is its simplicity. The equations are almost completely decoupled. Using initial conditions, equations (1) and (2) can be solved for M (t) and F (t). These functions are then input into equation (3), which is solved for Z(t). Now Z(t) goes into equation 5, etc... Thus, the dynamics can be studied locally, equation by equation. However, it may be of use to extend the model by adding complexity to better model specific aspects of the entire within-vector process. As an example, we look more closely at the fertilization process. When fertilization has been studied in vitro, it has been noted that male gametes often

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bind to previously fertilized female gametes; as many as 12 male gametes bound to a previously fertilized female gamete has been observed [6]. It appears to be unknown whether or not such an effect is important in vivo, but that remains a distinct possibility. At a minimum, male gametes bound to such a female would have little or no opportunity to fertilize another female gamete, so if we think of M (t) as the number of “free” male gametes, these bound gametes should be removed from consideration. We would expect the number of such bound gametes to be proportional to the number of interactions between free male gametes and fertilized female gametes, thus proportional to both M (t) and Z(t). Thus, equation (1) becomes: dM = −aM − rM F − sM Z, (10) dt where s (Number−1 ×day−1 ) is the constant of proportionality and measures the rate at which male gametes cling to zygotes. Note that the addition of this new term couples equation (3) to equations (1) and (2). Since this effect has not been observed in vivo, there is little to suggest reasonable ranges for the parameter s. The modification in equation (1) is a reasonable first approximation of the bound gamete phenomenon, but there is an obvious problem. In the basic model, Z represents both the zygote and ookinete stages of the life cycle process, yet a male gamete can only bind to a zygote, not an ookinete. One way to address this issue is to add an ookinete equation based on what is known about this part of the process. Upon fertilization of the male and female gametes (which occurs fairly quickly within an hour of gamete formation [3]), zygotes Z, are formed, which will either transform into motile ookinetes via meiosis, at rate σz = 1/(transformation time) (per day), or dies (without transformation) via digestion by proteases secreted from the mosquito’s midgut epithelium at rate µz = 1/(life time of zygotes) (per day). These proteases also digest early undifferentiated ookinetes and also gametocytes and gametes that fail to transform into zygotes and hence ookinetes [1, 3]. The resulting formed motile ookinetes, migrate through the blood meal crossing the peritrophic matrix (PM)1 and invading the midgut epithelium [1, 3, 21], where they (ookinetes) transform into early oocysts, a mechanism that is still quite unclear [3, 4]. If we let E(t) be the ookinete population (in numbers), we get a new equation dE = σz Z − µE E − σE E, dt

(11)

where µE (per day) is the death rate of oocysts and σE (per day) is the rate of transformation from oocyst to ookinete; a modification in the Oocyst equation (5) to obtain dO = σE E − µo O − k (t) O, dt

(12)

and a new value for σz , namely 1/(the mean transition time from Zygote to Ookinete). One of the questions our basic model does not answer is why m, the fraction of male gametocytes to all gametocytes, is significantly less than 0.5 in nature. Starting with various 1

PM is a thick extracellular sheath that completely surrounds the blood meal. Because of its thickness, it posses a major partial and natural barrier against parasite invasion of the midgut and can be a potential for control. Modification to the PM may lead to a complete barrier to malaria infection [1, 8]

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initial values of G0 , we varied m with the aim of maximizing sporozoite production. In all cases, the basic model gave optimal values of m that were very close to 0.5. While including the bound gamete effect in our model did drop sporozoite production significantly, it had little effect on the optimal value of m. What did significantly decrease the optimal m value was the inclusion in equation (1) of an M 2 term, that is, dM = −aM − rM F − δM 2 . dt

(13)

This could be thought of as a male gamete competition term, with rate δ (Number−1 ×day−1 ). There is, however, no experimental evidence suggesting that such competition, as modeled above, does exist. Certainly, one of the oversimplifications of the basic model is that it treats the blood meal as a passive deliverer of G0 . But, there is considerable evidence that the blood meal also delivers human immune factors that interfere with the within-vector processes [14]. Modeling this effectively requires more experimental evidence than we have. Assuming the immune factors attack gametes, we might model this by making a and b functions of the blood meal. But, we need to know much more to decide on functional dependencies. For example, how does the strength of the interference correlate with G0 , if at all? And is there an asymmetry in the interference in terms of male gametes and female gametes? We note that such an asymmetry might help explain the observed value of m, but we know of no evidence that such an asymmetry exists. There is still much to be learned before this part of the process can be modeled effectively. Many candidate vaccines against malaria are in development [20, 34], and the prospect of transmission blocking vaccines (to prevent parasite development in the mosquito midgut) is being discussed [4, 20]. There is evidence that antibodies against zygote or ookinete surface proteins can block transmission of malaria by preventing further development of the parasite inside the mosquito vector [20], which can lead to a reduction in the oocyst density per mosquito [19] and hence lower sporozoite load. Our model can be extended to include the probable effects of such vaccines, hopefully to gain a better understanding of how those vaccines might affect mosquito infectivity. As mechanisms that affect sporozoite load and/or infectivity continue to be elucidated (see for example [2, 4, 18, 23]), we are hopeful that our model can be extended to include them as well.

7.

Conclusion and Discussion

A model has been developed to study the dynamics of the within-vector part of the P. falciparum life cycle. Analysis of data from the basic model 7 using a base set of empirical data shows that the model dynamics conform well to the dynamics described in the biological literature [3, 4, 15], see Figures 3-5. The sporozoite load, which we took as a measure of vector infectivity, was computed and this number also conformed to biological literature. In addition to gaining a better understanding of the within-vector dynamics, one of our goals was to investigate mechanisms for significantly reducing sporozoite load. Hence, a sensitivity analysis of the basic model was carried out to determine the relative importance of some of the parameters and initial values in that regard. From the results, it was observed

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that reducing the initial gametocyte abundance (see Figure 7) led to a more than linear reduction in the sporozoite load and hence mosquito infectivity. This suggest that in order to control the spread of malaria from human to host and from host to human, targeting the gametocyte abundance in the human population by taking gametocidal drugs to reduce the load will lead to a significantly lower infectivity of the mosquito. We also found that reducing the fertilization rate will lead to a reduction in the sporozoite load, but not as significantly, Figure 8. However, ways of targeting this parameter are not apparent to us. Further experimental research investigating the fertilization process might well elucidate such mechanisms. Increasing gamete death/failure rates also produced a reduction in the sporozoite load. However, significant impact was achieved only when major reductions of the male or female gametes lifespan were made, Figure 9. Factors from the human host that enter the mosquito in the blood meal might accomplish this task, via drugs ingested by humans or as the result of a vaccine. Certainly, this is an approach that is worth investigating. An ideal control strategy will be one that targets all these parameters to yield a much smaller sporozoite load. This is important since almost all the sporozoites that make it to the salivary glands are infective [4]. The basic model does not address a variety of other approaches to decreasing infectivity, such as some of the strategies for transmission blocking vaccines. However, the basic model can be extended in several ways. Some of these extensions (see Section 6.) are currently under investigation.

References [1] Abraham, E.G. and Jacobs-Lorena, M. (2004). Mosquito midgut barrier to malaria parasite development. Insect Biochemistry and Molecular Biology, 34 ((7), pp. 667671. [2] Akhouri, R.R., A Sharma, A. Malhotra, P. and Sharma, A. (2008). Role of Plasmodium falciparum thrombospondin-related anonymous protein in host-cell interactions. Malaria Journal, 7 (63), http://www.malariajournal.com/content/7/1/63. [3] Baton, L.A. and Ranford-Cartwright, L.C. (2005). Spreading the seeds of millionmurdering death: metamorphoses of malaria in the mosquito. TRENDS in Parasitol., 21 (12), pp. 573-580. [4] Beier, J.C. (1998). Malaria parasite development in mosquitoes. Annu. Rev. Entomol., 43, pp. 519–543. [5] Beier, J.C. and Vanderberg, J.P. (1998). Sporogonic development in the mosquito. In: Malaria: Parasite Biology, Pathogenesis, and Protection, (Sherman, Irwin W, ed.). ASM Press, Washington, DC., pp. 49-62. [6] Carter, R. and Graves, P.M. (1988). Gametocytes. In: Malaria: Principles and Practice of Malariology (Wernsdorfer, W.H. and McGregor, I., eds.). Churchill Livingstone, Edingburgh. Vol 1, pp. 253-306.

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[7] Diebner, H. H., Eichner, M., Molineaux, L., Collins, W. E., Jeffery, G. M. and Dietz, K. (2000). Modelling the transition of asexual blood stages of Plasmodium falciparum to gametocytes. J. Theor. Biol., 202 (2), pp. 113-127. [8] Dinglasan, R.R., Devenport, M., Florens, L., Johnson, J.R., McHugh, C.A., Donnelly-Doman, M., Carucci, D.J., Yates 3rd, J.R., Jacobs-Lorena, M. (2009). The Anopheles gambiae adult midgut peritrophic matrix proteome. Insect Biochemistry and Molecular Biology, 39 (2), pp. 125-134. [9] Drakeley, C.J., Secka, I., Correa, S., Greenwood, B.M. and Targett, G.A.T. (1999). Host haematological factors influencing the transmission of Plasmodium falciparum gametocytes to Anopheles gambiae s.s. mosquitoes. Tropical Medicine and International Health, 4 (2), pp. 131-138. [10] Drexler, A.L., Vodovotz, Y., and Luckhart, S. (2008). Plasmodium development in the mosquito: biology bottlenecks and opportunities for mathematical modeling. TRENDS in Parasitol., 24 (8), pp. 333-336. [11] Encyclopedic reference of Parasitology, Biology. Structure. Function. (2001). (Heinz Mehlhorn ed. ), 2nd Ed, Springer, pp. 381. [12] Ghosh, A. Edwards, M.J. and Jacobs-Lorena, M. (2000). The journey of the malaria parasite in the mosquito: Hopes for the new century. Parasitology Today, 16 (5), pp. 196-201. [13] Gouagna, L.C., Mulder, B., Noubissi, E., Tchuinkam, T., Verhave, J.P., Boudin, C. (1998). The early sporogonic cycle of Plasmodium falciparum in laboratory-infected Anopheles gambiae: an estimation of parasite efficacy. Tropical Medicine & International Health, 3 (1), pp. 21-28. [14] Gouagna, L.C., Ferguson, H.M., Okech, B.A., Killeen, G.F., Kabiru, E.W., Beier, J.C., Githure, J.I. and Yan, G. (2004). Plasmodium falciparum malaria disease manifestations in humans and transmission to Anopheles gambiae: a field study in Western Kenya. Parasitology, 128 (3), pp. 235-243. [15] Hillyer, J.F., Barreau, C. and Vernick, K.D. (2007). Efficiency of salivary gland invasion by malaria sporozoites is controlled by rapid sporozoite destruction in the mosquito haemocoel. Int. J. Parasitol. 37 (6), pp. 673-681. [16] McKenzie, F.E., Bossert, W.H. (1997). The dynamics of Plasmodium falciparum blood-stage infection. J. Theor. Biol., 188 (1), pp. 127-140. [17] McKenzie, F.E., Bossert, W.H. (1998). The Optimal production of gametocytes by Plasmodium falciparum. J. Theor. Biol., 193 (3), pp. 419-428. [18] M´enard, R. (2000). The journey of the malaria sporozoite through its hosts: two parasite proteins lead the way. Microbes and Infection, 2 (6), pp. 633-642.

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[19] Miura, K., Keister, D.B., Muratova, O.V., Sattabongkot, J., Long, C.A. and Saul, A. (2007). Transmission-blocking activity induced by malaria vaccine candidates Pfs25/Pvs25 is a direct and predictable function of antibody titer. Malaria Journal, 6(107), http://malariajournal.com/content/6/1/107. [20] Mlambo, G., Maciel, J. and Kumar, N. (2008). Murine model for assessment of Plasmodium falciparum Transmission-Blocking Vaccine using uransgenic Plasmodium berghei parasites Expressing the target antigen Pfs25. Infection and Immunity, 76 (5), pp. 2018-2024. [21] Moreira, C.K., Marrelli, M.T. and Jacobs-Lorena M. (2004). Gene expression in Plasmodium: from gametocytes to sporozoites. Int. J. Parasitol, 31, pp. 1431-1440. [22] Nwakanma, D., Kheir, A., Sowa, M., Dunyo, S., Jawara, M., Pinder, M., Milligan, P., Walliker, D., Babiker, H.A. (2008). High gametocyte complexity and mosquito infectivity of Plasmodium falciparum in the Gambia. Int J Parasitol. 38 (2), pp. 219-227. [23] Pancake, S.J., Holt, G.D., Mellouk, S. and Hoffman, S.L. (1992). Malaria sporozoites and circumsporozoite proteins bind specifically to sulfated glycoconjugates. J. Cell Biol. 117, pp. 1351–1357. [24] Pichon, G., Awono-Ambene, H.P. and Robert, V. (2000). High Heterogeneity in the number of Plasmodium falciparum Gametocytes in the bloodmeal of mosquitoes fed on the same human. Parasitology, 121, pp. 115-120. [25] Pumpuni, C.B, Mendis, C. and Beier J.C. (1997). Plasmodium yoelii sporozoite infectivity varies as a function of sporozoite loads in Anopheles stephensi mosquitoes. J. Parasitol., 83, pp. 652-655. [26] Robert, V., Le Goff, G., Toto, J.C., Essong, J. and Verhave, J.P. (1994). Early sporogonic development in local vectors of Plasmodium falciparum in rural Cameroon. Mem. Inst. Oswaldo Cruz, 89 (2), pp. 23-26. [27] Sinden, R.E. (1983). The cell biology of sexual development in Plasmodium. Parasitology, 86, pp. 7-28. [28] Sowunmi, A., Balogun, S.T., Gbotosho, G.O. and Happi, C.T. (2008). Plasmodium falciparum gametocyte sex ratios in children with acute, symptomatic, uncomplicated infections treated with amodiaquine. Malaria Journal, 7 (169), http://www.pubmedcentral.nih.gov/picrender.fcgi?artid=2542388&blobtype=pdf. [29] Talman, A.M., Domarle, O., McKenzie, F.E., Ariey, F. and Robert, V. (2004). Gametocytogenesis: the puberty of Plasmodium falciparum. Malaria Journal, 3 (24), pp. 3-24. [30] Targett, G., Drakeley, C., Jawara, M., von Seidlein, L., Coleman, R. Deen, J., Pinder, M. Doherty, T., Sutherland, C., Walraven, G. and Milligan, P. (2001). Artesunate reduces but does not prevent post treatment transmission of Plasmodium falciparum to Anopheles gambiae. Journal of Infectious Diseases, 83 (15), pp. 1254-1259.

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[31] Teboh-Ewungkem, M.I. (2009). Malaria control: The role of local communities as seen through a mathematical model in a changing population-Cameroon. In: Advances in Disease Epidemiology, (J.M. Tchuenche and Z. Mukandavire, eds), Nova Science Publishers, pp. 101-138. [32] Teboh-Ewungkem, M.I., Poddra, C.N. and Gumel, A.B. (2009). Mathematical study of the role of gametocytes and an imperfect vaccine on malaria transmission, Bulletin of Mathematical Biology, in press. [33] Tchuinkam, T., Mulder, B., Dechering, K., Stoffels, H., Verhave, J.P., Cot, M., Carnevale, P., Meuwissen, J.H. and Robert, V. (1993). Experimental infections of Anopheles gambiae with Plasmodium falciparum of naturally infected gametocyte carriers in Cameroon: factors influencing the infectivity to mosquitoes. Trop. Med. Parasitol., 44 (4), pp. 271-276. [34] Vekemans, J. and Ballou, W.R. (2008). Plasmodium falciparum malaria vaccines in development. Expert Rev Vaccines, 7 (2), pp. 223-240. [35] Vlachou, D., Schlegelmilch, T., Runn, E., Mendes, A., Kafatos, F.C. (2006). The developmental migration of Plasmodium in mosquitoes. Curr. Opin. Gen. Dev., 16, pp. 384-391. [36] West, S.A., Reece, S.E. and Read, A.F. (2001). Evolution of gametocytes sex ratios in malaria and related apicomplexan (protozoan) parasites, TRENDS in Parasitol., 17 (11), pp. 525-531. [37] West, S.A., Smith, T.G., Nee, S. and Read, A.F. (2002). Fertility insurance and the sex ratios of malaria and related hemospororin blood parasites. J. Parasitol., 88 (2), pp. 258–263. [38] World Health Organization (2008). Malaria, http://www.who.int/mediacentre/factsheets/fs094/en/. [39] World Malaria Report (2008). World Health Organization (WHO). http://www.who.int/malaria/wmr2008/malaria2008.pdf.

In: Infectious Disease Modelling Research Progress ISBN 978-1-60741-347-9 c 2009 Nova Science Publishers, Inc. Editors: J.M. Tchuenche, C. Chiyaka, pp. 197-227

Chapter 7

M YCOBACTERIUM T UBERCULOSIS T REATMENT AND THE E MERGENCE OF A M ULTI - DRUG R ESISTANT S TRAIN IN THE L UNGS G. Magombedze1,∗, W. Garira1,†, E. Mwenje2,‡ and C.P. Bhunu1,§ 1 Department of Applied Mathematics 2 Department of Applied Biology National University of Science and Technology P.O Box AC 939, Ascot Bulawayo, Zimbabwe

Abstract We develop a model for two Mtb strains, (i) drug resistant strain, and (ii) drug sensitive strain in this chapter. Our results suggest that the onset of Mtb infection by the drug resistant strain mainly depends on the level of resistance of the resistant strain to the TB drugs and emergence of mono-therapy is unlikely to occur in a three TB drug regimen. This study show the effects of DOTs compliance. We determine the number of misses that are permissible before Multi-drug resistant TB emerges. Our simulations suggest that if a certain number of prescribed doses are missed then either (i) the sensitive strain will take more time to clear, while the resistant strain will cause a latent infection, or (ii) treatment will fail to clear both strains. Only high levels of adherence will clear both strains. Our results demonstrate that maintenance of high concentration of both Isoniazid (INH) and Rifampicin (RMP) is required for treatment to be a success.

Keywords: Mycobacterium tuberculosis (Mtb), Multi-drug resistant TB (MDRTB), Chemotherapy, Mutation, Adherence. ∗

E-mail address: E-mail address: ‡ E-mail address: § E-mail address: †

[email protected] [email protected] [email protected] [email protected]

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Introduction

Tuberculosis (TB) continues to be a leading cause of mortality in spite of the availability of an effective chemotherapeutic regimen (Pandey and Khuller, 2004; Cohen and Murray, 2004). World Health Organisation (WHO) reports reveal that Multi-drug resistant TB (MDRTB) is a substantial problem in every region evaluated (Cohen and Murray, 2004). The fact that a TB patient needs to take multiple anti-tuberculosis drugs daily for at least 6 months is largely responsible for patient non-compliance and therapeutic failure. The cost and complexity of detecting and treating drug-resistant TB and the perception that drugresistant TB are less likely to cause epidemics than drug sensitive ones have mainly been the pivot for emergence and assimilation of MDRTB (Cohen and Murray, 2004; Rattan et al., 1998). Full supervision of the taking of medicament has been recognised for many years as necessary to obtain high cure rates in the TB chemotherapy and has been accorded high priority as directly observed treatment strategy (DOTs) by WHO. DOTs may be given daily or intermittently (2 or 3 times a week). Intermittent therapy was introduced when it was shown in controlled clinical trials that therapeutic serum levels of the various antituberculosis drugs were maintained even when medications were given only 2 or 3 times a week (Hershfield, 1999). Despite the availability of highly effective regimens, cure rates may not be satisfactory (Pandey and Khuller, 2004; Hershfield, 1999). Patients often do not take the prescribed drugs regularly or long enough to achieve cure. A potentially more serious problem than non-compliance is partial adherence to a prescribed regimen. When some drugs are selectively discontinued, there is an increased risk of acquired drug resistance (Balganesh et al., 2004; Jawahar, 2004). To help avoid the problem of MDRTB that are difficult to cure and costly to treat DOTs must be followed. DOTs aid adherence to the treatment regimen prescribed to the patients, in which a health care provider watches that patient swallows each dose of medication. DOTs is important in the treatment of tuberculosis because it allows for monitoring of the number of doses that an individual has taken, drawing attention immediately to those who have missed treatment and thus alerting the health care worker in charge of the particular case that the patient may be absconding from treatment. The phrase ”MDR” in mycobacteriology refers to simultaneous resistance to at least Isoniazid (INH) and Rifampcin (RMP) with or without resistance to other drugs (Rattan et al., 1998). Genetic and molecular analysis of drug resistance in Mtb suggests that resistance is usually acquired by the bacilli either by alteration of the drug through overproduction of the target or MDRTB results primarily from accumulation of mutations in individual drug target genes (Ratten et al., 1998). Therefore, mainly Mtb drug resistance can only occur through chromosomal mutations (Rattan et al., 1998; Gillespie, 2002). The rate at which resistance emerges differ for all the anti-tuberculosis agents (Ratten et al., 1998; Gillespie, 2002). The probability of resistance is very high for less effective anti-tubercular drugs and is proportional to the bacterial load (Rattan et al., 1998; Gillespie, 2002). The mutation rate, rather than the mutation frequency, is the most reliable measure, as it records the risk of mutation per cell division rather than the proportion of mutant cells. Mutation rates for RMP and INH are 3.32 × 10−9 and 2.56 × 10−8 mutations per bacterium per cell division, respectively (Rattan et al., 1998; Gillespie, 2002). It is assumed that since

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mutations conferring drug resistance are chromosomal, then the likehood of a mutant being simultaneously resistant to two or more drugs is the product of individual probabilities; thus the probability of MDR is multiplicative. Resistance to a drug does not confer any selective advantage to the bacterium unless it is exposed to that drug. RMP, along with INH, is the mainstay of chemotherapy for treatment of TB and resistance to either drug represents a serious impediment to successful therapy (Morlock et al., 2000; Ridzon et al., 1998), RMP Dosage and Resistance: RMP is a potent agent against actively dividing intracellular and extracellular organisms and has activity against semi-dormant bacilli (Howe, 1993; Hershfield, 1999; Jandini et al., 1979; Jandini et al., 1980). It works primarily by inhibiting DNA-dependent RNA polymerase, blocking RNA transcription. It is usually given as a daily oral dose of 10mg/kg. Resistance to RMP is associated with mutation in the gene coding for the beta subunit of RNA polymerase (rpoB). Telenti et al., (1993) demonstrated that at least 95% of RMP-resistant isolates have mutations in rpoB and that the mutations are clustered in an 81-bp region. Because RNA polymerase is an essential enzyme, there must be a limit number of possible mutations that confer RMP-resistant and retain polymerase activity, and this is reflected in the very low rate of mutation to resistance. INH Dosage and Resistance: INH is the most commonly used anti-tuberculosis drug. It is highly effective against Mtb, especially actively dividing bacilli (Howe, 1993; Hershfield, 1999; Jandini et al., 1979; Jandini et al., 1980). It is usually given orally, although parenteral preparations are available. The usual daily dose is 5mg/kg for adults and 10mg/kg for children. The mechanism conferring INH resistance are complex and not completely understood (Ratten et al., 1998). However evidence, suggest INH-resistance is a result of modification of KatG, partial or total deletions, point mutations. or insertions, leads to the abolition or diminution of catalase activity and high-level resistance to INH. Catalase activity is essential in activating INH to the active hydrazine derivative. A deficiency in enzyme activity produces high level resistance and is found in more than 80% of INHresistant strains. Alternativley, low level resistance can be caused by point mutations in the regulatory region of inhA operon, resulting in overexpression of inhA (Rattan et al., 1998; Gillespie, 2001). To date many models have been developed to describe the interaction of macrophages, T cells and the Mtb pathogen in the lungs (Magombedze et al., 2006a; Wigginton and Kirschner, 2001; Sud et al., 2006; Segovia et al., 2004; Gammack et al., 2004). Apart from these models few models have been done that considers the dynamics of Mtb infection in the lungs and in the draining lymph node (Marino et al., 2004; Magombedze et al., 2009). Magombedze et al., (2006b) developed an Mtb model that demonstrate the administration of the first line drugs in the treatment of Mtb infection at cell level. However, to the best of our knowledge no work has been done to model the dynamics of the emergence of drug resistant strains, but at population level a number of models have been developed in this respect (Jung et al., 2002; Castillo-chavez and Feng, 1997; Blower and Gerberding, 1998). The novel part of our work is that we develop a model to consider two Mtb strains in the lungs, (i) a drug sensitive strain (wild type) and (ii) a drug resistant strain that emergences as result of poor adherence or compliance. First we use ODEs then we couple the ODEs with impulsive differential equations. The development of MDRTB is mainly a result of poor adherence that is attributed primarily to improper prescriptions and patient non-compliance.

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In this chapter the effects of INH and RMP in the first line drug regimen are considered. Two strains of Mtb are assumed: a drug sensitive strain and a drug resistant strain that is resistant to both INH and RMP. The aim of this study is to understand, (i) the dynamics that lead to the emergence/occurrence of a MDRTB strain during TB treatment with first line drug regimen, (ii) the effects of treatment adherence to patients following DOTs and determine the levels of drug sensitivity to the drug resistant strain that leads to the emergence and resistance of MDRT-TB. We present a two strain model in section 2 and its analytical analysis is carried in section 3, while its numerical analysis in section 4. In section 5 we introduce impulsive differential equations and discussion of results is done in section 6.

1.1.

Model Development

We present a model that is a modification of our earlier studies (Magombedze et al., 2006a; Magombedze et al., 2006b). These studies (Magombedze et al., 2006a; Magombedze et al., 2006b) involved the dynamics of a single Mtb strain in the lungs (Magombedze et al., 2006a) and the treatment of Mtb infection with the first line TB drug regimen of INH, RMP and (Pyrazinamide) PZA. We modify the model in Magombedze et al., (2006b) by incorporating a Mtb mutant strain that arise due to drug pressure. We develop a model that models the dynamics of a multi-drug resistant strain, that is resistant to the first two drug of the first line drug regimen (INH and RMP). We assume that when there is no administration of the TB drugs then the drug resistant strain will not emerge. Since a drug resitant result due to chromosomal mutations that result from (i) resistance that is acquired by the bacilli by alteration of the drug through overproduction of the target, or (ii) that results primarily from accumulation of mutations in individual drug target genes (Ratten et al., 1998; Gillespie, 2002). The interaction of the human immune system and the Mtb pathogens during administration of the first line drugs, that is INH, RMP and PZA, is assumed by ten cell populations, that can be divided into three separate catergories. (i) Macrophages: resting macrophages (MR ), activated macrophages (MA ), macrophages infected by the Mtb drug sensitive strain (wild strain) (MIs ), macrophages infected by the drug resistant strain (MIr ), (ii) The Mtb pathogen: the Mtb drug sensitive strain (TBEs ), intracellular Mtb sensitive strain (TBIs ), Mtb drug resistant strain (TBEr ), intracellular resistant strain (TBIr ), and (iii) T cells: CD4+ Helper T cell (T ) and Mtb specific CTLs (C). The total bacterial burden TB is given by the sum of intracellular and extracellular bacterial particles of the drug sensitive and resistant strains (T B = TBIr + TBIs + TBEr + TBEs ). 1.1.1.

Macrophages

dMR (t) dt


= βm + αr MA (t) + ωr (MIs (t) + MIr (t)) + σMR (t) −βis

T

BEs (t)MR (t)



− βir

T

BEr (t)MR (t)

TBEs (t) + SE TBEr (t) + SE  T (t)  B + µd MA (t), −ωMR (t) TB (t) + SA



TB (t)  TB (t) + SR

− αMR (t)

dMIs (t) dt

dMIr (t) dt

dMA (t) dt 1.1.2.

Mycobacterium Tuberculosis Treatment... 201   C(t)  T BEs (t)MR (t) − k2s MIs (t) − k3s MIs (t) = βis TBEs (t) + SE C(t) + Gn    MIs (t) T (t) −k4s − k5s MIs (t)C(t) − µM Is MIs (t), 1 + b0 TBIs (t) T (t) + AT   C(t)  T BEr (t)MR (t) = βir − k2r MIr (t) − k3r MIr (t) TBEr (t) + SE C(t) + Gn    MIr (t) T (t) −k4r − k5r MIr (t)C(t) − µM Ir MIr (t), 1 + b0 TBIr (t) T (t) + AT  T (t)  B = ωMR (t) − µa MA (t) − µd MA (t). (1) TB (t) + SA

Drug Sensitive Bacteria Strain

dTBEs (t) dt

dTBIs (t) dt

MIs (t) T (t) )( ) 1 + b0 TBIs (t) T (t) + AT +k5s NC MIs (t)C(t)] + (1 − me )(1 − y1 γs )(1 − x1 ǫs )γ4s TBEs (t) C(t) TBEs (t) ) − γ5s TBEs (t)( ) −βis N1 MR (t)( TBEs (t) + SE C(t) + Gn −γ1s TBEs (t)MA (t) − γ2s TBEs (t)MR (t) − (γs + ǫs )TBEs (t), h (t) TBI s ) = (1 − y2 γs )(1 − x2 ǫs )ϕs N MIs (t)(1 − h TBIs (t) + (N MIs (t))h + E

= (1 − mi )ξs [N k2s MIs (t) + k4S NT (t)(

−ξs k2s N MIs (t) − ξs k3s N2 MIs (t)(

C(t) ) C(t) + Gn

TBEs (t) ) TBEs (t) + SE MIs (t) T (t) −ξs k4s NT ( )( ) − ξs k5s NC MIs (t)C(t) 1 + b0 TBIs (t) T (t) + AT −(µg + κo + xǫs + yγs )TBIs (t).

+βis N1 MR (t)(

1.1.3.

Drug Resistant Bacteria Strain

dTBEr (t) dt

MIs (t) T (t) )( ) 1 + b0 TBIs (t) T (t) + AT +k5s NC MIs (t)C(t)] + me (1 − y1 γs )(1 − x1 ǫs )γ4s TBEs (t)  T (t)   MIr (t) ξr [N k2r MIr (t) + k4r NT 1 + b0 TBIr (t) T (t) + AT +k5r NC MIr (t)C(t)] + (1 − y1 γr )(1 − x1 ǫr )γ4r TBEr (t)  T   C(t)  BEr (t) −βir N1 MR (t) − γ5r TBEr (t) TBEr (t) + SE C(t) + Gn −γ1r TBEr (t)MA (t) − γ2r TBEr (t)MR (t) − (γr + ǫr )TBEr ,

= mi ξs [N k2s MIs (t) + k4s NT (t)(

(2)

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dTBIr (t) dt

1.1.4.

 = (1 − y2 γr )(1 − x2 ǫr )ϕr N MIr (t) 1 −

h (t) TBI r



h (t) + (N M (t))h + E TBI Ir r  C(t)  −ξr [k2r N MIr (t) − k3r N2 MIr (t) ] C(t) + Gn  T  BEr (t) +βir N1 MR (t) TBEr (t) + SE  T (t)   MIr (t) −ξr [k4r NT 1 + b0 TBIr (t) T (t) + AT −k5r NC MIr (t)C(t)] − (µg + κo + xǫr + yγr )TBIr . (3)

CD4+ T Lymphocyte Cells and CD8+ T Cytotoxic Cells

dT (t) dt dC(t) dt

MA (t) + αT (MIs (t) + MIr (t))  T (t) − µT T (t), MA (t) + αT (MIs (t) + MIr (t)) + ST  (M (t) + α (M (t)) + M (t))T (t)C(t)  T Is Ir A − µC C(t). (4) = S2 + p2 MA (t) + αT (MIs (t) + MIr (t)) + SC = S1 + p1



where ξs = (1 − γs )(1 − ǫs )(1 − κo ) and ξr = (1 − γr )(1 − ǫr )(1 − κo ) with γs > γr , ǫs > ǫr or ǫr = ds γs and γr = ds γs (0 ≤ ds ≤ 1) and we assume that x, x1, x2, y, y1, y2 are the same for the sensitive strain and the resistant strain. The description of the model dynamics, parameters and administration of TB drugs closely follow that in our earlier studies (Magombedze et al., 2006a; Magombedze et al., 2006b). In equation (1) the recruitment of resting macrophages to the site of infection is due to macrophages infected by both the drug resistant and sensitive strain at the same rate of ωr . Resting macrophages are infected by Mtb at rates βis and βir , by the sensitive strain and resistant strain, respectively. The parameters, (i) k2s and k2r represent bursting of the sensitive and resistant strain infected macrophages, (ii) k3s and k3r represent the granulysin CTL effector mechanisms that kill intracellular bacteria (Schlunger, 2001; Schlunger and Rom, 2004; Flynn et al., 1995; Flynn 2004) in MIs and MIr , respectively. (iii) k4s and k4r represent apoptosis of MIs and MIr by CD4+ T helper cells, (vi) k5s and k5r represent lytic killing of MIs and MIr killing by Mtb specific CTLs (Schlunger, 2001; Schlunger and Rom, 2004; Flynn et al., 1995; Flynn 2004) and µM Is and µM Ir are the natural death rates of infected macrophages infected by the sensitive strain and resistant strain, respectively. We assume that release of bacterial particles due to bursting of infected cells as intracellular bacterial burden increases, from apoptosis, and due to lytic killing will result in equal release of bacterial particles from the intracellular to the extracellular environment. That is, bursting will release N particles, apoptosis NT particles and lytic killing NC particles irrespective of whether the cells are infected by the drug sensitive or resistant strain. The rate of mutation is modelled by the parameter mi (in the intracellular environment) and by me in the extracellular environment, such that when bursting occurs a fraction (1 − mi ) of N , NT and NC bacterial particles will remain sensitive to the drug, and also a fraction (1−me ) bacterial particles multiplying in the extracellular environment will remain sensitive to the TB drugs while the other fractions (mi and me ) develop drug resistance. ξs is the combined

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efficacy of the TB drugs to the Mtb sensitive strain and ξr to the resistant strain. γs , ǫs are the efficacies of INH and RMP to the Mtb drug sensitive strains while γr , ǫr are the reduced efficacies of INH and RMP to the drug resistant strain. κ0 is the efficacy of PZA both to the drug sensitive and resistant strains. We assume that PZA has the same efficacy for the two strains since our model considers a multi-drug resistant strain that is resistant both to INH and RMP but without resistance to PZA. The parameters ϕs and ϕr represent the rates of intracellular bacterial multiplication of the drug sensitive and resistant strains, respectively. Proliferation of CD4+ helper T cells and Mtb CTLs depend on the density of the sum of activated (MA ) and infected (MI = MIs + MIr ) macrophages at rates p1 and p2 , respectively.

2.

Model Analysis

The system of equations (1-4) has an initial condition given by MR (0) = βαm ≥ 0, MIs (0) = MIr (0) = MA (0) = TBEs (0) = TBEr (0) = TBIs (0) = TBIr (0) ≥ 0, T (0) = µST1 ≥ 0, C(0) = µSC2 ≥ 0 Since the model monitors human cell populations, all the variables and parameters of the model are non-negative. Based on biological considerations the system of equations (1-4) will be studied in the following region,  (5) D = (MR , MIs , MIr , MA , TBEs , TBEr , TBIs , TBIr , T, C) ∈ ℜ10 +

The following theorem assures that the system of equations (1-4) is well posed such that solutions with non-negative initial conditions remain non-negative for all 0 < t < ∞, and therefore makes biological sense. Theorem 1 The region D ⊂ ℜ10 + is positively invariant with respect to the system of equations (1-4) and a non-negative solution exist for all time 0 < t < ∞. Proof: From the first equation of (1), we get  dMR βis TBEs βir TBEr ωTB  MR (6) ≥− α+ + + dt TBEs + SE TBEr + SE TB + SE β T

β T

dMR is BEs ir BEr B let ψM R = α + TBE + TBE + TBωT +SE , therefore, dt ≥ −ψM R MR , integrating s +SE r +SE we get Z ∞ Z ∞ 1 ψM R dt (7) dMR ≥ − MR 0 0

Which evaluates to ln MR ≥ −ψM R t + K, that simplifies to MR (t) ≥ Ke−ψM R t , substituting the initial conditions, the following relationship is obtained MR (t) ≥ MR (0)e−ψM R∗ t ,

(8)

R t βis TBEs (τ ) βir TBEr (τ ) (τ )
where ψM R∗ = αt+ 0 TBE dτ . MR (t) → 0 as t → ∞, + TBE + TBωT(τB)+S E s (τ )+SE r (τ )+SE therefore MR (t) > 0. This implies that at any finite time moment, MR (t) is positive. This

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analysis can be followed for MI(s,r) , MA , T and C, hence the following expressions. dMI(s,r) dt dMA dt

≥ −ψM I(s,r) MI(s,r) , ≥ −(µa + µd )MA , dT dt dC dt

≥ −µT T,

≥ −µC C,

(9)

where
ψM I(s,r) = k2(s,r) + µM I(s,r) t Z t  T (τ ) C(τ ) + k4(s,r) + k5(s,r) C(τ ) dτ. + k3(s,r) C(τ ) + Gn (T (τ ) + AT )(1 + b0 TBI(s,r) (τ )) 0 Integrating expressions in (9) and applying initial conditions the following expressions are obtained. −ψM I(s,r) t

MI(s,r) (t) ≥ MI(s,r) (0)e

MA (t) ≥ MA (0)e−(µa +µd )t T (t) ≥ T (0)e−µT t

C(t) ≥ C(0)e−µC t

(10)

This follows that MIs (t) ≥ 0, MIr (t) ≥ 0, MA (t) ≥ 0, T (t) > 0, C(t) > 0, these variables are positive at any finite time moment. Note, if MI(s,r) (t) ≥ 0 at any time t, therefore TBI(s,r) (t) ≥ 0 at any time moment t, since TBI(s,r) (t) ≈ N MI(s,r) (t). TBI (t) is the measure of the intracellular bacterial burden in infected macrophages (MI (t)), this implies that if MI (t) = 0, then TBI (t) = 0 and maximum TBI occurs when MI is maximum. From the first equation of (2), we get  dTBE ψTBE2  ≥ − ψTBE1 + TBE dt TBE + SE  ψ 2 TBE1 TBE + (ψTBE1 SE + ψTBE2 )TBE , (11) ≥ − TBE + SE   C where ψTBE1 = γs +ǫs +γ1 MA +γ2 MR +γ5 C+G and ψTBE2 = βi N1 MR . Separating n variables and integrating both sides Z ∞  TBE + SE dTBE ≥ −t + K (12) 2 + (ψ ψTBE1 TBE TBE1 SE + ψTBE2 )TBE 0 K is a constant of integration. Applying partial fractions, the integral on the right hand side is equivalent to Z ∞  TBE + SE dTBE = 2 + (ψ ψTBE1 TBE TBE1 SE + ψTBE2 )TBE 0 Z ∞ 1  ψ ψTBE1 TBE1 SE + ψTBE2 dTBE (13) − SE TBE ψTBE1 SE + ψTBE2 + ψTBE1 TBE 0

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Evaluating and simplifying the left hand side of (13), we get the following result TBE (t)(1 − ψTBE1 TBE (0)ψTBE3 e−ψTBE3 t ) ≥ (ψTBE1 SE + ψTBE2 )TBE (0)ψTBE3 e−ψTBE3 t , (14) where ψTBE3

=

ψ

TBE1 SE

+ ψTBE2 

SE

.

Since TBE (t) = 0 at t = 0, that is, TBE (0) = 0. Therefore, TBE (t) > 0, at any time t > 0. This analysis gives the same result for TBEs and TBEr . We analyse system of equations (1-4) by determining its steady states. A steady state of a system is a point in phase space for which the system will not change in time. Solving this system of equations gives at least four steady states ˆ where A uninfected state (H), ˆ = ( βm , 0, 0, 0, 0, 0, 0, 0, S1 , S2 ). H α µT µC

(15)

And an endemically infected state ¯ = (M ¯ R, M ¯ Is , M ¯ Ir , M ¯ A , T¯BEs , T¯BEr , T¯BIs , T¯BIr , T¯, C). ¯ H Infection with Mtb gives two possible disease outcomes that is latency and active disease. The equilibrium value of resting macrophages is given by

¯R = M

(βm ¯ β T s ( T¯ is BE BEs +SE

¯ A + ωr M ¯ A) ¯ I ) + µd M + αr (M +

βir T¯BEr T¯BE +SE

+α+

ω T¯B T¯B +SA

−

σ T¯B ) T¯B +SR

.

(16)

The equilibrium value of resting macrophages depends on the rates at which new resting macrophages are coming into the site of infection through natural supply, deactivation of Mtb activated macrophages and recruitment due to the two Mtb strains, and on the rate at which they are lost through infection with the Mtb pathogens. Mtb infected macrophages are given by ¯I M (s,r)

¯R βi(s,r) T¯BE(s,r) M

= (T¯BE(s,r) +SE )(k2(s,r) +

¯ k4 T¯ k3 C (s,r) (s,r) ¯ +k5(s,r) C+µ + M I(s,r) ) ¯ ¯ (1+bo TBI )(T¯ +AT ) C+Gn (s,r)

. (17)

The level of Mtb infected macrophages depends on the rate at which the Mtb pathogen infects resting macrophages and the combined rate at which the immune components are eliminating them and their natural death. Increasing the dividing term (increasing the effectiveness of the immune components) reduces the level of infected macrophages. This indicates that the way to control the Mtb infection is to have a robust immune response mechanisms that can eradicate the infection. The endemic equilibrium value of activated macrophages is given by ¯A = M



¯ R T¯B ω  M . µa + µd T¯B + SA

(18)

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The endemic equilibrium value of activated macrophages is shown by expression (18) to depend on the density of the bacterial load that is on the site of infection and on the population of resting macrophages. Also, it is affected by the rate at which resting macrophages are activated (ω) and are deactivated due to a set of cytokines that are released during Mtb infection as well as their natural death. The equilibrium value of extracellular bacteria at the endemically infected state is given by T¯BE(s,r)

=

¯ R )+ −(R1 +R2 SE −βi(s,r) N1 M

q

¯ R )2 −4R1 R2 SE (R1 +R2 SE −βi(s,r) N1 M 2R2

,

(19)

where   = ξr R1r + mi ξs R1s + me (1 − y1 γs )(1 − x1 ǫs )γ4s ,

R1

  ¯ C ¯ A − γ2r M ¯ R − (γr + ǫr ) , R2 = (1 − y1 γr )(1 − x1 ǫr )γ4r − γ5r C+G − γ1r M ¯ n   ¯ I T¯ M ¯ Ir C¯ , R1r = N k2r + k4r NT (1+b T¯ r)(T¯+A ) + k5r NC M 0 BIr T   ¯ I T¯ M ¯ Is C¯ . R1s = N k2s + k4s NT (1+b T¯ s)(T¯+A ) + k5s NC M 0 BIs

T

But, R1 = (1 − mi )ξs R1s for the drug sensitive strain and in R2 the subscript r is replaced with subscript s. These expressions show that the equilibrium value of the drug resistant strain depend on the additional source of bacterial particles from mutation of the drug sensitive strain. Bursting of infected cells and release of bacterial particles from the intracellular bacteria through apoptosis and lytic killing by CD4+ T cells and Mtb specific CTLs, as well as extracellular bacterial multiplication increase extracellular bacteria load. Direct killing of bacterial particles, killing by activated and resting macrophages reduce the extracellular bacterial load. Administration of TB drugs reduce the amount of bacterial particles that are added to the extracellular environment from the intracellular environment by the factors ξr and ξs , reduce rate of bacterial multiplication and also directly kill the Mtb pathogen. At the endemically infected state the equilibrium value of intracellular bacteria is evaluated from 3 2 T¯BI + λ2 T¯BI + λ1 T¯BI(s,r) + λ0 = 0. (s,r) (s,r)

where λ0 =


 1 ¯ I )2 + E ξo k2 N M ¯I (N M (s,r) (s,r) (µg + κo + xǫo + yγo ) ¯I M
C¯
T¯
(s,r) ¯I +ξo k3 N2 M + ξ k N o 4 T (s,r) C¯ + Gn T¯ + AT 1 + bo T¯BI (s,r)

T¯BE(s,r)

¯ I C¯ − βi N1 M ¯R +ξo k5 NC M (s,r) T¯BE(s,r) + SE  ¯I −ϕ(s,r) (1 − y2 γo )(1 − x2 ǫo )N M , (s,r)

¯ I )2 + E, λ1 = (N M (s,r)




(20)

λ2

Mycobacterium Tuberculosis Treatment...  1 C¯
¯I ¯I = ξo k2 N M + ξ k N M o 3 2 (s,r) (s,r) (µg + κo + xǫo + yγo ) C¯ + Gn ¯I M
T¯
(s,r) +ξo k4 NT + ξo k5 NC 1 + bo T¯BI T¯ + AT

207

(s,r)

T¯BE(s,r)

¯R −βi N1 M T¯BE(s,r) + SE




.

(21)

Using the cubic formulae the equilibrium value of intracellular bacteria is given by 1 = − λ2 + (H + O), 3 1 1 = − λ2 − (H + O) − 3 2 1 1 = − λ2 − (H + O) + 3 2

T¯BI(s,r) 1 T¯BI(s,r) 2 T¯BI(s,r) 3 where

1p 3(O − H), 2 1p 3(O − H), 2

q p √ √ 3 3 H = R + D, O = R − D, D = Q3 + R2 , Q

=

3λ1 −λ22 , 9

R=

9λ1 λ2 − 27λ0 − 2λ32 . 54

(22)

(23)

There are three possible values which represent the value of intracellular bacteria at the endemically infected state which are mathematically correct but biologically not all of them feasible. We take the positive value only. Therefore, T¯BI(s,r)

= T¯BI(s,r) 1 .

(24)

Parameters in the equation for TBI are altered in the presence of the drug regimen to smaller values. The set of expressions (21) show how the drugs alter parameters that determine the equilibrium value of TBI . This effectively reduces intracellular bacterial population and as a result the equilibrium value of TBI is reduced. At the endemically infected state the equilibrium value for CD4+ T cells (T ) and CD8+ T cells (C) are given by T¯ =

¯ A + αT M ¯ I ) + S1 ST S1 (M ¯ A + αT M ¯ I ) + µT ST , (µT − p1 )(M

(25)

¯ A + αT M ¯ I ) + S2 SC S2 (M ¯ A + αT M ¯ I ) + µC SC . (µC − p2 T¯)(M

(26)

and C¯ =

The equilibrium values of CD4+ T cells and Mtb specific CTLs depend on the density of activated and infected macrophages, which are cells that are responsible for the secretion of cytokines that induce the proliferation and amplification of these cell categories.

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2.1.

Reproductive Ratio

The reproduction ratio in this case, where there are two Mtb strains is defined by the spectral radius of the next generation matrix. This gives rise to two reproduction ratios for the resistant and wild type strains. We use the next generation approach to compute R0 (Diekmann et al., 1990). Let Fi be the rate of new infections into compartment i, Vi+ (x) be the rate of transfer of susceptible cells into i by all other means and Vi− (x) be the transfer out of i (Vi = Vi− − Vi +). For the system of equations (1)-(4) F and V are given as follows    TBEs MR βis TBE s +SE    TBEr MR   βir TBE +SE  r     0       0 F = ,   0       0     0   0 

  k2r MIr +      V=        

where





C  C+Gn  C k3r MIr C+G n

k2s MIs + k3s MIs





+ k5s MIs C + µM Is MIs



  + k5r MIr C + µM Ir MIr    v1   v2  .  v  3   TB ω − MR TB +SA + µa MA + µd MA     MA +αT (MIs +MIr  −S1 − p1 MA +αT (MI +MI )+ST T + µT T  s r    (MA +αT (MIs )+MIr )T C −S2 − p2 MA +αT (MI +MI )+SC + µC C +

s



r

MIs T )( ) + k5s NC MIs C] 1 + b0 TBIs T + AT −(1 − me )(1 − y1 γs )(1 − x1 ǫs )γ4s TBEs TBEs +βis N1 MR ( ) TBEs + SE C ) + γ1s TBEs MA + γ2s TBEs MR + (γs + ǫs )TBEs , +γ5s TBEs ( C + Gn MIs T = −mi ξs [N k2s MIs + k4s NT ( )( ) + k5s NC MIs C] 1 + b0 TBIs T + AT −me (1 − y1 γs )(1 − x1 ǫs )γ4s TBEs

v1 = −(1 − mi )ξs [N k2s MIs + k4S NT (

v2



M Is T  1+b0 TBIs  T +AT M T k4r 1+b0 TIrBI T +AT r

+ k4s

v3

Mycobacterium Tuberculosis Treatment... 209    MIr T −ξr [N k2r MIr + k4r NT + k5r NC MIr C] 1 + b0 TBIr T + AT  T   C  BEr −(1 − y1 γr )(1 − x1 ǫr )γ4r TBEr + βir N1 MR + γ5r TBEr TBEr + SE C + Gn +γ1r TBEr MA + γ2r TBEr MR + (γr + ǫr )TBEr ,  
TB = −βm − αr MA + ωr (MIs + MIr ) − σMR TB ) + SR     T   T TBEr MR B BEs MR + βir + αMR + ωMR +βis TBEs + SE TBEr + SE TB + SA −µd MA .

(27)

Therefore,

F V −1

  0 F1 0 V11 0 0 0  0 0 F2  0  ,   0 V22 0   V31 0 V33 0  0 0 0 V41 V42 V43 V44 0 0 0

 0 0 = 0 0

where F1

=

F2

=

V11

=

V22

=

V31

=

V33

=

V41

=

βis

M  R

SE M  R βir SE

1 T C + k k2s + k3s C+G 4s T +A + k5s C + µM Is n T 1 T C + k k2r + k3r C+G 4r T +A + k5r C + µM Ir n T T + k5s NC C) (1 − mi )ξs (N k2s + k4s T +A T T C + k4s T +A + k5s C + µM Is )Q1 (k2s + k3s C+G n T ˆR (βis N1 M SE



+

ˆ C γ5s C+G ˆ n

1 ˆ + γ2s MR + (γs + ǫs ) − (1 − me )(1 − y1 γs )(1 − x1 ǫs )γ4s )

V31 (me (1 − y1 γs )(1 − x1 ǫs )γ4s )  1 T + k5s NC C)) T + AT V44 T + k4r T +A + k 5r NC C) T

−V11 (−mi ξs (N k2s + k4s V42

=

V43

=

ξr (N k2r

T C + k4r T +A + k5r C + µM Ir )Q2 (k2r + k3r C+G n T ˆR (βis N1 M SE

+

ˆ C γ5s C+G ˆ n

me (1 − y1 γs )(1 − x1 ǫs )γ4s × ˆ R + (γs + ǫs ) − (1 − me )(1 − y1 γs )(1 − x1 ǫs )γ4s )) + γ2 s M 1

ˆ

ˆ

C R ˆ R + (γr + ǫr ) − (1 − y1 γr )(1 − x1 ǫr )γ4r ) (βir N1 M + γ2 r M + γ5r C+G ˆ SE n

V44

=

ˆR (βir N1 M SE

+

ˆ C γ5r C+G ˆ n

1 ˆ + γ2r MR + (γr + ǫr ) − (1 − y1 γr )(1 − x1 ǫr )γ4r )

(28)

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where ˆ

ˆ

C R ˆ R + (γs + ǫs ) − (1 − me )(1 − y1 γs )(1 − x1 ǫs )γ4 ), Q1 = (βis N1 M + γ2s M s ˆ SE + γ5s C+G n

Q2 =

ˆ

ˆ

C R ˆ R + (γr + ǫr ) − (1 − y1 γr )(1 − x1 ǫr )γ4 ). (βir N1 M + γ2r M r ˆ SE + γ5r C+G n

The derivatives of F, (DF(H)) and V, (DV(H)) are evaluated and partitioned to give F and V that are 4 × 4 matrices. To find the reproduction ratio, we evaluate the eigenvalues of the matrix given by F V −1 . Therefore, R0 = ρ(F V −1 ) where ρ(F V −1 ) denotes the spectral radius of matrix (F V −1 ). When evaluated this gives the following reproduction ratios, The reproduction numbers of the sensitive and resistant strain are given as follows RT Bs

=

RT Br

=

R1s , R2s R3s R1r . R2r R3r

(29) (30)

Where, R1s = R2s = R3s = R1r = R2r = R3r =

ˆR Tˆ (1 − mi )ξs βis M ˆ (N k2s + k4s NT + k5s NC C), SE Tˆ + AT Tˆ Cˆ + k5s Cˆ + µM Is , + k4s k2s + k3s Cˆ + Gn Tˆ + AT ˆR βis N1 M γ5s Cˆ ˆ R + γs + ǫs − (1 − me )(1 − y1 γs )(1 − x1 ǫs )γ4s , + + γ2s M SE Cˆ + Gn ˆR Tˆ ξr βir M ˆ (N k2r + k4r NT + k5r NC C), SE Tˆ + AT Tˆ Cˆ + k5r Cˆ + µM Ir , + k4r k2r + k3r Cˆ + Gn Tˆ + AT ˆR M Cˆ ˆ R + γr + ǫr − (1 − y1 γr )(1 − x1 ǫr )γ4r . βir N1 + γ5r + γ2r M SE Cˆ + Gn

Therefore, R0 = max(RT Bs , RT Br ) The disease reproduction number is the maximum of the two reproduction numbers • If RT Bs < 1, and RT Br < 1, both the Mtb infections from either the drug resistant or the drug sensitive strains will be abortive. This implies that TB treatment will eradicate TB disease and will prevent the emergence of drug resistant strains. • If RT Bs > 1, and RT Br < 1, then Mtb infection will be due to the drug sensitive strain and treatment will not eradicate infection. The drug resistant strain will be abortive. • If RT Bs < 1, and RT Br > 1, then TB therapy will only contain the drug sensitive strain while the resistant strain will cause an infection that will proceed to cause latent TB or active TB.

Mycobacterium Tuberculosis Treatment...

211

• If RT Bs > 1, and RT Br > 1, then both Mtb strains will not be contained by treatment, but will cause infections. The two strains will co-exist. The dominant strain will cause the infection, therefore, the disease reproduction number is the maximum of the two. Expression (29) shows that RT Bs depends on CD4+ T helper T cells, Mtb CTLs, and macrophages. Whereby increasing the mechanisms/activities and the populations of these cell categories will reduce the value of RT Bs . On the other hand increasing the parameters βis , k2s , and γ4s will favour fast progression of Mtb infection. This ratio is similar to the result obtained in Magombedze et al., (2006b). The difference is the inclusion of the mutation parameters (mi and me ) that arise during TB treatment due to TB drug pressure, that force the wild strain to evolve mechanisms to dodge/escape or avoid elimination by the TB drugs. This mutation gives birth to a drug resistant strain. While on the other hand, RT Br measures the progression of the drug resistant mutant strain. This measure depends on the same cell variables and parameters as the sensitive strain (T cells, macrophages, pathogen’s infection and induced burst rate of infected macrophages, and rate of bacterial replication). However, this strain thrive on the fact that it is less sensitive to TB drugs or has a degree of resistance to the TB drugs. Co-existence of the two strains only occur when the efficacy values γs , ǫs , and κ0 are low such that TB drug administration cannot push RT Bs to a value less than one and when the resistant strain is resistant to the TB drugs such that, the TB drugs will fail to prevent their emergence and the establishment of the resistant strain infection. 2.1.1.

Global Stability Conditions for the Disease-Free Equilibrium

We adopt the method of Castillo-chavez et al., (2002) and re-write the set of equations (1-4) in the form: dX dt dZ dt

= F (X, Z), = G(X, Z).

(31)

With G(X, 0) = 0, where X ∈ ℜ4 denotes the number of uninfected cells, and Z ∈ ℜ6 denotes the number of infected cells including latent and infectious cells. U0 = (X ∗ , 0) denotes the disease-free equilibrium of the system. The conditions (H1) and (H2) below must be met to guarantee global asymptotic stability ∗ H1): for dX dt = F (X, 0), X is globally asymptotically stable ˆ ˆ H2): G(X, Z) = AZ − G(X, Z), G(X, Z) ≥ 0 for (X, Z) ∈ Ω where A = DZ G(X ∗ , 0) is an M-matrix (the off diagonal elements of A are non-negative) and Ω is the region where the model makes biological sense. If the above two conditions are satisfied, then, the following theorem holds. Theorem 2 (Castillo-chavez et al., 2002): The fixed point U0 = (X ∗ , 0) is a globally stable equilibrium of (31) provided that R0 < 1 and that assumptions (H1) and (H2) are satisfied.

212

G. Magombedze, W. Garira, E. Mwenje et al.

For the set of equations (1-4), we set X = (MR , MA , T, C): the set of resting macrophages, activated macrophages, CD4+ helper T cells and Mtb specific CTLs and ZG = (MIs , MIr , TBEs , TBEr , TBIs , TBIr ): the Mtb infected macrophages and the Mtb pathogens in the extracellular and intracellular environment. We computed F (X, 0), A ˆ (= DZ G(X ∗ , 0)) and G(X, Z) given by,  A11 0 A13 0 0 0  0 A22 0 A24 0 0  βm − αMR    A31 0 A33 0  0 0  0 ,    ,A =  F(X, 0) =   A A A A 0 0 S1 − µT T  41 42 43 44    A51 0 0 0 A55 0  S2 − µC C 0 A62 0 0 0 A66 



where

k4s Tˆ + k5s Cˆ − µM Is ), Tˆ+AT k Tˆ + ˆ 4r + k5r Cˆ − µM Ir ), −(k2r + T +AT k N Tˆ ˆ (1 − mi )ξs (N k2s + 4ˆs T + k5s NC C), T +AT

A11 = −(k2s + A22 =

ˆ k 3s C ˆ C+G n ˆ k 3r C ˆ C+G n



+

A31 = A33 = (1 − me )(1 − y1 γs )(1 − x1 ǫs )γ4s ˆ ˆ β N M γ C ˆ R − (γs + ǫs ), − is 1 R − 5s − γ2s M ˆ C+G n k N Tˆ ˆ mi ξs (N k2s + 4ˆs T + k5s NC C), T +AT k N Tˆ ˆ ξr (N k2r + T4ˆr+AT + k5r NC C), T SE

A41 =

A42 = A43 = (1 − me )(1 − y1 γs )(1 − x1 ǫs )γ4s , ˆ ˆ β N M γ C ˆ R − (γr + ǫr ) A44 = − ir 1 R − 5r − γ2r M SE

ˆ C+G n

+ (1 − y1 γr )(1 − x1 ǫr )γ4r , A51 = (1 − y2 γs )(1 − x2 ǫs )N ϕs

ˆ k N C k N Tˆ ˆ − ξs (k2s N + ˆ3s 2 + 4ˆs T + k5s NC C), C+Gn T +AT A62 = (1 − y2 γr )(1 − x2 ǫr )N ϕr ˆ k N C k N Tˆ ˆ − ξr (k2r N + 3r 2 + 4r T + k5r NC C), ˆ C+G n

Tˆ+AT

A55 − (µg + κ0 + xǫs + yγs ), A66 = −(µg + κ0 + xǫr + yγr ),

and  ˆ 1 (X, Z) G G   ˆ 2 (X, Z) ˆ  G (X, Z) ˆ G(X, Z) =  ˆ 3 , G4 (X, Z)   ˆ 5 (X, Z) G ˆ 6 (X, Z) G 

where

A13 = A24 =

ˆR βis M SE , ˆR βir M SE ,

Mycobacterium Tuberculosis Treatment...

ˆ1 G

213

  C¯ Cˆ  ¯ ¯ ˆ − + k M C − C 5s Is C¯ + Gn Cˆ + Gn  Tˆ  T¯ ¯I − +k4s M s (1 + b0 T¯BIs )(T¯ + AT ) Tˆ + AT  ˆR  ¯R M M − , +βis T¯BEs ¯ SE TBE + SE

¯I = k3s M s



(32)

s

ˆ2 G

ˆ3 G

ˆ4 G

ˆ5 G

ˆ6 G

  C¯ Cˆ  ¯ I C¯ − Cˆ + k M − 5 r r C¯ + Gn Cˆ + Gn  T¯ Tˆ  ¯I +k4r M − r (1 + b0 T¯BIr )(T¯ + AT ) Tˆ + AT  ¯R ˆR  M M , +βir T¯BEr ¯ − SE TBEr + SE h   Tˆ T¯ ¯ I k4 NT = (1 − mi )ξs M − s s (1 + b0 T¯BIs )(T¯ + AT ) Tˆ + AT  i  C¯ Cˆ  ¯ I C¯ − Cˆ + γ1 M ¯ A + γ5 M ¯I +k5s M − s s s s C¯ + Gn Cˆ + Gn   ˆR ¯R M M ¯R − M ˆ R ), − + γ2s (M +βis T¯BEs ¯ SE TBEs + SE   i h  Tˆ T¯ ¯ I C¯ − Cˆ ¯ I k4 NT + k M − = ξr M 5 r r r r (1 + b0 T¯BIr )(T¯ + AT ) Tˆ + AT  C¯ Cˆ  ¯I +γ5r M − r C¯ + Gn Cˆ + Gn   Tˆ T¯ ¯ I k4 NT +mi ξs M − s s (1 + b0 T¯BIs )(T¯ + AT ) Tˆ + AT  ¯R ˆR  M M ¯ A + βi T¯BE ¯R − M ˆ R ), +γ1r M − + γ2r (M r r SE T¯BEr + SE h  Tˆ  T¯ ¯ I k4 NT − = ξs M s s (1 + b0 T¯BIs )(T¯ + AT ) Tˆ + AT  i   ¯ I C¯ − Cˆ + k5 M ¯ I C¯ − Cˆ +k5s NC M s s s   ¯ I 2X , +(1 − y2 γs )(1 − x2 ǫs )ϕs N M s ¯I = k3r M r



(33)

(34)

(35)

(36)

  i h  Tˆ T¯ ¯ I C¯ − Cˆ ¯ I k4 NT − + k M = ξr M 5 r r r r (1 + b0 T¯BIr )(T¯ + AT ) Tˆ + AT     ¯ I C¯ − Cˆ + (1 − y2 γr )(1 − x2 ǫr )ϕr N M ¯ I 2X , +k5r M (37) r r T2

where X = T 2 +(NBI . From the calculated endemic equilibrium values and MI )2 +E BI comparing them with the disease free equilibrium states we note that, ˆR ≥ M ¯ R . This follows that, Mˆ R ≥ ¯ M¯ R , ¯ T¯ ≥ Tˆ , T¯ ≥ Tˆ, C¯ ≥ C, M ˆ SE T +S T +A BEs

E

T

T +AT

214

G. Magombedze, W. Garira, E. Mwenje et al.

T¯ (1+b0 T¯BIs )(T¯+AT )

≥

Tˆ Tˆ+AT

,

¯ C ¯ C+G n

≥

ˆ C . ˆ C+G n

ˆ 1 (X, Z), G ˆ 2 (X, Z), G ˆ 3 (X, Z), G ˆ 4 (X, Z), G ˆ 5 (X, Z), G ˆ 6 (X, Z)) ≥ 0. ˆ Z) ≥ 0, if (G G(X,     ˆ C ˆ1 ¯ Is ¯ C¯ ¯ ¯ ˆ − (i) G ≥ 0 if, k3s M + k M C − C ≥ 5s Is ˆ n  C+G  C+Gn   ˆR ¯ ˆ T¯ ¯ Is − ˆT + βis T¯BEs T¯ MR+S − M k4s M SE , (1+b0 T¯BIs )(T¯+AT ) T +AT BEs E    ¯ T¯ Tˆ C ˆ 3 ≥ 0 if, (1 − mi )ξs M ¯ Is k4s NT − − (ii) G + γ 5 ¯ ¯ ¯ s ˆ+AT C+G T (1+b )( T +A ) n T 0 BI T s i     ¯ ˆ ˆ C ¯ A T¯BEs ≥ k5s M ¯ Is C¯ − Cˆ +βis T¯BEs ¯ MR − MR +γ2s T¯BEs (M ¯R− +γ1s M ˆ SE TBEs +SE C+G n ˆ R ), M     ˆ ¯ ˆ ¯ ¯ Ir ¯ C¯ − Cˆ (iii) G4 ≥ 0 if, ξr MIr k4r NT ˆ T − (1+b T¯ T )(T¯+A ) +γ5r M + ˆ C+Gn T +AT T  0 BIr  C+Gn ¯ ˆ T ¯ Is k4s NT ¯ A T¯BEr ≥ k5s M ¯ Is C¯ − Cˆ + − (1+b T¯ T )(T¯+A ) + γ1r M mi ξs M T s   Tˆ+AT   0 BI ¯ Ir C¯ − Cˆ + βir T¯BE ¯ M¯ R − Mˆ R + γ2r (M ¯R − M ˆ R ), k5r M r T SE BE  i   r +SE h ¯ ˆ ¯ Is N2 k5s ¯ C¯ − Cˆ ˆ 5 ≥ 0 if, ξs M C − C + (1 − y2 γs )(1 − + k N (vi) G 5 C s ˆ C+Gn C+G n     ¯ Tˆ ¯ Is 2X ≥ ξs M ¯ Is k4s NT − (1+b T¯ T )(T¯+A ) . x2 ǫs )ϕs N M Tˆ+AT 0 BIs T ˆ 2, G ˆ 6 follows closely the analysis of G ˆ 1 and G ˆ 5 , replacing the subThe analysis of G ˆ2, G ˆ6. scripts s with r gives similar results for G ˆ ˆ Therefore, if G(X, Z) ≥ 0, then, H is globally stable. This implies that the administration of TB drugs will eradicate and contain the two Mtb strains. Otherwise, if this condition ˆ may not be globally asymptotically stable. That is, either the drug fails to hold, then H resistant strain or the drug sensitive will cause an infection that will either result in latent TB or active TB or both strains co-exist and compete within the host.

3.

Numerical Simulations

In this section we use numerical simulations to investigate the effects of the first line TB drugs on the development of a TB multi-drug resistant strain. A fourth order Runge-Kutta scheme is used to simulate the results. We carry out our simulations up to 200 days (≈ 6 months), which is the recommended period of TB treatment with the first line regimen drugs. Our simulations are targeted to (i) determine the levels/degree of drug sensitive of the resistant strain that can cause the emergence and permanent infection with the drug resistant strain, (ii) the effects of reduced drug efficacy values on the propagation of the resistant strain, and (iii) the progression of a mono drug resistant strain. The on-set of Mtb mutant strains that are less sensitive to the TB drugs could arise as a result of, (a) poor compliance or adherence to the prescribed drug dosage scheme, (b) reduced drug concentration due to drug mal-absorption or reduced drug bio-availability in HIV/TB co-infected individuals, hence the reason why the multi-drug TB resistant strains are rampant in HIV/TB co-infected populations, and (c) drug pressure; as a result of TB drug administration, the Mtb pathogen will evolve mechanisms that enable it to survive in the new environment created by the TB drugs. This is normally associated with the pathogen sacrificing some gene components or evolution of chromosomal changes to components that are primarily targeted by the drug or

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mutations in some gene components (Rattan et al., 1998; Gillespie, 2002; Morlock et al., 2000; Ridzon et al., 1998). The initial conditions for all the numerical simulations are MR (0) = 17500.0, MIs (0) = 12500.0, MIr (0) = 0.0, MA (0) = 2000.0, TBEs (0) = 35000000000.0, TBIs (0) = 800000.0, TBEr (0) = 0.0, TBIr (0) = 0.0 T (0) = 10000.0, and C(0) = 325.0. We use the following parameter values for the infection rate, rate of bursting of infected macrophages, rate of extracellular and intracellular bacterial multiplication of βir = 0.3, k2r = 0.3, γ4r = 0.13255 and ϕr = 0.375 for the drug resistant strain, respectively. We used smaller values for these parameters since gene components alteration due to drug administration result in a less virulent Mtb strain, while the rest of other parameters are assumed to be equal to those of the drug sensitive strain for the reason of simplifying our numerical simulations. However, this does not affect the scope of our investigations. Parameters used in these simulations are in Table 2 while drug parameters are found in Table 1. Table 1. Table of drug efficacy values and drug parameters. Name γs ǫs κo x x1 x2 y y1 y2

Value 0.7 0.2 0.025 0.135 0.165 0.125 0.15 0.145 0.135

Defination Efficacy value for Isoniazid Efficacy value for Rifampicin Efficacy value for Pyrazinamide Factor for ǫs effectiveness in TBI killing ǫs TBE multiplication hindrance factor ǫs TBI multiplication hindrance factor Factor for γs effectiveness in TBI killing γs TBE multiplication hindrance factor γs TBI multiplication hindrance factor

Reference [21, 22] [21, 22] [21, 22] [3] [3] [3] [3] [3] [3]

Figure 1 shows the dynamics of the drug sensitive and resistant strains during the administration of the first line TB drugs. The administration of TB drugs demonstrate the eradication of the drug sensitive strain, (i) TBEs is eradicated within the first 14 days of therapy (Figure 1(d)), (ii) clearance of TBIs and MIs within one to two months of therapy, (iii) recovery of the macrophage population. These results are similar to results in our previous studies (Magombedze et al., 2006b). However, Figure 1(a), unlike in Magombedze et al., (2006b), shows recovery of the macrophages population only within the first 100 to 120 days (≈ 4 months) of TB therapy. But after about 120 days the macrophage population starts to decline. This decline of macrophages is associated with the onset of TB drug resistant strain as noticed in Figure 1(d). Figure 1(d) shows that TBEr starts from zero, but reach its peak within the first 10 days of TB drug treatment. Then, it will start to decline to very low levels and seems to be getting cleared from the 40th day of TB treatment, but it starts to arise again from the 80th day of treatment. These simulations correspond to a degree of drug sensitivity of 40% to the drug mutant strain. Note, the window period from the time the resistant strain seem to be cleared to the time it emerges again corresponds to the time patients under treatment starts to enjoy the benefits of treatment, symptoms disappear and their sputam may test negative. This is the window period when most patients stop taking

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their medications. Our simulations suggest this is also the time mutant strains are most likely to emerge and cause a permanent infection. Therefore, it is necessary for health care providers to ensure that DOTs is followed. The early emergence and the delayed clearance of the mutant strain is an indication of how important it is for patients to closely follow DOTs for 6 months. Our numerics suggest that the emergence and the persistence of the drug resistant strain depends on the degree or levels of sensitivity of the resistant strain to the TB drugs. This suggets that if the resistant strains are highly insensitive to the drugs, then emergence of the resistant strain could happen earlier and could be even stronger. The emergence and persistence of TBEr is associated with the increase in the intracellular bacterial load of TBIr , increase in MIr levels and a marked increase in the population of CD4+ helper T cells, Mtb specific CTLs and activated macropahges.

3.1.

Efficacy Levels

Using the same degree of drug sensitivity of the drug resistant strain we carried out numerical experiments to determine the impact of reduced drug efficacy values on the on-set and progression of the drug mutant strain. Generally, we observed that as the efficacy values of the drugs decrease, the levels in the extracellular and intracellular environments of the drug mutant strains increase. We notice interesting results in Figure 2 (b) and (e), that there are more infected macrophages by TBEr and the corresponding intracellular bacterial burden of TBIr from efficacy 1 levels instead of the expected efficacy 2 bacterial burdens since efficacy 2 has the lowest efficacy values than efficacy 1 and regimen efficacy. This result is unexpected, but this could be connected to the issue of co-existence and competition between the sensitive strain and the resistance strain. Because as efficacy levels decrease, the TB drug regimen fails to clear even the drug sensitive strain (Magombedze et al., 2006b), such that in the efficacy 2, the sensitive strain will not be cleared. The sensitive strain will then compete with the drug resistant strain, hence, lower bacterial burdens than those from efficacy 1. In efficacy-1 drug efficacy level, the sensitive strain takes a prolonged time to get cleared, but efficacy 2 does not eradicate the sensitive strain (this result is also shown in Magombedze et al., (2006b)). These numerics demonstrate that, the administration of drugs that could result in lower efficacy values as a result of poor adherence, drug mal-absorption and reduced drug bio-availability that is noticed in HIV/TB co-infected individuals can cause a problem of emergence of resistant strains and co-existence of the drug resistant strains and the wild type strain. This complicates the case of TB treatment and could result in the transmission of drug resistant strains in a population that are difficult to control.

3.2.

Drug Sensitivity Levels

In determining the degree of drug sensitivity of the drug mutant strains during TB drug administration, we varied the parameter ds (0 < ds ≤ 1), that measures the degree of sensitivity of the mutant strain. We observe that as the value of ds decreases (resistance of mutant strain to the drugs increase), there is a rapid and strong emergence of the resistant strains. Our simulations suggest that there should be a value of drug sensitivity between ds = 0.4 and ds = 0.5 which is a critical drug sensitivity value such that if ds is below that value, then the resistant strain will be eradicated. However, this is rather

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Table 2. Table of parameters used in the model Name βm αr ωr σ SR βis , βir α ω SA k2s , k2r µM I µa N N1 γ1 γ2 γ4s , γ4r ϕs , ϕ r S1 SE µd k3 NT NC N2 k4 bo k5 γ5 S2 ST AT αT p1 p2 µT µC E Gn µg SC

Value 5000 0.05 0.4 0.01 1000000 0.4, 0.3 0.011 0.03 500000 0.4, 0.3 0.011 0.011 50 25 0.000000125 0.0000000125 0.1 0.49 100 1000000 0.3 0.000000125 40 40 30 0.000000125 500000 0.00000185 0.85 100 1500000 1000 0.3 0.03 0.01 0.01 0.68 10 1000 0.011 1500000

Defination MR source Recruitment due to MA Recruitment due to MI Proliferation of MA Saturation limit of TB Infection rate Death Maximal activation Sat constant of activation MR Bursting Death Death Burst Size TBE size that cause chronic infection TBE killing by MA TBE killing by MR TBE multiplication TBI multiplication source of T TBE saturation limit MA deactivation TBI killing by C (CTL granules) TBE size killed by T TBE size killed by C TBI size killed by C (CTL granules) Apoptosis Apoptosis inhibition factor Lysis TBE killing by C (CTL granules) source of C Cytokine induced T Sat Limit Half-sat of T for Apoptosis Recruitment Proliferation of T Proliferation of C Death Death Bounding value CTL granules saturation constant TBI killing MR immune mechanisms Cytokine induced C Sat limit

Units MR cm−3 day day −1 scalar day −1 TB cm−3 day −1 day −1 day −1 TB cm−3 day −1 day −1 day −1 TBI MI−1 TBI MI−1 (cm3 MI−1 )day (cm3 MI−1 )day day −1 day −1 day −1 TBE cm−3 day −1 day −1 TBI MI−1 TBI MI−1 TBI MI−1 day −1 scalar day −1 day −1 C cm−3 day −1 T cm−3 day −1 (cm3 T −1 )day day −1 day −1 day −1 day −1 day −1 scalar (cm3 T −1 )day day −1 Ccm−3 day −1

Reference [2, 3, 16] [2, 3, 16] [2, 3, 16] [2, 3, 16] [2, 3, 16] [2, 3, 16] [2, 3, 16] [2, 3, 16] [2, 3, 16] [2, 3, 16] [2, 3, 16] [2, 3, 16] [2, 3, 16] [2, 3, 16] [2, 3, 16] [2, 3, 16] [2, 3, 16] [2, 3, 16] [2, 3] [2, 3, 16] [2, 3] [2, 3] [2, 3] [2, 3] [2, 3] [2, 3] [2, 3] [2, 3] [2, 3] [2, 3] [2, 3] [2, 3] [2, 3] [2, 3] [2, 3] [2, 3] [2, 3] [2, 3] [2, 3] [2, 3] [2, 3]

a hypothetical value, since ds is a parameter that represents many factors that contribute to reduce drug sensitivity, include poor compliance and adherence, reduced drug absorption and bio-availability in HIV/TB co-infection cases. Figure 3 demonstrates that as the percentage or degree of drug sensitivity increases, the levels of MIr , TBEr , TBIr reduce,

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Table 3. This table shows the efficacy values for the drugs used to obtain results in Figure 2 Key (Efficacy) regimen efficacy low efficacy-1 low efficacy-2

γo 0.7 0.415 0.2

ǫo 0.2 0.15 0.06

κo 0.025 0.015 0.01

while macrophage population increases. Decreases in MIr , TBEr , TBIr is associated with decreases in activated macrophages, CD4+ helper T cells and Mtb specific CTLs.

3.3.

Mono Drug Resistance

We carry simulations of a rare possibility of the occurrence of mono resistance to single TB drugs during administration of a three drug regimen. We only considered a INH mono resistant strain and an RMP mono resistant strain, then we made a comparison of these two mono resistant strains with the INH/RMP resistant strain. With 40% sensitivity for all the strains to the TB drugs. At this level of drug sensitivity, all the mono resistant strains will be cleared but it takes more time to clear the INH mono-resistant strain than the RMP mono-resistant strain. This suggests that the development of mono-resistance during administration of a TB three drug regimen is unlikely to occur. Since INH and RMP are the most potent drugs in the TB first line regimen, therefore occurrence of INH/RMP resistance compromise the option for TB first choice drugs. This might result in opting for the second line drugs that require a longer time of administration that can span from 9 months to 12 months. In these simulations, we used the regimen efficacy values. Our simulations in Figure 2 showed that reducing drug efficacy values increase the levels of drug resistant strains. Result in Figure 4 also suggest that reducing drug efficacy levels might also support the development of mono-resistance.

4.

The Impact of Adherence on the Emergence of MDRTB-strains

Using impulsive differential equations and treating drug parameters γs and ǫs as concentration functions instead of fixed efficacy values as we did in the simulations in section 3, we notice that drug concentration is given by the following equations. dCnγ,ǫ (t) dt

= −wCnγ,ǫ (t),

γs , ǫs =

Cnγ,ǫ (t) . IC50 + Cnγ,ǫ (t)

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Cn (0) = 0 at the dosing times, the drug concentration increases instantaneously by the single dose amount Cnm , if a dose is taken Cn (t+ k) =

m Cn (t−1 k ) + Cn , t = 0, τ, 2τ , dose is taken,  Cn (t−1 ), t = 0, τ, 2τ, dose missed. k



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Doses are taken at fixed times 0, τ, 2τ, ..., where τ is a constant dose interval. Between doses, the drug concentration decays exponentially with decay rate w, determined by the drug half-life t 1 and w = 2 lnt 2 . 2 We carry numerical simulations for the system of ODEs (1-4) together with equations (38-40). Since TB drugs can be administered twice or thrice a week [8], we used a dose interval of 3.5 days (τ = 3.5). This follows that if a patient take a dose at time τ = 0, then, the second dose will be at τ (after 3.5 days). Missing one dose after taking the first dose implies that the next dose will be at time 2τ (=7 days) and so on. In this section we seek to find when and how many doses a patient should miss before emergence of a resistant strain and the impact of poor adherence to the drug sensitive strain Complete adherence of the prescribed treatment strategy achieve the required treatment results and prevent the emergence of a drug resistant strain. Missing one dose results in long clearance of sensitive strain and facilitates emergence of a resistant strain (as shown in Figure 5). Missing of two or more doses fail to clear both strains. This results in coexistence of strains in the human body.

5.

Discussion

In this chater, we presented a model of two Mtb strains, (i) the drug resistant strain, (ii) the wild type strain. In our earlier studies (Magombedze et al., 2006a) we addressed the issue of CTLs in the lungs, we then extended the work to understand the treatment of Mtb in the lungs with the first line drug regimen (Magombedze et al., 2006b). Our studies showed that when DOTs is closely followed, the first line drug regimen is effective in controlling the Mtb infection within the stipulated time. This study addresses the issue of emergence and the on-set of Mtb infection from a drug resistant strain during the treatment of a drug sensitive strain. Our analysis show that the on-set of an Mtb infection with a multi-drug resistant strain is a daunting task or an unlikely event. Emergence of Mtb resistance should be accompanied by a number of tasks that have a compound effect on the level of drug sensitive of the resistant strain. If the level of drug sensitive is a function of drug concentration that can be reduced by poor compliance or poor adherence or due to drug mal-absorption and bio-availability in HIV/TB co-infected individuals, then, reduced drug concentration corresponds to reduction in the drug sensitivity parameter, that in turn favours the onset of an multi-drug resistant strain. Our simulations demonstrate that if the patient is highly compliant, then, both strains will be cleared. Reduced drug efficacy values support the on-set of TB resistance and poor clearance of both strains. Considering the dynamics of INH and RMP mono-resistant strains, it appears that the occurrence of mono-resistance is an unlikely event during administration of a TB three drug regimen.

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Figure 5. Graphs showing the impact of TB drug adherence during TB DOTs: (a) Mtb sensitive strain infected macrophages, (b) Mtb resistant strain infected macrophages, (c) sensitive Mtb strain, (d) resistant Mtb strain, (e) intracellular sensitive Mtb strain and (f) intracellular resistant Mtb strain.

As drug resistance increases, emergence and persistence of an MDRTB-strain is eminent. The main result from this study is that even if high efficacy values are used, but if

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factors that reduce drug concentration, bio-availability of drugs at the site of infection increase, the onset of an MDRTB-strain is most likely to occur. Also our studies shows that if the drugs are not strong enough, both strain will co-exist. Co-existence and dominance of the drug sensitive strain will result in low levels of the MDRTB-strain. TB disease will be due to the dominant strain and transmission of a drug sensitive strain is better because it is not difficult to treat. Numerical results in section 4 suggest that a strongly adherent patient has no risk of developing MDRTB-strain and treatment is successful. Missing some doses or treatment compliance failure aids treatment failure and emergence of MDRTB-strains. Missing one dose systematically through the prescribed 6 months of treatment results in the development of a resistant strain that causes a latent TB infection and increasing the number of misses will increase the risk of development and transmission of MDRTB-strains. High levels of adherence to TB treatment are required to prevent occurrence/emergence of MDRTB strains that can either cause latent or active TB due to the MDRTB-strain. We recommend strong adherence to TB treatment as a way to close all avenues for emergence of MDRTB resistant strains. Since TB drug bio-availability and absorption is greatly affected in HIV/TB co-infected individuals, therefore TB treatment in such a case is associated with emergence of TB resistant strains. TB treatment in AIDS patients is a real concern and is an area where medical research efforts should be directed.

References [1] Castillo Chavez C. et al., 2002. On the computation of R0 and its Role on global stability. IMA.125 , 229-250. [2] Magombedze G, Garira W, Mwenje E. 2006. Modelling The Human Immune Response Mechanisms To Mycobacterium tuberculosis Infection In The Lungs. Math. Biosci. Eng. 3(4), 661-682. [3] Magombedze G, Garira W, Mwenje E. 2006. Mathematical Modeling of Chemotherapy of Human TB Infection. Bio. Sys. 14(4), 509-553. [4] Magombedze G, Garira W, Mwenje E. 2009. The role of dendritic cells and other immune mechanisms during human infection with Mycobacterium tuberculosis. I. J. Biomaths. 2(1), 69-105. [5] Castillo-chavez C. and Feng Z. 1997. To treat or not to treat: The case of tuberculosis. J. Math. Bio. 35, 629-656. [6] Blower S.M and Gerberding J.C. 1998. Understanding, predicting and controlling the emergence of drug-resistant tuberculosis: a theoretical framework. J. Mol. Med. 76, 624-636. [7] Diekmann O et al., 1990. On the definition and computation of the basic reproduction number R0 in models for infectious diseases in heterogeneous population. J. Math. Biol. 28, 365-382.

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[8] Hershfield E. 1999. Tuberculosis: 9. Treatment. CMAJ. 161(4), 405-410. [9] Howe M. 1993. Aids Information Newsletter: Aids Information Center. Aids Service VA Headquarters. 202, 273-276. [10] Jung, E., Lenhart, S., Feng, Z.S. 2002. Optimal Control of Treatments in a Two-Strain Tuberculosis Model. Discrete and Continuous Dynamical System-Series B. 2(4). [11] Telenti A., Imboden P., Marchesi F., Lowrie D., Cole S., Colston M.J., et al., 1993. Dectection of rifampicin-resistance mutations in Mycobacterium tuberculosis. Lancet. 341, 647-650. [12] Balganesh, T.S., Balsubramanian, V., Anand Kumar, S. Drug Discovery for Tuberculosis: Bottlenecks and path forward. 2004. Current Science. 86(1). [13] Jawahar M.S. 2004. Current Trends in Chemotherapy of Tuberculosis. Indian. J. Med. Res. 120, 398-417. [14] Hafner R et al., 1997. Early Bactericidal Activity of Isoniazid in Pulmonary Tuberculosis: Optimazition of Methodology. Am. J. Respir. Crit. Care. Med. 156, 918-923. [15] Wigginton, J. E., Kischner, D. 2001. A model to predict cell mediated immune regulatory mechanisms during human infection with Mycobacterium tuberculosis. J. Immunol. 166(3), 1951. [16] Marino, S,. Kirscher, D. 2004. The human immune response to Mycobacterium tuberculosis in lung and lymph node. Journal of Theoretical Biology 227, 463-486. [17] Flynn JL, Goldstein MM, Chan J, Triebold KJ, Pfeffer K, Lowenstein CJ, et al., 1995. Tumor necrosis factor-a is required in the protective immune response against Mycobacterium tuberculosis in mice. Immunity. 2, 561-72. [18] Flynn JL. 2004. Immunology of tuberculosis and implications in vaccine developments. Tuberculosis. 84, 93-101. [19] Schluger NW, and WN Rom. 1998. The Host Immune Response to Tuberculosis. Am. J. Respir. Crit. Care. Med. 157, 679-691. [20] Schluger NW. 2001. Recent advances in our understanding of human host response to tuberculosis. Respir. Res. 2, 157-163. [21] Jindani A. 1979. The effect of single and multiple drugs on the viable count of M.tubeculosis in sputum of patients with pulmonary tuberculosis during the early days of treatment. Thesis, University of London. [22] Jindani A, Aber VR, Edwards EA, Mitchison DA. 1980. The early bactericidal of drugs in patients with pulmonary tuberculosis. Am Rev Respir Dis. 121, 939-49. [23] Sud D, Bigbee C, Flynn JL, Kirschner DE. 2006. Contribution of CD8+ T Cells of Mycobacterium tuberculosis Infection. J. Immunol. 176, 4296-4314.

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[24] Segovia-Juarez JL, Ganguli S, Kirschner D. 2004. Identifying control mechanisms of granuloma formation during M. tuberculosis infection using an agent-based model. Journal of Theoretical Biology 231, 357-376. [25] Gillespie S.H. 2002. Evolution of Drug Resistance in Mycobacterium tuberculosis: Clinical and Molecular Perspective. Antimicrobial Agents and Chemotherapy. 46:267274. [26] Morlock G.P., Plikaytis B.B., Crawford J.T. 2000. Characterization of Spontaneous, In Vitro-selected, Rifampin-Resistant mutants of Mycobacterium tuberculosis Strain H37Rv. Antimicrobial Agents and Chemotherapy. 44:3298-3301. [27] Rattan A., Kalia A., Ahmad N. 1998. Multidrug-Resistant Mycobacterium tuberculosis: Molecular perspectives. Emerging Infectious Diseases. 4:195-209. [28] Pandey R., and Khuller G.K. 2004. Subcutaneous nanoparticle-based antitubercular chemotherapy in an experimental model. Journal of Antimicrobial Chemotherapy. 54:266-268. [29] Lipsitch M., Levin B.R. 1998. Population dynamics of tuberculosis treatment: Mathematical models of the roles of non-compliance and bacterial heterogeneity in the evolution of drug resistance. International Journal of Tuberculosis and lung Diseases. 2:187-199. [30] Cohen T., and Murray M. 2004. Modelling epidemics of multi-drug resistant M.tuberculosis of heterogeneous fitness. Nature Medicine: Letters. 10:1117-1121.

In: Infectious Disease Modelling Research Progress Editors: J.M. Tchuenche, C. Chiyaka, pp. 229-251

ISBN: 978-1-60741-347-9 © 2009 Nova Science Publishers, Inc.

Chapter 8

MATHEMATICAL MODELING FOR TUMOR GROWTH AND CONTROL STRATEGIES Sanjeev Kumar1,a, Deepak Kumar2,b and Rashmi Sharma1,c a

Department of Mathematics, cDepartment of Pharmacy, Dr. B. R. Ambedkar University, I.B.S., Khandari Campus, Agra-282002, India b Department of Mathematics, M. R. International University, Faridabad, India

Abstract A tumor or neoplasm (literally meaning ‘new growth’) is a mass of tissue that grows faster than normal in an uncoordinated manner, and continues to grow after the initial stimulus has ceased. The purpose of this chapter is to describe the tumor and its type (benign and malignant), symmetrical and asymmetrical view of a vascular tumor, angiogenesis process of tumor and vascular tumor. The tumors are basically of two types, benign tumor and malignant tumor, and modeling of tumor growth has to be very clear about the symmetrical view of avascular tumor as well as asymmetrical view of avascular tumor. This chapter will cover the angiogenesis process of tumors and also discuss about the vascular tumor. It will also consider the causes of cancer as well as the different type of tumor treatment therapies. How the mathematical modeling will be useful in the development of biological research is also shown through this chapter. In the newly progress of the mathematical models of cancerous growth, consider a procedure for cancer therapy which consists of interaction between immune response (immune cells) and tumor cells without any specific drug. This process is modeled as a system of tumor cell density and tumor necrosis factor. The purpose of this chapter is to establish a thoroughly accurate mathematical analysis of the model and to explore the concentration of tumor cell and immune response. A model of chemotherapy that serves the procedure for tumor treatment, which contains tumor cell energy and specific dose of Adriamycin is also considered in this chapter. The tumor cell energy depends upon tumor cell density and the Adriamycin is a particular drug that works against tumor cell energy. Problem in the form of partial differential equations for the tumor cells density, tumor cell energy and the effect of Adriamycin is modeled here. a

E-mail address: [email protected]. Corresponding author. E-mail address: [email protected] c E-mail address: [email protected] b

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Sanjeev Kumar, Deepak Kumar and Rashmi Sharma As far as the idea of tumor treatment is concern, one may go for the combination of immunotherapy and viral therapy against tumor cells. Initially, consider the patterns of injection for cancer therapy that consists of injecting replication-competent viruses and infect the tumor cells. Such type of work may generalize the effect of injected replicationcomponent virus throughout the tumor in the presence of immune response. Now by considering the concept of immunotherapy, a new mathematical model of radioimmunotherapy for tumor treatment is also considered in this chapter. In the proposed model, we worked with the linear and exponential spatial dependence of tumor parameters and formulate the dose distributions model of radiotherapy with immune response for tumor treatment.

1. Introduction A cell divides in a most controlled manner, if not, it loses its control and deviates from the normal rule and thus causes chaos in the respective area. This loss of control over cell division and unnecessary production of the contiguous mass of cells is called tumor. Tumors are classified as benign and malignant although a clear distinction is not always possible. Table. Difference between benign and malignant tumors Benign Slow growth Cells well differentiated Usually encapsulated No distant spread Recurrence is rare

Malignant Rapid growth Cells poorly differentiated Not encapsulated Spread via lymph, blood, body cavities Recurrence is common

1.1. Avascular Tumor (Solid Tumor) Growth • • • • • •

Avascular, i.e., no blood vessels Nutrient obtained via diffusion Cells in the center starve Cells on the periphery thrive Cells in the interior are quiescent Growth limited to a few mm in diameter

Figure 1.1. Avascular Tumor.

Mathematical Modeling for Tumor Growth and Control Strategies

1.2. View of Avascular Tumor

Figure 1.2. Two-dimensional view of Tumor.

Figure 1.3. Three-dimensional view of Tumor.

Figure 1.4. Three-dimensional asymmetries view of Tumor.

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In vitro observations suggest that in the early stages, solid tumors remain approximately spherical as they grow. Therefore, the analysis focuses on the existence and stability of radially symmetric solutions of the model equations. We believe that our derivation of this simplified model indicates how models of avascular tumor growth arise as special cases of more physically based models. If all asymmetric perturbations ultimately disappear, then, we conclude that the tumor will simply grow as a radially symmetric mass.

1.3. Angiogenesis • •

Formation of new blood vessels from the existing vasculature Hypoxia induces a chemical cascade which stimulates endothelial cells of near by vessels to aggregate, proliferate, and migrate towards the tumor

Figure 1.5. The process of angiogenesis.

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1.4. Vascularized Tumor Penetrated by Capillaries

Figure 1.6. Vascular Tumor.

1.5. Cause of Cancer In order to support continued growth, the tumor’s vascular system persistently remodels itself. Hence, angiogenesis is an on-going process, continuing indefinitely until the tumor is removed or killed, or until the host dies. a) Hereditary Changes • Several relatives with cancer • Cancers that occur at an earlier age than normal • Multiple or bilateral cancers: for example, a person with breast cancer who also develops ovarian cancer b) Somatic and Multiple Mutations This is because of accumulation of several mutations within a cell that increase with age. For example, a retinoblastoma results from multiple mutations in the persons who have inherited one of the mutations that can cause cancer. c) Environmental Factors • Age: cancer is most common among people over the age of 50. • Diet: high-fat, high-cholesterol diets are proven risk factors for several types of cancer, particularly colon cancer. • Obesity: although no clear link has been established, research indicates obesity may be a contributing factor to some cancers. • Cigarettes greatly increase the lung cancer risk, even among non-smokers forced to inhale passive smoker. Other tobacco products, like pipes and chewing tobacco, are linked to cancers of the mouth, tongue and throat. Most of these are the cause of cancer in India. • Long-term exposure to chemicals like asbestos, radon and benzene. • Immune system diseases like AIDS can make one more susceptible to some cancers.

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Sanjeev Kumar, Deepak Kumar and Rashmi Sharma d) Viruses The viruses that contain DNA as their genetic material and induce tumor are called tumor viruses; for example, papova viruses, adenoviruses, herpes viruses, poxviruses and hepatitis B virus. The viruses that contain RNA as genetic material and cause tumors in humans and other animals are called RNA tumor viruses. All the RNA tumor viruses are put into retrovirus group. e) Chemical Carcinogens Chemicals such as benzpyrene, benzanthracene and aflatoxin B1 had no specific affinity for DNA, but their carcinogenicity is due to their binding with certain proteins resulting in activation of proteins.

1.6. Treatments In medical science, researchers have developed an arsenal of standard treatment practices that have proven effective on many different types of cancer. Depending on the type of disease, these treatments are used alone or in combination to either control cancer cell growth or to eliminate the disease entirely. a) Surgery Surgery is the oldest form of cancer treatment. About 60% of cancer patients will undergo surgery, either by itself or in combination with other therapies. b) Chemotherapy Chemotherapy uses powerful drugs to kill cancer cells, control their growth, or relieve pain symptoms. Chemotherapy may involve one drug, or a combination of two or more drugs, depending on the type of cancer and its rate of progression. c) Radiation Therapy Also known as radiotherapy, this treatment uses large doses of high-energy beams or particles to destroy cancer cells in a specifically targeted area. Radiation damages the internal chemical structure of cancer cells, which keeps them from multiplying. d) Gene Therapy The focus of most gene therapy research is the replacement of a missing or defective gene with a functional, healthy copy, which is delivered to target cells with a "vector." Viruses are commonly used as vectors because of their ability to penetrate a cell’s DNA. These vector viruses are inactivated so they cannot reproduce and cause disease. e) Viral Therapy In this therapy, the rapid advances are being made in the engineering of replicationcompetent viruses to treat cancer. Adenovirus is a mildly pathogenic human virus that propagates prolifically in epithelial cells, the origin of most human cancers.

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f)

Immunotherapy A tumor is an abnormal growth of cells from a lump or mass of transformed cells. This abnormal growth occurs due to loss of regulatory control on cell division and the genes responsible for certain characteristics of the cell have undergone mutation.

2. An Ordinary Mathematical Model for Tumor Growth 2.1. Temporal Model A simple and time-tested approach to modeling tumors is the use of differential equations for the total number of cells. This model ignores the spatial aspects of the disease.

dc = λc , dt

(2.1)

λ (= 5.04 × 10 −4 ) is the proliferation rate of the tumor. This model assumes that the tumor grows exponentially at the rate λ in the

where c is the number of tumor cells at time t, and absence of treatment.

Figure 2.1. The tumor cells with respect to time, with the 3000 hours study of a vascular tumor growth.

2.2. Spatio-temporal Model a) Spherical Model: Murray (1990) formulated the problem of modeling tumor growth as a density equation, which can be expressed as follows: The rate of change of tumor cell population = the diffusion (motility) of tumor cells +the net proliferation of tumor cells

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∂c = ∇.( D∇c) + λc , ∂t

or mathematically

(2.2)

where c(x,t) designates the tumor cell density at x radial distance from the origin and at time t, λ is the proliferation rate and ∇ defines the spatial gradient operator, D is the diffusion coefficient representing the active motility of tumor cells. The numerical values of the above parameters are as follows: D= 5.4 × 10

−3

mm2/h,

λ = 5.04 × 10 −4 /h, N 0 = 1.19 × 10 6

cells, t 0 = 1.18 × 10 4 h

The initial and boundary condition for the brain tumor is as determined by the Burgess et al., (1997). Now we set: 2

N 0 eλ ( t0 +t ) e − x 4 Dt0 c ( x, t ) = . 8(π Dt0 )3 2

(2.3)

2

Initially, the tumor cell density is c( x,0) =

N 0 e λt0 e − x / 4 Dt0 8(πDt 0 ) 3 / 2

Figure 2.2. The surface plot tumor cell density with respect to time t and radius x.

The above figure shows the surface plot of tumor cell density with respect to time t and radius x, with a consideration of time 0 to 3000 h and radius 0 to 1 mm. It can be partially be interpreted as: the growth of tumor cell density (as shown in the above figure) shows that initially, the tumor cell density is 1.99 × 10 6

boundary of the tumor it is 1.98 × 10 .

6

cells at the center of tumor while at the

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b) Cylindrical Model: The physical representation of a solid mass tumor is a cylindrical capillary surrounded by a cylindrical mass of tumor cells. Therefore, we are able to assume an axi-symmetric geometry. Figure 2.3 depicts the 3D representation of the capillary surrounded by tumor, with the boxed-out section representing the region of focus in our model. Figure 2.4 illustrates the 2D-axisimmetric geometry, which is split into Region 1 (capillary) and Region 2 (tumor).

Figure 2.3.

Figure 2.4.

The growth of a tumor in a rigid walled cylindrical duct is examined in order to model the initial stages of a tumor cell expansion in ductal carcinoma in situ of the breast. The concentrations of tumor cell is expressed as c(r , z , t ) , where r and z being the distances in the radial and axial directions, respectively, and t the time. In this region, there is connective −3

term while the diffusion coefficient of tumor cell Dc is 5.4 × 10 mm2/h while the proliferation rate of cells is

λ = 5.04 × 10−4 /h . The initial size of the tumor at its source

6

point is N 0 = 1.19 × 10 cells.

⎛ ∂ 2 c 1 ∂c ∂ 2 c ⎞ ∂c ⎟ + λc = Dc ⎜⎜ 2 + + ∂t r ∂r ∂z 2 ⎟⎠ ⎝ ∂r

(2.4)

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Sanjeev Kumar, Deepak Kumar and Rashmi Sharma If axial diffusion is neglected, then this equation becomes

⎛ ∂ 2 c 1 ∂c ⎞ ∂c ⎟ + λc = Dc ⎜⎜ 2 + ∂t r ∂r ⎟⎠ ⎝ ∂r

(2.5)

⎧1 0 < r < 1 c(r , t ) = ⎨ r >1 ⎩0

(2.6)

We take as initial condition

Using the symmetry of the capillary about the axis, the radial component of the concentration gradient must vanish along r = 0 so that, in the capillary region,

∂c = 0 at r = 0 ∂r ∂c = 0 at r = 1 ∂r

and

(2.7)

Figure 2.5 shows the evolution of tumor cell concentration with the radius r (0-1 mm) and the time (0-3000 h). This figure represents the study of cylindrical capillary surrounded by a cylindrical mass of tumor cells, and we observe through the figure that the tumor cell 6

concentration is 1.99 × 10 cells initially. After 3000 h, the tumor cell density is 9 × 10 cells.

Figure 2.5. Evolution of tumor cell concentration with radius and time.

6

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2.3. Conclusions Here a discussion was made about the two different, but interrelated mathematical models of avascular tumor growth. The field has developed over the past twenty years, in response to new biological results and deficiencies in the existing models. Here, one may believe that the mathematical modeling will play an increasingly important role in helping biomedical researchers to gain useful insights into different aspects of solid tumor growth. At the same time, by studying such biological systems, it could be possible to generate new models that either extends the existing theories or independent mathematical interest. Whilst there are some claims of the discovery of the cure for tumor, we are fully optimistic that in the near future, medical scientists will have the necessary knowledge to develop effective treatments for individual tumor patients.

3. A Model for the Interaction between Tumors Cell Density and Immune Response With the recent development of the mathematical models of cancerous growth, consider a procedure for cancer therapy, which consists of interaction between immune response (immune cells) and tumor cells without any specific drug. Here, the main focus of study is to observe the effects of interaction between tumor cell density (TCD) and immune response (TNF) without a specific drug.

3.1. Mathematical Model According to Murray’s formulation (2.2), we have

∂c = ∇.( D∇c ) + λc ∂t

(3.1)

If i(x,t) define the immunotherapy treatments, then assuming the loss proportional to the amount of therapy at a given time, the equation model can be written as : The rate of change of tumor cell population = the diffusion (motility) of tumor cells + the net proliferation of tumor cells - loss of tumor cells due to immune response (TNF) . Or mathematically we may write,

∂c = ∇.( D∇c ) + λc − αci ∂t

∂i = β ci − γi 2 ∂t

(3.2)

(3.3)

The initial and boundary conditions for the brain tumor are determined as in Burgess et al., (1997).

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Sanjeev Kumar, Deepak Kumar and Rashmi Sharma Finally, we have 2

N 0eλ (t0 +t ) e − x / 4 Dt0 e −α ti c ( x, t ) = 8(π Dt0 )3/ 2 where initially, the tumor cell density is c( x ,0 ) =

N 0 e λt0 e − x

2

(3.4)

4 Dt0

8( πDt 0 )3 2

β = 0.048mm 3 /cells.h and γ = 1.6 ml/ng.h 3.2. Solution Figure 3.1 shows the TCD at different radius x with respect to time in the absence of immune response. A study of the 100 hours of tumor growth and radius of tumor up to 10mm is made, and after 50 hours, comparing the figures in the presence of immune response, the TCD is 6

less when immune response is not working. The TCD is 0.75 ×10 cells (approximately) in the absence of immune response.

Figure 3.1. Tumor cell density computed at different radiuses of the gliomas with respect to time in absence of immune response (TNF density).

In figure 3.2, suppose that the immune killing rate α is working against the tumor growth and its value is 0.05 mm3/ng.h. This shows the TCD as a function of radius x and time t. With respect to time, one may observe that TCD is going to decrease more rapidly when immune response is working, also observe that TCD in the presence of immune response is

0.60 ×106 cells (approximately).

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Figure 3.2. Tumor cell density computed at different radiuses of the gliomas with respect to time in presence of immune response (TNF density) when immune killing rate α = 0.05 .

3.3. Conclusion The approach in this work is to develop the simplest model that can show the interaction between TCD and immune response (TNF). Herein, the tumor is not symmetric, and no drug is administered, thus, we have a simple possible model, which can show the tumor growth with and without immune response of the body. The model shows that the radius of brain tumor increases with respect to time. Hence, in the presence of immune response, the TCD behavior is instinctive. The TCD curves show the non-monotonic behavior with immune response (TNF). In summary, we have formulated and analyzed a simple model that shows the interplay between a growing tumor cell density (TCD) and immune response (TNF), and the result suggests that, although TCD is responsible for growth of tumor, the immune response inhibits a direct tumor growth.

4. A Model for Tumor Treatment with Chemotherapy Consider a model that represents the procedure for tumor treatment, which consists of tumor cell energy and specific dose of Adriamycin. The tumor cell energy depends upon tumor cell density and the Adriamycin which is a specific drug that works against tumor cell energy. Here, a model in the form of partial differential equations formulation is designed for the tumor cells density, tumor cell energy and the effect of Adriamycin.

4.1. Introduction The energy in all cells originates from the energy of the sun’s light quanta, which is convicted by photosynthetic plants into cellular molecules. These plants are used as food sources by various microorganisms and animals. Life is an energy process. It takes energy to operate

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muscles, extract wastes, make a new cell, and heal wounds, even to think. It is in an organism’s cells that all this energy is spent. Adriamycin (Doxorubicin), an active medicine against many cancers is one of the most recommended chemotherapy drugs, having been in use for decades. Adriamycin is clear, orange-red powder or liquid, which is administrated intravenously only. It is most commonly used in treatment of all cancers (Breast, Stomach, Lymphomas etc…). Adriamycin is a drug that interrupts the cell cycle, effectively stop the cell growth, and it degrades rapidly in solution. A fluorometric method was developed to determine the precise dose use in treatments. According to a recent research Doxil appears to be as effective as Adriamycin for treating women with metastatic breast cancer. Doxil was less likely to cause heart problems, less of white blood cells, vomiting, and hair loss.

4.2. The Model Mathematically, for an untreated tumor, the dynamical equation may reasonably be quantified by a single partial differential equation

∂c = ∇.J + λ c ∂t where J = D∇μ (c) . The gradient of the potential

(4.2.1)

μ produces a flux J which is

proportional to ∇μ . D is the Fickian diffusion coefficient representing the active motility of tumor cells, which may depend on x, t and c. The tumor spread is assumed to be spherically symmetric in this model, and x measures the distance from the center.

∂c = ∇.[ D∇μ (c)] + λ c ∂t

(4.2.2)

Spatial distribution of cells, an energy density n(c), which is an internal energy per unit volume of an evolving spatial pattern and the total energy N(c) in a volume is associated by the following equations:

N (c) = ∫ n(c) dx V

(4.2.3)

The small variation in energy δ N , is the work done in small variety states by an amount δ c , the variation derivative δ N

μ (c ) =

δ c defines a potential μ (c) . Therefore,

δN = n′(c) δc

(because δc → 0 )

(4.2.4)

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243 2

The internal energy density is usually given by the quadratic n(c) = c 2 . In this context, we see that tumor cell energy depends on the tumor cell density, thus,

μ (c ) = c .

Therefore, choose the following functional form for the tumor cell c as discussed by Burgess et al., [1997].

c ( x, T ) =

N 0eλ ( t 0 +T ) e − 8(πDt0 )

x2

3

4 Dt 0

(4.2.5)

2

The initial and boundary conditions for the Adriamycin resistance, tumor cell density are:

c = c0 ( x )

at t=0

c = cmax .

at x=0

c = c*

at x=1

Now, introduce the dose of adriamycin bind with cells and it can prevent the repair of DNA. We can use Adriamycin against repair of DNA of tumor cells; where the term α is the effect of adriamycin on tumor.

∂c = ∇.( D∇c) + λc − αc ∂t

(4.2.6)

4.3. Numerical Result

(a)

(b)

Figure 4.1. (a) shows the tumor cell density without effect of adriamycin at a distance 0 to 1mm from the center of the tumor and the time 0-3000 h, while 4.2(b) shows the tumor cell density with time.

Figure 4.1 shows the surface plot tumor cell density with respect to time t and radius x, where the time varies from 0 to 3000 h and radius is 0 to 1 mm. The growth of tumor cell

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density in above the above plot is initially 1.99 × 10 cells. After 3000 h, the tumor cell density is almost same as at the center and boundary of the tumor, in which the effect of adriamycin is neglected ( α = 0 ).

(a)

(b)

Figures 4.2. (a) and (b) shows the tumor cell density with the effect of dose adriamycin α = 0.0001 / h and α = 0.00025 / h at a distance 0 to 1mm from the center of the tumor and time range 0-3000 h respectively. 6

Figure 4.2 (a) shows that the tumor cell density is 1.99 × 10 cells initially and after 6

3000 h of treatment, it increases up to 6.75 × 10 cells, with the effect of adriamycin being α = 0.0001 /h. Figure 4.2 (b) shows that after the treatment, the tumor cell density is

4.2 × 106 cells, the effect of adriamycin being α = 0.010 /h. 4.4. Conclusions An enzyme (Adenosine Triphoshate ATP) provides chemical energy for the cell and this enzyme ATP release energy by releasing a phosphoric acid radical. Then, energy derived from the cellular nutrient causes the acceptor molecule and phosphoric acid to recombine to form new ATP. The entire process continues over and over again. Without that energy, blood vessels cannot grow to the site of a tumor, and without the nutrient supply in blood, tumors cannot grow larger than a pinhead. This model does not account the behavior of dead tumor cells. The numerical simulation mainly involved the study of the effects on tumor cell survival of the dimensionless parameter α , which encapsulates the extent of penetration of the drug. The simulation emphasizes that drug penetration is a crucial factor in determining drug effectiveness. The growth of tumor in a spherical shape has been examined in order to describe the initial stages. Here, we introduced a tumor cell energy model and used tumor cell density with the effect of adriamycin to describe the movement of tumor cells. As observed for different values of effect of adriamycin. Therefore, the numerical results suggest that the

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tumor cell density decreases with the effect of adriamycin, because tumor cell energy fully depends on the tumor cell density.

5. A Model of Radioimmunotherapy for Tumor Treatment There is rapidly growing interest in the potential for synergistic, clinically relevant therapeutic responses by combining radiation therapy with immune response. A new mathematical model of radioimmunotherapy for tumor treatment is described. In this model, we work with the linear and exponential spatial dependence of tumor parameters and formulate the dose distributions model of radiotherapy with immune response for tumor treatment.

5.1. Introduction Radiation therapy is a certain type of energy to kill cancer cells and shrink tumors. Radiotherapy destroys cells in the area being treated by damaging their genetic material. As we know, the radiation damages both kinds of cells, cancer cells and normal cells, therefore, the goal of radiation therapy is to damage as many cancer cells as possible, while limiting harm to nearby healthy tissue. The body’s immune system work to defend it against disease and infection. Typically, when a cancer cell arises in the body, the body’s immune system recognizes it as abnormal and destroys it before it starts spreading. Immunotherapy is the manipulation of the immune system in order to prevent or treating a disease. One of the most well known examples of immunotherapy is the use of vaccines to prevent infectious disease. However, in case of tumor, immunotherapy is being investigated for treatment of cancer. Unlike traditional therapies for tumor which act directly on the tumor cells, immunotherapy is designed to help a patient’s immune system, which also works against the tumor cells, and is an experimental treatment strategy for tumor. Radioimmunotherapy is an experimental, internal radiation treatment in which radioactive isotopes attached to antibodies from the tumor cells, which are injected into the body. The bloodstream carries the antibodies to the tumor where the isotopes attack and kill malignant cells. The division of radiation oncology is investigating a new type of radiation therapy that targets radiation to the tumor using monoclonal antibodies. This form of therapy is called radioimmunotherapy.

5.2. Mathematical Model This model is based on the work of Levin-Plonik et al., (2004). Since we are modeling the radioimmunotherapy, one must include a term for immune response. A model for the special case of radiotherapy scenario in the presence of immune response is designed. In this model, take a linear quadratic model retaining only the linear term for tumor control while neglecting the quadratic term. In the presence of immune response, we formulate the dose distributions of radiotherapy, which optimize tumor cure probability (TCP) in tumor. The dose distribution rate is assumed to be high enough so that the dose distribution is delivered instantaneously.

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G

Let p (r ) be the tumor cell density at time t , and p0 the initial tumor cell density, the

G

dose distribution is represent by x(r ) , the tumor cure probability after dose distribution has

G

been delivered, and the internal body immune response is represented by e( r ) . The given evolution equation of tumor cell density is

p = p0 exp [ −α x + γ t − ke] , where

(5.2.1)

α is the rate of cell killed due to radiosensitivity, γ is the rate of tumor cell division

and k is the rate of cell killed due to immune response. Now,

T C P = exp ⎡− ⎣

∫

p d 3r ⎤ = e x p ⎡ ∫ p 0 e (−α ⎦ ⎣

x + γ t − ke )

d 3r ⎤ ⎦

(5.2.2)

G

Here, the aim is to obtain the dose distribution x(r ) that maximizes TCP while keeping it invariant, therefore,

∫ Wxd r = ∫ Wd r 3

x

(5.2.3)

3

G

where x represents the generalized average tumor dose distribution and the term W ( r ) is a weighting function. G Let W ( r ) = 1 , then, we have

∫ xd r = D , = 3

x

(5.2.4)

V

where D represents average tumor dose distribution and V is the tumor volume.

5.2.1. Linear Spatial Dependence Suppose that the spherical symmetry of tumor as well as the functions α, on radius of the tumor. The functions for

α, γ & k

γ& k

depends

are taken from the work of Levin-

Plonik et al., (2004).

⎛ ⎝

α = α 0 ⎜1 −

r ⎞ α1r ⎟+ R⎠ R

(5.2.5)

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r ⎞ γ 1r ⎟+ R⎠ R

(5.2.6)

r ⎞ kr ⎛ k = k0 ⎜ 1 − ⎟ + 1 ⎝ R⎠ R

(5.2.7)

⎛ ⎝

γ = γ 0 ⎜1 −

The above equations (5.2.5), (5.2.6) and (5.2.7) show linear increase/decrease in

α , γ & k when α 0 , γ 0 & k 0 are the values at r = 0 , while α 1 , γ 1 & k1 are the values at r = R.

Figure 5.1 (a). Plot of tumor cell density for three different values of radiotherapies dose distributions with immune response.

Figure 5.1 (b). Plot of tumor cell density for three different values of radiotherapies dose distributions without immune response.

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Sanjeev Kumar, Deepak Kumar and Rashmi Sharma

Figure 5.1(a) represents the tumor cell density at different dose distribution of radiotherapy drug, considering the value of dose distribution is D=1 which is shown through 6

(*) in figure. Here, the tumor cell density at the centre of the tumor is 3.4 × 10 cells, which is 6

maximum while at the boundary of the tumor it is 2.8 × 10 cells, which is minimum. Now, consider that the value of the dose distribution is D=1.5 as shown through (o) in the figure. 6

Here, the tumor cell density at the centre of tumor is 2.9 ×10 cells, which is maximum, and 6

at the boundary of tumor it is 2.3 ×10 cells, a minimum. The distribution for D=2 is represented by (.) in the figure. Here, the tumor cell density at the centre of tumor has a 6

6

maximum of 2.6 × 10 cells and a minimum of 1.9 × 10 cells at the boundary of the tumor. Figure 5.1(b) shows the tumor cell density at different dose distribution of radiotherapy drug when immune response is not working. If the dose distribution D=1, as shown through 6

(*) in the figure, then the tumor cell density at the centre of tumor is 8.2 ×10 cells which is 6

maximum while at the boundary of tumor, the minimum number of cells is 7.4 × 10 . If the dose distribution is D=1.5, which is represented by (o) in the figure, the tumor cell density at 6

the centre of the tumor is 7.2 × 10 cells, which is maximum and at the boundary of the tumor 6

it is 6.4 × 10 cells, which is the minimum. If the dose distribution is D=2, which shows 6

through (.) in figure, the tumor cell density at the centre of the tumor is 6.4 × 10 cells, which 6

is maximum and at the boundary of the tumor, its minimum is 5.1× 10 cells.

5.2.2. Exponential Spatial Dependence Another functional form for α , γ and k as given by Levin-Plotnik et al., (2004) is

α = α 0 exp[r / d ]

(5.2.8)

γ = γ 0 exp[− r / c ]

(5.2.9)

k = k 0 exp[r / f ]

(5.2.10)

Figure 5.2(a) represents the tumor cell density at different dose distribution of radiotherapy drug. The dose distribution for D=1 is represented by (*) on the figure. Here, the 6

tumor cell density at the centre of tumor is 3.4 × 10 cells, which is the maximum while at the 6

boundary of tumor; it is 1.5 × 10 cells (minimum). When the dose distribution is D=1.5 as shown by (o) on the figure, the tumor cell density at the centre of tumor has a maximum value 6

6

of 1.4 × 10 cells, and a minimum value at the boundary of 2.3 ×10 cells. For the distribution D=2, which is represented by (.) in the figure, the tumor cell density at the centre 6

of tumor is 2.6 ×10 cell, which is maximum, while at the boundary of the tumor it has a 6

minimum value of 1.0 × 10 cells.

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Figure 5.2. (a). Plot of tumor cell density for three different values of radiotherapies dose distributions with immune response.

Figure 5.2. (b). Plot of tumor cell density for three different values of radiotherapies dose distributions without immune response.

Figure 5.2 (b) shows the tumor cell density at different dose distribution of radiotherapy drug when the immune response is not working. If the dose distribution D= 1, as shown by 6

(*) in the figure, then, the tumor cell density at the centre of tumor is 8.2 × 10 cells 6

(maximum), while at the boundary of tumor is 7.9 × 10 cells (minimum). If the dose distribution is D=1.5 as represented by (o) in the figure, the tumor cell density are

7.2 ×106 cells (maximum) and 6.6 × 10 6 cells (minimum), respectively at the centre and the

boundary of the tumor. If the dose distribution is D=2 as shown by (.) in the figure, the tumor 6

cell density at the centre of tumor is 6.4 ×10 cells which maximum and at the boundary of 6

the tumor it is 5.5 × 10 cells, which is minimum. The numerical result of the different functional forms of tumor parameters shows that α and γ works in the opposite directions. If the tumor cells in the center of tumor are hypoxic, then, they are less radiosensitive, therefore, they require more dose at the center of the tumor.

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The results of the exponential spatial dependency for α , γ and k are almost similar to the linear spatial dependency form. The numerical results show that tumor cell density and dose distribution are not very much sensitive to the different functional forms of the tumor parameters. This mathematical study of tumor parameters is helpful to understand the behavior with respect to the radius of the tumor.

5.3. Discussion The radioimmunotherapy program is a multidisciplinary effort involving basic and clinical scientists. Within the next few years, radioimmunotherapy will soon become part of the standard therapies offered to the patients. Investigators are actively evaluating the use of radioimmunotherapy in patients undergoing bone marrow transplantation, to determine whether this form of radiation can complement and/or replace traditional forms of radiation in these patients. The biological effect of radioimmunotherapy is most commonly assessed in terms of absorbed radiation dose. In tumor, conventional dosimetry methods assume a uniform radionuclide and calculate a mean dose throughout the tumor. However, the vasculature of solid tumors tends to be highly irregular and the systemic delivery of antibodies is therefore heterogeneous. Here, a study of the tumor treatment by radiotherapy in the presence of immune response is investigated. A more detailed knowledge of tumor parameters will also be helpful in order to assess the relative benefits of delivering dose distribution in tumor treatment. The study compares the tumor cell density with different dose distributions of radiotherapy in the presence and absence of immune response. The comparison of the results shows some differences, thus, we can say that the immune response is more helpful for the treatment of tumor. These trials are evaluating strategies to further improve the therapeutic index of this treatment. In conclusion, the dose distribution can be optimized in the tumor by selecting the appropriate immune response and radionuclide.

References [1] Eisen M. (1979): Mathematical models in cell biology and cancer chemotherapy. Lecture Notes in Biomathematics (30), Springer-Veralg. [2] Adam J. (1986): A simplified mathematical model of tumor growth. Math. Biosci., 81, 229-244. [3] Gleiberman A. S., Kudrjavtseve E. I., Sharovskaya Yu Yu, Abelev G. I., (1989): Synthesis of alpha-fetoprotein in hepatocytes is coordinately regulated with cell-cell and cell-matrix interactions. Mol. Biol. Med., 6, 95-107. [4] Murray J. D. (1990) Mathematical Biology. Springer Verlag, New York. [5] Ebert and Hobal (1996) Some characteristics of tumor control probability for heterogeneous tumor. Physics in Medicine and Biology, 41, 2725-33. [6] Bischoff J R et. al., (1996) An adenovirus mutant that replicates selectively in p53deficient human tumor cells. Science, 274, 373-376.

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[7] Burgess P. K., Kulesa P. M., Murray J. D. and Alvord E. C. Jr (1997): The interaction of growth rates and diffusion coefficients in a three-dimensional mathematical model of gliomas. J. Neuropath. Exp. Neurol., 56, 704. [8] Chaplain M. A. J. and Kuznetsov V. A., James Z. H. and Stepanova L. A. (1998): Spatio-temporal dynamics of the immune system response to cancer. Mathematical Models in Medical and Health Science. (M. A. Horn, G. Simonett & G. Webb, eds). Nashville, TN: Vanderbilt University Press, 1-20. [9] Chen G. and Goeddel, D.V.(2002) TNF-R1 signaling: a beautiful pathway. Science, 296(6673), 1634-5. [10] McDougail S.R., Andrson A.R.A., Chaplain M.A.J. & Sherrat J.A. (2002). Mathematical modeling of flow through vascular Networks: Implication for Tumorinduceds Angiogenesis and chemotherapy Strategies. Bulletin of Mathematical Biology, 64, 673-702 [11] Swanson K. R., Bridge C., Murray J. D. and Alvord E. C. Jr (2003): Virtual and real brain tumors: using mathematical modeling to quantify glioma growth and invasion. J. Neurological Science, 216, 1-10. [12] Byrne H. and Preziosi L. (2003): Modeling solid tumor growth using the theory of mixtures. 1: Math Med Biol., 20(4), 341-66. [13] Lawrence M.W., Joseph T. W. and Kirn H. D. (2003): Validation and analysis of a mathematical model of a replication-component oncolytic virus for cancer treatment: implications for virus design and delivery. Cancer Research, 63, 1317- 1324. [14] Newwman W., Lazareff J. (2003) A mathematical model for self-limiting tumors. J. Thero. Biol., 222, 361-371. [15] Swanson K. R., Bridge C., Murray J. D. and Alvord E. C. Jr (2003) Virtual and real brain tumors: using mathematical modeling to quantify glioma growth and invasion. J. Neurological Science, 216, 1-10. [16] Friedman A. and Tao Y. (2003): Analysis of a model of a virus that replicates selectively in tumor cells. J. Math. Biol., 47, 391-423. [17] Lawrence M. W., Joseph T. W. and Kirn H. D. (2003): Validation and analysis of a mathematical model of a replication-component oncolytic virus for cancer treatment: implications for virus design and delivery. Cancer Research, 63, 1317-1324. [18] Chaplain M. A. J., Matazavinos A., and Kuzetsov (2004) Mathematical modeling of the spatio- temporal response of cytotoxic T lymphocytes to a solid tumor. 21, 1-34. [19] Rao C.V. (2004): An Introduction to Immunology. Narosa Publishing House, Delhi (India). [20] Andrew M. S. (2005): Radioimmunotherapy of Prostate Cancer: Does Tumor Size Matter? Journal of Clinical Oncology, 23, 4567-4569 [21] Anderson M. D. (2006): Tumor Treatment. The University of Texas, Cancer Center.

In: Infectious Disease Modelling Research Progress ISBN 978-1-60741-347-9 c 2009 Nova Science Publishers, Inc. Editors: J.M. Tchuenche, C. Chiyaka, pp. 253-260

Chapter 9

W ITH - IN H OST M ODELLING : T HEIR C OMPLEXITIES AND L IMITATIONS Gesham Magombedze∗ Department of Applied Mathematics, National University of Science and Technology Box AC 939 Ascot, Bulawayo, Zimbabwe

Abstract Ordinary differential equations (ODEs) are mainly used in the study of control of human infectious diseases at population level and at cellular level. Modelling of with-in host (cellular level) dynamics is coupled with a number of challenges. Often times, simple systems of ODEs are used to model with-in host disease dynamics. Model simplicity that is often accompanied by over simplifying assumptions is preferred at the expense of complicated systems with realistic assumptions. The intricate dynamics of the human immune system coupled with the infecting pathogen’s diverse invasive mechanisms makes the mathematical modelling of with-in host dynamics complex and difficult to comprehend. Also, it is difficult to categorize with-in host model in the well known categories of population models: (i) SIR models and (ii) predator-prey models. In most current with-in host models, both attributes of SIR models and predator-prey models are noticed. Bringing in a balance between simple models that are less realistic but complimented with rigorous mathematical analysis and realistic complicated models complimented with rigorous numerical analysis with less analytical rigor can give more informative results; since mathematical models must aid in explaining biological phenomena that are not well understood by biologists.

Keywords: With-in Host modelling, Michaelis-Menten function, Predator prey modelling. ∗

E-mail address: [email protected]

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Gesham Magombedze

Introduction

Mathematical models have been used to study the transmission of infectious diseases for centuries. Mostly, SIR models have been used vastly back dating to 1760 when Daniel Bernoulli used mathematical methods in an effort to tame small pox. The majority of models that have been developed, mainly study the population epidemic dynamics of malaria, small pox, tuberculosis (TB), measles, and most recently HIV/AIDS. Within host modelling of these diseases is relatively new in the field of disease modelling with substantial models that has been developed starting around 1985 to 1990 and afterwards. Most recent with-in host models (Magombedze et al., 2008; Magombedze et al., 2009; Shiri et al., 2005; Sub et al., 2006; Marino and Kirschner, 2004; Wordaz et al., 2000; Wordaz et al., 2007) seek to enhance the understanding of the immune reponse mechanisms, pathogen invasion mechanisms, the best way of treating HIV infection, Mycobacterium tuberculosis (Mtb) infection, the dynamics of HIV/Mtb co-infection. HIV and Mtb are the leading infectious diseases among other infectious killer diseases (Friedland et al., 2006; Schluger and Rom, 1998; North and Jung, 2004; Weiden et al., 2000; Honda et al., 1998). Over the decades, there have been subsequent discoveries to enhance the understanding of the immune system; which is a highly regulated system which has evolved to provide the human body with substantial defenses against pathogenic organisms. This has been followed by an explosion of experimental data in this area and development of new molecular and cellular techniques to enhance understanding of the immune system. Most of the new techniques have allowed the isolation of the process or cell under study so that the results can be readily interpretable. At the present time, however, there is an emerging need to understand the system as it functions as a whole and the language of mathematics is the one best suited for this purpose. There has been relatively little interest, on part of experimental immunologists, in this area, probably due to the complexity of the mathematics necessary for the work. However, there seem to be a change in the situation, since there has been a great insight in the area of HIV derived from mathematical models of HIV infection. It has been intriguing to immunologist, how the HIV infection follows a long latent period that is followed by a rapid increase in the viral load that leads to AIDS and death of patients. The use of mathematical models has been instrumental in understanding of such biological phenomena and formed the basis of the first HIV models (Nowak et al., 1991). As a result, mathematical models have been found useful in challenging existing paradigms of AIDS and are now an integral part of the HIV research effort. This is also true for other infection like Mtb, malaria, etc.

2.

What Is with-in Host Modelling

With-in host (cell level) modelling is the study of infectious diseases with-in the human body. Instead of looking at infected individuals as vehicles of disease transmission as in population models, the disease causing micro-organism (pathogen) is considered. Cell level modelling is also known as immunological modelling because it involves the study of the intricate dynamics of humoral and adaptive immune response mechanisms. In a complete with-in host mathematical model, cell populations (which include cells that are susceptible

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255

to infection), infecting pathogen, immune cells (cells that combat the infection), cytokines and chemokines (soluble factors that are involved in priming the immune cells) which are referred to as the biological variables are transformed into mathematical variables. These mathematical variables must carry the biological properties of the biological variables under study in a sound with-in host model. Then, these variables are used to construct functions (ODEs) that describe the interactions and relationships between the different variables. After the mathematical model has been calibrated, it is then analysed or is used in biological investigations from which biological results are obtained analytically or numerically. This helps to prove a solid frame work upon which to build experiments and generate hypothesis beyond verbal or graphical reasoning where clear understanding of the phenomenon under investigation can be obtained. With-in host models can assist in, (i) identifying important immune players necessary in the control of the infection, (ii) revealing attributes of the pathogen under study that makes it able to invade the immune system, (iii) suggesting directions of treatment strategies, (iv) predicting the success or failure of a treatment strategy, (v) enforcing clarity of thought, and (vi) modelling events that are beyond resolution of current laboratory instruments and enhance biological insight where biological laboratory experiments are limited. Mathematical models can also serve distinct purposes. They can be used to analyse experimental results and provide predictions and suggestions for follow up experiments, or they can attempt to synthesize knowledge and prove a theoretical framework for the interpretation of existing paradigms (Morel, 1998).

3.

Categorisation of Disease Models

Most disease modellers use a deterministic approach that can either follow the class of SIR models or predator prey models. SIR models constitute categorisation of the human population into the class of (i) susceptibles (S), (ii) infectious (I) and recovered (R) individuals. Simple SIR models can further be modified to SEIR models depending on the dynamics of the disease under study, where E is the class of individuals exposed to the disease but that are not yet infectious. Further more, they can also be simplified to consider the S and I classes only, where the R class is ignored for example in diseases like influenza and TB. In these diseases, individuals who are infectious do not develop permanent immunity, but move back into the susceptible class. On the other hand, in predator prey models, population dynamics that are governed by the laws of ecology are modelled. Also, some ideas of SIR modelling can be incorporated in predator prey models or vise-versa in modelling complex population dynamics. In with-in host models, both aspects of SIR and predator prey models are noticed and tend to compliment each other. Its unreasonable to categorise cell level models completely under the brackets of SIR and predator-prey models. Since with-in models involve disease dynamics within the human body, the dynamics of the disease causing organisms, hormones, proteins, cytokines and chemokines that are generated when the human immune system encounters the invading organism. The invading pathogen can infect certain type of cells (macrophages in the case of Mtb infection or CD4+ T cells in HIV infection), these cells can be categorised as susceptible, the infected cells as infected, but these cells cannot be labeled as infectious since they do not infect other cells. Unlike in the population dynamics, infected individuals infect susceptible people. At the cell level dynamics, the disease etiology only carries the potential to infect susceptible cells. This in-

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fection then triggers the development of the immune response that is comprised of effector T cells, activated cells that assist in cabbing the infection either as antigen presenting cells or immune cells that directly kill the foreign bodies. Therefore, the human body (the host) can be taken as the eco-system and the immune cells can be considered as the predators that are continually updated (proliferate) according to the density of the pathogen (the prey) (Antia et al., 2003; Zhivkov and Waniewsk, 2003; Kuang, 2007). The specific immune response is a clonal response to specific antigens via cytotoxic and humoral reactions, as a result, various functions were proposed to describe immune system stimulation. A bilinear relationship (Lotka-Voltera type of interaction) may be the simplest assumption to represent pathogen depended immunity growth. However, as cells multiply, there must be some limitation on the population growth and as such, a Michaelis-Menten function that levels off is frequently assumed. The Michaelis-Menten function (similar to the Holling type II of modelling in predator-prey models) has been used in many recent studies (Shiri et al., 2005; Sud et al., 2006; Chiyaka et al., 2008; Magombedze et al., 2006a; Magombedze et al., 2008) to model proliferation of immune cells, recruitment of susceptible cells and infection of cells in response to the pathogen density at the site of infection. Therefore, disease dynamics at cell level commands selection of terms that follow the Hill’s equation, Michaelis-Menten’s terms, logistic growth terms and saturation terms that closely mimic cell proliferation, pathogen growth in the intracellular environment, densities of cytokines at the site of infection, infection inhibition and cell mediated apoptosis.

4.

Challenges of with-in Host Modelling

The first challenge in with-in host models is the selection of modelling terms in model equations. Selection of linear terms is the first temptation one is faced with in modelling the biological phenomena involved. Modelling cell population growth, pathogen growth (for example bacterial growth) which is normally logistic, cell proliferation, immune stimulation which is depended on the density of cytokines, require the use of terms that follow the Hill’s equation, Michaelis-Menten’s terms, logistic growth terms and saturation terms that closely approximate the behaviour of such processes. Therefore, the use of linear terms to model such interactions will not be accurate. However, the use of the non-linear terms result in the construction of complex model equations. A limitation is faced when many saturation terms are used in the equations of the model. In an attempt to make the model more realistic and biological, by use of saturation terms, a complicated mathematical model that is highly non-linear and difficult to handle mathematically is developed. Also, using more linear terms enable the development of simpler and mathematically manageable models that are easy to analyse, but that are far from representing the correct biological dynamics. One of the biggest challenge in with-in host modelling is encountered in trying to capture all the immune-pathogen interactions that are involved. Realistic models carry more realistic assumptions with elaborate cell-cell interactions, which result in complicated mathematical models. The proposed models by Marino and Kirschner, 2004 and Wigginton and Kirschner, 2001 are such examples in the modelling of Mycobacterium tuberculosis (Mtb). These studies present mathematical models of Mtb infection in the lungs and in the

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draining lymph node (DLN). In these models, T cells (combined CD4+ and CD8+ T cells, primed and unprimed), Mtb bacteria (extracellular and intracellular), macrophages (resting, activated and infected), dendritic cells (immature and mature), cytokines and chemokines interactions are captured and explained in a comprehensive way. Capturing of such exclusive dynamics result in complicated systems of ODEs in the modelling of the biological dynamics. Marino and Kirschner, 2004 and Wigginton and Kirschner, 2001 used two comprehensive systems of ODEs to elaborately model what happens in the lungs and DLN during Mtb infection in order to predict the occurrence of Mtb latency and active disease and to determine the immune parameters that can control the disease outcomes. The merits of these models are that, (i) they incorporate most of the important cell categories and cytokines that are involved in Mtb infection and (ii) terms that follow the Hill’s equation, Michaelis-Menten’s terms, logistic growth terms and saturation terms that closely mimic cell proliferation, pathogen growth in the intracellular environment, densities of cytokines at the site of infection, infection inhibition and cell mediated apoptosis are used. However, this gives realistic complicated mathematical models with a more precise biological picture of what happens during infection that is difficult to handle mathematically. In such models (Marino and Kirschner, 2004; Wigginton and Kirschner, 2001; Sud et al., 2006), the power of mathematical analysis is paralysed, for instance it is difficult to calculate the disease equilibrium states, establish the stability analysis of the disease steady states for these models and the authors of the models did not even bother to calculated the disease reproduction number which is a very important threshold value in disease control. That is no form of mathematical analysis was employed in these models to enhance the biological understanding of the disease infection and immune dynamics. However, these studies (Marino and Kirschner, 2004; Wigginton and Kirschner, 2001; Sud et al., 2006) come up with brilliant results from numerical analysis. In such complex models, we notice that the need for mathematical analysis is sacrificed in place of complicated mathematical models that are more realistic and that are difficult to analyse. Eventually modelers have to rely on numerical simulations to obtain results. Nevertheless, biological understanding of the disease is enhanced, while the possible mathematical insight that could have been earned remains concealed. In the study of Mtb infection in the lungs and DLN, simpler models have been developed (Magombedze 2006a, 2006b, 2008a) in which the role of cytokines were not considered, but were captured indirectly through saturation terms of cell populations they stimulate, unlike in Marino and Kirschner (2004), Wigginton and Kirschner (2001) and Sud et al. (2006). This enables development of less complex models that are more easier to handle mathematically, but however with less representation of biological facts or with more unrealistic assumptions. The advantages of doing so are that, (i) mathematical analysis tools can be employed to bring insight into the biological dynamics under study, and (ii) numerical simulations can also be used to compliment results from mathematical analysis. Studies by Magombedze (2006a, 2006b, 2008) showed similar results as posited by Marino and Kirschner (2004), Wigginton and Kirschner (2001) and Sud et al. (2006), which demonstrates that quality results can be obtained from simpler models with help of mathematical analysis. In Magombedze (2006a, 2006b, 2008), mathematical analysis (complimented with numerical simulations) is used to determine the most critical immune

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players and pathogen attributes that are behind occurrence of Mtb latency and active TB just as from the results obtained in Marino and Kirschner (2004), Wigginton and Kirschner (2001) and Sud et al. (2006). Apart from that, the disease reproduction number was easily calculated, stability analysis carried out and disease steady states were estimated but still the models are complex to permit rigorous mathematical analysis. These models demonstrate that similar biological result can be obtained from simpler mathematical models. The other challenge of with-in host modelling is the unavailability of data for model parameters. Building of big complicated models normally result in use of many parameters that might have not been obtained from biological experiments. In many cases, when parameters are unavailable, they are estimated from animal models and when there are no animal models to help with parameter estimation, they are manipulated. This is normally followed by sensitivity and uncertainty analysis of the parameters. In particular, the studies (Marino and Kirschner, 2004; Magombedze et al., 2008) involve the use of more than thirty estimated parameters without biological references. Whenever data is not available, use of many unknown parameter values removes confidence in the results obtained. In most cases this problem is difficult to resolve and could be a possible source of a chain of inaccurate finding in many mathematical models. Since mathematicians have a habit of referencing biological parameters from other mathematical papers where the parameters were used even though they were estimated from sources that are not biological. The normal temptation a bio-mathematician is faced with is to develop a smaller model with a good number of known parameters (derived from biological data or with biological references) and with a few new estimated parameters. Then, he uses this model to gain confidence in the new parameters from numerical simulations and comparison with results from a model with a correct set of parameters. Their intension is to use these new parameters later on in bigger models, how smart? This is where immunologists and biologists have problems with mathematicians because the results even though close could be mis-leading. On the other hand, the mathematician is happy and hopes that the immunologist can now take over his findings and hypotheses and test them in the laboratory.

5.

Discussion

In general, complex models, (i) are difficult to follow or to understand, as a result, the intended beneficiaries will not benefit, (ii) limit the use of mathematical tools and the desired insight that was to be derived from mathematical analysis will never be obtained unless otherwise and (iii) give realistic models that closely follow the biology involved that can only be analysed by means of numerical simulations. The more complicated and bigger the model is, the difficult the model is to analyse. On the other hand, simplistic models are easier to handle mathematically, to understand and follow, but unfortunately carry many unrealistic assumptions such that their use might not be a true representation of the well known biological phenomena. Hence, there is a need to balance the two, that is to include more realistic assumptions that make mathematical analysis possible yet trying to maintain the model readable and understandable than rather to use too simple models with unrealistic assumptions or using complicated big models that are difficult to comprehend and analyse

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but that yield similar result that can be obtained from simpler models. Therefore, the more assumptions that have to be put into the model, the harder it is to be confident about the conclusions. On the other hand, a well designed model can test different assumptions and provide important new insights into questions that cannot be readily answered experimentally.

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[12] Magombedze G., Garira W , Mwenje E. Modelling the immuno-pathogenesis of HIV1 infection and the effect of multi-drug therapy: The role of Fusion inhibitors in HAART. Math. Biosci. Eng. 5(3), 485-504, 2008. [13] North R. R., Jung Y. J. Immunity to tuberculosis. Annu. Rev. Immunol. 22:599-623, 2004. [14] Nowak M. A., Andersin R. M., Mblean A. R., et al. Antigenic diversity thresholds and the development od AIDS. Science. 254, 963-969, 1991. [15] Schluger N. W., and Rom W.N. The host immune response to tuberculosis: State of the art. Am. J. Respir. Crit. Care. Med. 157:679-691, 1998. [16] Shiri T., Garira W., Musekwa S. D. A two-Strain HIV-1 Mathematical Model To Assess The Effects Of Chemotherapy On Disease Parameters. Math. Biosci. Eng. 2, 811832, 2005. [17] Sud D., Bigbee C., Flynn J. L., Kirschner D. E. Contribution of CD8+ T cells of Mycobacterium tuberculosis infection. J. Immunol. 176:4296-4314, 2006. [18] Weiden M., Tanaka N., Qiao Y., Zhao B. Y., et al. Differentiation of Monocytes to Macrophages Switches the Mycobacterium tuberculosis Effect on HIV-1 Replication from Stimulation to Inhibition: Modulation of Interferon Response and CCAAT/Enhancer Binding Protein β Expression. J. Immunol. 165: 2028-2039, 2000. [19] Wigginton J. E., Kischner D. A model to predict cell mediated immune regulatory mechanisms during human infection with Mycobacterium tuberculosis. J. Immunol. 166(3):1951, 2001. [20] Wodarz D and Nowak M.A. Immune responses and viral phenotype: do replication rate and cytopathicity influence virus load? J. Theor. Med. 2, 113-127, 2000. [21] Wodarz D, Sierro S and Klenerman P. Dynamics of killer T cell inflation in viral infections. J. R. Soc. Interface. 4, 533-543, 2007. [22] Zhivkov P., and Waniewski J. Modelling tumour-immunity interactions with different stimulation functions. Int. J. Appl. Math. Comput. Sci. 13(3):307-315, 2003.

INDEX A abdomen, 154 abnormalities, 184 absorption, 219, 225 acceptor, 244 access, 36, 87, 122, 152, 155 accessibility, 159 accounting, 165 acid, 32, 244 Acquired Immune Deficiency Syndrome, 85, 86, 87, 101, 102, 112, 113, 114, 119, 120, 121, 122, 123, 124, 125, 129, 158, 225, 233, 254, 260 acquired immunity, 160, 162, 163, 166, 168, 173, 175, 176 activation, 219, 234 acute, 171, 195 adenovirus, 250 adenoviruses, 234 administration, 76, 89, 158, 199, 200, 202, 211, 214, 215, 216, 220, 223 adriamycin, 243, 244, 245 adult, 86, 154, 166, 174, 182, 194 adults, 37, 83, 152, 159, 169, 199 Africa, 33, 34, 78, 85, 86, 87, 101, 130, 131, 151, 152, 154, 157, 165, 169, 172, 174, 175, 177, 184 age, 1, 2, 15, 29, 33, 151, 152, 165, 177, 183, 233 agent, 199 agents, vii, 83, 130, 164, 198 agricultural, 165 aid, 77, 159, 198, 253 air, 32 alpha-fetoprotein, 250 alternative, 50, 53, 69, 155 alters, 259 amelioration, 112, 113, 130 anaemia, 154, 161 angiogenesis, 229, 232, 233, 251 animal models, 258 animals, 234, 241 Anopheles gambiae, 170, 184, 194, 195, 196 Anopheles mosquitoes, 162, 172, 174

antibiotic, 85, 129 antibiotic resistance, 129 antibody, 33, 195 anticoagulant, 155 antigen, 154, 160, 175, 195, 256 antigen presenting cells, 256 antigenic shift, 32 antimalarial drugs, 157, 168, 175 Antiretroviral, 86, 108, 131 antiretrovirals, 120, 124 antiviral, 31, 33, 34, 35, 36, 37, 73, 81, 82, 83 antiviral agents, 83 antiviral drugs, 31, 33, 34, 36, 37, 73, 83 apoptosis, 162, 202, 206, 256, 257 appetite, 134 application, 5, 174 Arabs, 134 argument, 41, 53, 99, 124 arithmetic, 65 Arizona, 167 artemisinin, 176 asbestos, 233 asexual, 154, 155, 158, 161, 168, 178, 194 Asia, 33, 177 Asian, 32 assessment, vii, 75, 195 assimilation, 198 assumptions, 3, 34, 36, 45, 103, 105, 114, 133, 134, 180, 211, 253, 256, 257, 258, 259 asymmetry, 192 asymptomatic, 113, 162, 163 asymptotic, 1, 5, 20, 22, 44, 85, 93, 105, 106, 118, 211 asymptotically, 21, 45, 59, 62, 67, 85, 92, 96, 105, 108, 117, 118, 137, 211, 214 ATP, 244 attacks, 133, 145, 146, 153, 163, 171 availability, 31, 34, 37, 159, 198 avian influenza, 31, 33, 84 awareness, 156

262

Index

B B cell, 162 B cells, 162 bacilli, 198, 199, 200 bacteria, 87, 202, 206, 207, 217, 218, 221, 222, 257 bacterial, 198, 200, 202, 203, 204, 206, 207, 211, 215, 216, 227, 256 bacterium, 198, 199 barrier, 191, 193 basal lamina, 185 Bayesian, 167 beams, 234 behavior, 171, 241, 244, 250 behaviours, 165 benefits, 36, 215, 250 benign, 229, 230 benzene, 233 bifurcation, 31, 35, 37, 45, 52, 53, 54, 69, 70, 72, 77, 78, 79, 85, 93, 94, 96, 107 bifurcation point, 69, 93, 94, 107 binding, 234, 259 biological systems, 239 birds, 32, 33 birth, 2, 29, 38, 88, 135, 136, 146, 147, 148, 163, 171, 211 birth rate, 2, 135, 147, 148 birth weight, 163 births, 15, 146 blocks, 163 blood, 134, 153, 154, 155, 156, 157, 158, 160, 161, 168, 170, 172, 173, 175, 177, 178, 179, 180, 183, 184, 185, 186, 187, 188, 189, 191, 192, 193, 194, 196, 230, 232, 242, 244 blood monocytes, 173 blood stream, 155, 178 blood transfusion, 153 blood vessels, 230, 232 bloodstream, 155, 245 bone marrow, 250 bone marrow transplant, 250 Botswana, 151 bottlenecks, 194 bottom-up, 170 boundary conditions, 239, 243 brain, 134, 135, 236, 239, 241, 251 brain damage, 134 brain tumor, 236, 239, 251 breast cancer, 233, 242 breeding, 154, 157, 172 Burundi, 152

C C++, 99, 111 Cameroon, 184, 195, 196 campaigns, 153

Canada, 133 cancer, 229, 230, 233, 234, 239, 245, 250, 251 cancer cells, 234, 245 cancer treatment, 234, 251 candidates, 159, 195 capacity, 16, 23, 152, 162 capillary, 155, 237, 238 carbon, 154 carbon dioxide, 154 carcinogenicity, 234 carcinoma, 237 catalase, 199 Catalase, 199 cavities, 230 CD8+, 202, 207, 226, 257, 259, 260 CDC, 32, 82 cell, 154, 155, 172, 175, 178, 179, 195, 198, 199, 200, 203, 207, 211, 226, 229, 230, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 254, 255, 256, 257, 260 cell cycle, 242 cell division, 198, 235, 246 cell growth, 234, 242 cell invasion, 154 cell surface, 154 changing population, 196 chaos, 230 chemical energy, 244 chemicals, 152, 233 chemokines, 255, 257 chemotherapy, 161, 168, 170, 197, 198, 199, 225, 226, 227, 229, 234, 241, 242, 250, 251, 259, 260 chewing, 233 childhood, 168 children, 32, 152, 153, 159, 165, 167, 171, 172, 174, 175, 177, 195, 199 China, 131, 134 chloroquine, 158, 159, 161, 168, 170, 171 chronic disease, 37 chronic diseases, 37 circulation, 33, 179 civil war, 152 classes, 2, 37, 42, 72, 74, 113, 135, 147, 160, 255 classical, 35, 53, 134, 135 climate change, 152, 173, 175 clinical presentation, 160 clinical symptoms, 162 clinical trial, 159, 198 clinical trials, 159, 198 coding, 199 co-existence, 53, 62, 99, 216 cohort, 16, 31, 49, 76, 131 colon, 233 colon cancer, 233 combined effect, 34 common symptoms, 32 communities, 48, 113, 121, 152, 156, 167, 196 community, 1, 2, 4, 36, 43, 46, 47, 76, 112, 113, 119, 121, 122, 151, 160, 164

Index competition, 161, 192, 216 complement, vii, 5, 250 complexity, viii, 159, 160, 190, 195, 198, 254 compliance, 40, 164, 197, 199, 214, 219, 223, 225 complications, 164, 165 components, 7, 8, 14, 49, 105, 159, 205, 214, 215 composition, 175 computation, 82, 129, 225 computing, 70, 77 concentration, 197, 214, 220, 223, 225, 229, 238 confidence, 188, 258 constant rate, 114 construction, 73, 256 consumption, 161, 171 continuity, 17, 23, 29 control, vii, 1, 10, 11, 12, 13, 14, 34, 35, 36, 37, 53, 76, 77, 82, 83, 84, 86, 87, 122, 123, 124, 125, 129, 130, 131, 134, 144, 151, 152, 153, 156, 157, 158, 159, 161, 163, 164, 165, 167, 169, 170, 171, 174, 175, 177, 178, 180, 191, 193, 196, 205, 216, 227, 230, 234, 235, 245, 250, 253, 255, 257 conversion, 185 coordination, 144, 156 corruption, vii, 1, 13 costs, 176 coughing, 32 couples, 191 coverage, 47, 99, 124, 171 craving, 135 crime, 134 critical value, 70 culture, vii, 133, 134, 168 cycles, 59, 155 cytokines, 206, 207, 255, 256, 257 cytotoxic, 160, 202, 251, 256

D danger, 33 data collection, 70 dating, 254 death, 2, 15, 16, 18, 29, 38, 39, 71, 72, 73, 75, 88, 89, 102, 114, 135, 136, 138, 146, 147, 148, 180, 185, 186, 188, 189, 190, 191, 193, 202, 205, 206, 254 death rate, 2, 15, 39, 71, 72, 73, 75, 88, 89, 102, 114, 136, 146, 147, 148, 180, 185, 186, 189, 191, 202 deaths, 32, 33, 70, 86, 125, 146, 152, 154, 177 decay, 223 decision makers, 34 decomposition, 134 defenses, 254 deficiency, 199 definition, 19, 72, 82, 129, 225 degenerate, 26 delivery, 250, 251 demand, vii demographic structure, 153

263

dendritic cell, 225, 257 density, 2, 15, 29, 161, 177, 189, 192, 203, 206, 207, 229, 235, 236, 238, 239, 240, 241, 243, 244, 245, 246, 247, 248, 249, 250, 256 dependent variable, 5, 162, 180 derivatives, 19, 25, 65, 70, 125, 127, 128, 176, 210 destruction, 146, 148, 157, 161, 194 deterrence, 14 developed countries, 86 developing countries, 33, 36, 73, 86 diets, 233 differential equations, 40, 89, 114, 136, 141, 150, 180, 199, 220, 235, 253 diffusion, 230, 235, 236, 237, 238, 239, 242, 251 digestion, 191 disaster, 87 Discovery, 146, 226 disease model, 254, 255 diseases, vii, 2, 36, 43, 104, 128, 134, 146, 151, 152, 158, 165, 170, 233, 254, 255 distribution, 82, 152, 160, 170, 242, 245, 246, 248, 249, 250 diversity, 162, 260 division, 198, 230, 235, 245, 246 DNA, 234, 243 dominance, 225 donors, 153 dosage, 84, 214 dosimetry, 250 dosing, 223 drug half-life, 223 drug resistance, vii, 36, 85, 87, 99, 100, 101, 102, 113, 114, 123, 124, 125, 129, 131, 158, 161, 164, 167, 169, 170, 173, 174, 198, 199, 202, 224, 227 drug therapy, 167, 259 drug treatment, 77, 215 drug use, 33, 36, 82, 83, 87, 124 drug-resistant, 82, 129, 158, 168, 176, 198, 225 drugs, 37, 44, 86, 87, 113, 122, 124, 152, 153, 156, 157, 158, 159, 168, 169, 175, 193, 197, 198, 199, 200, 202, 203, 206, 207, 211, 214, 215, 216, 220, 223, 225, 226, 234, 242 duration, 35, 36, 154, 163 dynamical system, 62

E earth, 134 ecology, 154, 255 economic development, 152 economic growth, 152 economic theory, 83 Education, 129 efavirenz, 259 egg, 154 eigenvector, 95, 107 elderly, 32, 37 emergency preparedness, 84

264

Index

emigration, 39 encapsulated, 230 endothelial cell, 232 endothelial cells, 232 energy, 229, 241, 242, 243, 244, 245 energy density, 242, 243 entertainment, 133 environment, 202, 206, 212, 214, 256, 257 enzymes, 158, 184 epidemic, vii, 31, 32, 34, 35, 36, 37, 46, 47, 48, 49, 71, 76, 83, 93, 113, 119, 120, 121, 122, 124, 128, 129, 133, 146, 152, 153, 160, 163, 165, 166, 170, 171, 172, 175, 254 epidemics, 47, 87, 124, 128, 151, 152, 176, 198, 227 epidemiology, vii, 83, 86, 130, 151, 153, 156, 160, 161, 162, 164, 167, 170, 174 epithelial cell, 234 epithelial cells, 234 epithelium, 180, 191 equality, 65 Equatorial Guinea, 174 equilibrium, 19, 23, 31, 42, 43, 44, 45, 49, 50, 52, 53, 55, 56, 57, 59, 62, 63, 67, 68, 69, 70, 73, 76, 77, 78, 79, 80, 81, 85, 91, 92, 93, 94, 95, 96, 97, 98, 99, 104, 105, 106, 108, 109, 110, 116, 117, 118, 136, 137, 138, 139, 140, 141, 142, 143, 144, 205, 206, 207, 211, 213, 257 equilibrium state, 213, 257 erythrocyte, 155, 161, 178, 179 erythrocytes, 155, 162, 178, 179 estimating, 169, 184 etiology, 255 Europe, 33 evolution, 72, 74, 82, 129, 130, 170, 214, 227, 238, 246 exclusion, 128 expansions, 6 exploitation, 152 exposure, 134, 160, 162, 163, 165, 166, 233 extinction, 72 eyes, 134

F factor H, 47 failure, 34, 98, 102, 181, 183, 184, 186, 188, 189, 190, 193, 198, 225, 255 falciparum malaria, 158, 161, 162, 167, 168, 169, 170, 174, 175, 194, 196 family, 32, 59 fatalities, 175 fatality rates, 152 fatigue, 32 feedback, 1, 10, 11, 13, 14 feeding, 157, 180 females, 14, 154, 180, 184 fertilization, 155, 174, 180, 181, 184, 188, 189, 190, 191, 193

fever, 32, 153, 154, 169 film, 134 films, 134 Finland, 78 fitness, 31, 47, 77, 227 flexibility, 146 flow, 2, 39, 81, 89, 102, 141, 251 fluid, 134 fluorometric, 242 folklore, 134 food, 158, 159, 241 France, 82, 134, 173 free will, 134 freedom, 140 fuel, 86 funding, 146, 156 fusion, 184 futures, 129

G games, 134 gamete, 167, 179, 181, 184, 185, 187, 189, 190, 191, 192, 193 gametes, 155, 179, 180, 181, 183, 184, 186, 188, 189, 190, 191, 192, 193 gametocytes, 155, 158, 161, 166, 168, 178, 179, 180, 183, 184, 186, 188, 189, 191, 194, 195, 196 gametogenesis, 179, 180, 183, 184, 186 gene, 199, 214, 215, 234 gene therapy, 234 generation, 179 genes, 131, 198, 200, 235 Geneva, 82, 170, 172 geographical variations, 165 Ger, 129 gestation, 29, 30 gland, 177, 189, 194 glioma, 251 gliomas, 240, 241, 251 Global Warming, 168 glutamate, 160 glycoconjugates, 195 goals, 192 government, 156 granules, 219 granulysin, 202 graph, 185 groups, 37, 151, 163 growth, vii, 14, 46, 152, 158, 229, 230, 232, 233, 234, 235, 236, 237, 239, 240, 241, 242, 243, 244, 250, 251, 256, 257 growth rate, 251 Guangzhou, 131 guidelines, 134 Guinea, 184 gut, 179

Index

H H1N1, 32, 33 H3N2, 32, 33 H5N1, 31, 33, 34, 82, 84 HAART, 260 Haemosporidia, 168 hair loss, 242 hantavirus, 128 harm, 158, 245 Harvard, 170, 172 headache, 32 health, 33, 34, 48, 73, 77, 81, 84, 85, 86, 101, 152, 156, 157, 158, 176, 198, 216 health care, 81, 152, 198, 216 health problems, 85 health services, 152 heart, 134, 242 helper cells, 202 hepatitis, 153, 175, 234 hepatitis B, 153, 175, 234 hepatocyte, 178 hepatocytes, 155, 160, 178, 250 herpes, 153, 234 herpes virus, 234 heterogeneity, 82, 130, 184, 227 heterogeneous, 82, 129, 153, 160, 225, 227, 250 HHS, 33, 82 hibernation, 134 high risk, 37, 177 high-fat, 233 highlands, 172, 176 high-level, 199 hips, 151 HIV, vii, 2, 82, 83, 84, 85, 86, 87, 101, 102, 105, 109, 110, 111, 112, 113, 114, 117, 118, 119, 120, 121, 122, 123, 124, 125, 128, 129, 130, 131, 151, 152, 153, 169, 214, 216, 219, 223, 225, 254, 255, 259 HIV infection, 86, 113, 124, 152, 169, 254, 255 HIV/AIDS, vii, 2, 85, 86, 87, 101, 102, 113, 119, 120, 123, 124, 125, 130, 151, 254 HIV-1, 131, 259, 260 homosexuals, 82, 130 Honda, 254, 259 Hong Kong, 32 hookworm, 174 hormones, 255 hospital, 36, 152 hospital death, 152 hospitalization, 73 hospitalized, 37, 39 host, viii, 87, 128, 152, 155, 156, 157, 159, 160, 161, 162, 163, 165, 166, 171, 172, 174, 175, 178, 179, 184, 193, 214, 226, 233, 253, 254, 255, 256, 258, 260 host population, 128, 160 House, 130, 251

265

human, 31, 32, 33, 36, 41, 70, 73, 77, 84, 88, 114, 133, 134, 135, 137, 143, 146, 152, 153, 154, 155, 156, 157, 159, 167, 170, 171, 173, 174, 175, 177, 178, 179, 180, 183, 187, 192, 193, 195, 200, 203, 223, 225, 226, 234, 250, 253, 254, 255, 256, 259, 260 humans, 32, 33, 70, 76, 88, 113, 128, 133, 135, 138, 142, 144, 146, 151, 153, 154, 158, 159, 163, 164, 175, 177, 183, 184, 193, 194, 234 humidity, 165 hydrazine, 199 hygiene, 37 hypothesis, 61, 161, 255 Hypoxia, 232 hypoxic, 249

I identification, 157 IMA, 225 immigration, 38, 81 immune cells, 229, 239, 255, 256 immune response, 84, 153, 157, 159, 160, 161, 167, 168, 169, 172, 174, 205, 226, 229, 230, 239, 240, 241, 245, 246, 247, 248, 249, 250, 254, 256, 259, 260 immune system, 32, 152, 153, 159, 161, 165, 175, 200, 245, 251, 253, 254, 255, 256 immunity, 32, 33, 36, 38, 39, 71, 72, 73, 83, 128, 143, 153, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 172, 173, 175, 255, 256 immunization, 34, 159 immunocompetence, 33 immunogenicity, 160, 162, 170 immunoglobulin, 173 immunological, 160, 168, 254, 259 immunologist, 254, 258 immunology, 129, 160, 170, 173 immunosuppression, 153, 163, 170 immunotherapy, 230, 235, 239, 245 impaired immune function, 165 implementation, 37, 86 impulsive, 133, 144, 145, 199, 200, 220 in situ, 237 in vitro, 166, 169, 173, 175, 190 in vivo, 191 incidence, 15, 35, 36, 47, 62, 76, 77, 83, 86, 129, 135, 152, 153, 154, 159, 163, 164 inclusion, 192, 211 income, 165 incubation, 113 incubation period, 113 independence, 6 independent variable, 180 India, 134, 229, 233, 251 Indian, 226 indication, 216 indicators, 15

266

Index

indices, 70, 71, 73, 77 induction, 160, 162, 172 inequality, 13, 15, 46, 65 infancy, 168 infant mortality, 157 infants, 157, 171, 172, 174 infection, 1, 18, 27, 31, 32, 33, 34, 36, 38, 39, 43, 47, 48, 69, 73, 77, 78, 86, 91, 92, 97, 98, 101, 104, 106, 108, 109, 113, 114, 118, 124, 129, 130, 133, 135, 136, 138, 139, 140, 142, 146, 152, 153, 154, 155, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 169, 172, 173, 174, 184, 191, 194, 197, 199, 200, 202, 205, 206, 210, 211, 214, 215, 216, 219, 223, 225, 226, 227, 245, 254, 255, 256, 257, 259, 260 infections, 31, 36, 39, 47, 76, 77, 87, 91, 113, 119, 122, 124, 125, 128, 141, 152, 153, 154, 155, 159, 162, 163, 167, 168, 171, 172, 195, 196, 208, 210, 211, 259, 260 infectious, vii, 18, 32, 34, 35, 36, 38, 39, 41, 70, 75, 77, 82, 87, 88, 89, 102, 113, 114, 129, 130, 133, 135, 146, 155, 159, 160, 162, 163, 173, 178, 180, 188, 211, 225, 245, 253, 254, 255 infectious disease, vii, 18, 32, 35, 36, 77, 82, 129, 130, 135, 159, 160, 173, 225, 245, 253, 254 infectious diseases, vii, 36, 77, 82, 129, 130, 135, 159, 160, 173, 225, 253, 254 infinite, 2 inflation, 260 influenza, vii, 31, 32, 33, 34, 35, 36, 37, 46, 47, 48, 70, 73, 75, 76, 77, 81, 82, 83, 84, 158, 255 influenza a, 32, 36, 83, 255 influenza vaccine, 33 infrastructure, 142, 156, 157 ingestion, 155, 156, 179, 180, 181, 182 inherited, 233 inhibition, 219, 256, 257 inhibitors, 82, 260 inhibitory, 259 initiation, 158, 178 injection, 173, 230 innovation, 73 insecticide, 156, 157, 169, 171, 174 insecticides, 152, 153, 154, 156, 159 insight, 254, 255, 257, 258 instability, 1 instruments, 255 insurance, 196 integration, 41, 204 intensity, 148, 152, 162 interaction, vii, 1, 2, 14, 15, 114, 137, 153, 155, 161, 162, 165, 199, 200, 229, 239, 241, 251, 256 interactions, 1, 14, 15, 27, 161, 165, 171, 191, 193, 250, 255, 256, 257, 260 interdisciplinary, vii interface, vii interference, 192 interferon, 259 interpretation, 13, 14, 18, 24, 255

interval, 18, 23, 41, 223 intervention, 34, 37, 44, 62, 66, 70, 72, 112, 119, 120, 121, 122, 123, 142, 151, 153, 164, 165 intervention strategies, 37, 70, 121, 122, 151, 153, 165 intravenously, 242 invasive, 253 investment, 156 irrigation, 152, 167, 171, 172 island, 174 isolation, 31, 34, 37, 73, 81, 254 isoniazid, 86, 87, 129 isotopes, 245 ITRC, 64

J Jacobian, 44, 55, 57, 59, 67, 93, 106, 137, 138, 141 Jacobian matrix, 44, 55, 59, 93 JAMA, 129, 173 Japan, 134 Jung, 199, 226, 254, 260

K Kenya, 172, 184, 194 kernel, 5, 26 killing, 157, 173, 202, 206, 215, 219, 240, 241 kinetics, 174 King, 133, 169

L Lafayette, 177 lamina, 185 language, 99, 111, 254 larva, 154 larval, 171, 175 latency, 162, 163, 205, 257, 258 law, 15, 18, 19, 24 laws, 255 lead, 76, 136, 146, 178, 191, 192, 193, 194, 200 life cycle, 154, 156, 158, 159, 171, 178, 179, 183, 187, 191, 192 life expectancy, 71 life forms, 135 life span, 2, 161 lifespan, 184, 185, 193 likelihood, 2, 37, 158 limitation, viii, 256 limitations, 34, 142 linear, 5, 6, 11, 20, 26, 43, 59, 60, 69, 79, 91, 106, 109, 117, 189, 193, 230, 245, 247, 250, 256 linear systems, 59 liver, 154, 155, 158, 160, 178 liver cells, 158

Index Livestock, 84 location, 184 London, 29, 130, 166, 169, 171, 226 long period, 156 longevity, 157 longitudinal study, 175 long-term, 159 loss of control, 230 Louisiana, 134 low risk, 177 lung, 226, 227, 233, 259 lung cancer, 233 lungs, vii, 134, 199, 200, 223, 256, 257 Lungs, 225, 259 Lyapunov, 1, 5, 6, 10, 29, 35, 63, 65, 83 Lyapunov function, 35, 63, 65, 83 lymph, 199, 226, 230, 257, 259 lymph node, 199, 226, 257, 259

M macrogamete, 155, 180 macrophage, 215, 220 macrophages, 199, 200, 202, 203, 204, 205, 206, 207, 211, 212, 215, 216, 217, 218, 220, 221, 222, 224, 255, 257, 259 maintenance, 197 malabsorption, 86 malaria, vii, 82, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 184, 191, 192, 193, 194, 195, 196, 254, 259 males, 14, 180 malignant, 229, 230, 245 malignant cells, 245 malignant tumors, 230 mammals, 32 management, 130, 157, 172 manifold, 85, 98, 106, 110 manipulation, 245 market, 33 Massachusetts, 170 maternal, 162, 170 mathematical biology, vii mathematical knowledge, vii mathematical methods, 254 mathematicians, 258 mathematics, vii, 254 Matrices, 83 matrix, 4, 5, 7, 8, 11, 12, 44, 46, 55, 56, 57, 58, 59, 61, 93, 107, 141, 191, 194, 208, 210 maturation, 155, 156, 185, 186 maturation process, 185 MBE, 131 MDR, 198, 199 meals, 154, 179 measles, 254

267

measures, 31, 32, 33, 35, 36, 37, 46, 47, 48, 49, 76, 77, 84, 86, 124, 156, 164, 191, 211, 216, 242 media, 84 medication, 198 medications, 198, 216 medicine, 36, 242 mefloquine, 158 meiosis, 185, 191 memory, viii, 162, 172 men, 166 merozoites, 155, 160, 161, 178 metabolism, 158 metastatic, 242 methane, 171 mice, 226 Michaelis-Menten function, 253, 256 Microbes, 194 microgamete, 180, 183 microorganisms, 241 Middle Ages, 134 migration, 88, 196 mitosis, 181, 185 MMW, 82 mobility, 171 model system, 3, 40, 41, 45, 49, 50, 53, 63, 76, 79, 90, 92, 93, 96, 100, 103, 105, 106, 113, 116, 117, 119 modeling, vii, 34, 77, 84, 161, 162, 166, 171, 177, 194, 229, 235, 239, 245, 251, 259 models, viii, 15, 34, 35, 36, 37, 44, 73, 77, 81, 82, 83, 84, 102, 113, 124, 128, 129, 130, 131, 135, 142, 143, 146, 150, 151, 158, 160, 161, 162, 164, 165, 168, 169, 170, 171, 172, 175, 178, 182, 199, 200, 225, 227, 229, 232, 239, 250, 253, 254, 255, 256, 257, 258, 259 molecular markers, 187 molecules, 159, 241 monoclonal, 245 monoclonal antibodies, 245 monocytes, 173 monograph, 156 mononuclear cell, 175 monotone, 2, 3, 4, 12, 13, 15, 20, 21, 22 morbidity, 32, 33, 70, 113, 153, 162, 165 mortality, 15, 32, 33, 70, 86, 88, 102, 113, 128, 129, 153, 157, 160, 162, 163, 165, 174, 182, 198 mortality rate, 15, 88, 102 mosquito bites, 164 mosquitoes, 154, 156, 157, 158, 162, 163, 164, 165, 171, 172, 173, 174, 183, 185, 187, 188, 193, 194, 195, 196 motivation, 34 mouth, 233 movement, 14, 154, 157, 244 Mozambique, 159 MSC, 32 multidisciplinary, 250 multiplication, 155, 156, 203, 206, 215, 219 muscle, 32

268

Index

muscles, 242 mutant, 87, 198, 199, 200, 211, 214, 215, 216, 250 mutant cells, 198 mutants, 227 mutation, 130, 198, 199, 202, 206, 211, 235 mutation rate, 198 mutations, 158, 159, 173, 198, 199, 200, 215, 226, 233 mycobacterial infection, 259 Mycobacterium, 86, 197, 199, 201, 203, 205, 207, 209, 211, 213, 215, 217, 219, 221, 223, 225, 226, 227, 254, 256, 259, 260

N natural, 15, 18, 38, 39, 87, 102, 114, 135, 138, 146, 152, 163, 164, 165, 169, 172, 174, 191, 202, 205, 206 natural resources, 152 natural selection, 87 Near East, 33 necrosis, 226, 229 negotiation, 170 neonatal, 162, 163 neoplasm, 229 nervous system, 134 Netherlands, 84 network, 84 neural network, 84 neuraminidase, 33, 82 New York, 83, 84, 129, 166, 171, 250 next generation, 35, 208 Nigeria, 1, 156, 171 non-human, 154 non-human primates, 154 nonlinear, 4, 7, 75, 83, 135, 180, 256 non-smokers, 233 normal, 32, 163, 229, 230, 233, 245, 258 nuclei, 184 nucleus, 155, 184 numerical analysis, 200, 253, 257 numerical computations, 37 nutrient, 244

O obesity, 233 observations, 232 oncology, 245, 251 oncolytic, 251 online, 84 operator, 5, 35, 236 operon, 199 oral, 199 orbit, 42 ordinary differential equations, 3, 38, 95 organism, 153, 242, 255

organizations, 156 oscillation, 81 oscillations, 161, 170 outpatient, 152, 165 ovarian cancer, 233 overproduction, 198, 200 oversight, viii ownership, 157 oxygen, 134

P P. falciparum, 154, 155, 156, 161, 162, 163, 177, 178, 179, 180, 183, 184, 186, 187, 192 Pacific, 33, 134 pain, 134, 234 pandemic, 31, 32, 33, 34, 35, 36, 37, 81, 82, 83, 84, 125 paper, 14, 81, 85, 87, 170, 180, 187 Papua New Guinea, 184 parameter, 3, 21, 22, 27, 35, 37, 44, 68, 70, 71, 72, 73, 95, 99, 102, 111, 113, 114, 135, 140, 145, 161, 169, 179, 184, 185, 189, 191, 193, 202, 215, 216, 219, 223, 258 parameter estimates, 169 parameter estimation, 70, 258 parasitaemia, 153, 154, 157, 161, 163, 169 parasite, 130, 152, 153, 154, 155, 156, 158, 159, 160, 161, 162, 163, 165, 169, 171, 174, 178, 179, 180, 181, 185, 186, 187, 188, 191, 192, 193, 194 parasitemia, 168 parasites, 153, 154, 155, 156, 157, 158, 159, 160, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 173, 175, 181, 186, 195, 196 parenteral, 199 partial differential equations, 3, 229, 241 particles, 200, 202, 206, 234 passive, 192, 233 passive smoke, 233 pathogenesis, 153 pathogenic, 31, 234, 254 pathogens, 87, 128, 200, 205, 212 pathology, 161 pathophysiology, 154 patients, 85, 86, 102, 112, 113, 131, 153, 163, 169, 171, 183, 198, 200, 215, 216, 225, 226, 234, 239, 250, 254, 259 peptide, 170 per capita, 2, 88, 89, 113 perception, 198 periodic, 33, 35, 50, 59, 62, 67, 68, 77, 84, 150 permit, 144, 258 Persia, 134 personal, 164 perturbation, 19 perturbations, 19, 232 pharmacodynamics, 176 phase space, 205

Index phenotype, 260 phenotypes, 161 phenotypic, 161 Philadelphia, 81 photosynthetic, 241 Physicians, 81 physiological, 160 placebo, 168 placental, 153 plague, 134 planning, 36 plants, 241 plasma, 179 plasma membrane, 179 plasmodium, 173, 183, 184 Plasmodium falciparum, vii, 154, 165, 166, 167, 168, 169, 170, 171, 173, 174, 175, 177, 179, 188, 193, 194, 195, 196 Plasmodium malariae, 154 Plasmodium ovale, 154 Plasmodium vivax, 154, 172 play, 33, 77, 239 pneumonia, 32 point mutation, 173, 199 political parties, 146 polygamy, 29 polygons, 68 polymerase, 199 polymorphism, 162 polynomial, 51, 81, 141 poor, 34, 48, 87, 89, 152, 156, 157, 162, 199, 214, 216, 219, 223 poor health, 156, 157 population, 2, 14, 15, 16, 18, 27, 29, 30, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 43, 46, 47, 49, 68, 72, 73, 76, 81, 88, 91, 98, 99, 100, 101, 102, 111, 112, 113, 114, 119, 120, 121, 122, 123, 128, 130, 135, 138, 140, 142, 144, 146, 147, 152, 158, 160, 161, 162, 163, 164, 165, 166, 167, 169, 177, 178, 181, 182, 185, 186, 191, 193, 199, 206, 207, 215, 216, 220, 225, 235, 239, 253, 254, 255, 256, 259 population group, 165 population growth, 256 population size, 88, 102, 114, 130, 162 positive correlation, 114 poultry, 31, 33 powder, 134, 242 power, 6, 7, 46, 152, 257 predators, 256 pre-existing, 32 pregnancy, 163 pregnant, 157, 162, 165, 169 pregnant women, 157, 162, 165, 169 preparedness, 82, 83 pressure, 200, 211, 214 prevention, 33, 76, 153, 157, 158, 159, 180 preventive, 32, 36, 75, 76, 129, 172, 174 primigravidae, 171 priming, 255

269

probability, 18, 77, 102, 135, 136, 160, 198, 199, 245, 246, 250 production, viii, 162, 165, 178, 182, 183, 192, 194, 230 productivity, 152, 157, 165 program, 47, 76, 77, 78, 250, 259 programming, 99, 111 proliferation, 175, 207, 235, 236, 237, 239, 256, 257, 259 promote, 173 propagation, 214, 217, 218, 221, 222 property, 141 prophylactic, 37, 130, 157 prophylaxis, 36, 37 proportionality, 181, 191 proteases, 184, 185, 191 protection, 33, 161, 164, 173 protein, 154, 160, 172, 173, 193, 259 proteins, 192, 194, 195, 234, 255 proteome, 194 protozoan parasites, 154 puberty, 195 public, vii, 34, 37, 46, 47, 62, 73, 84, 129, 152, 153 public health, vii, 34, 37, 46, 47, 62, 73, 129, 152, 153 pupa, 154

Q quanta, 241 quarantine, 31, 34, 36, 73, 81, 133, 140, 142, 143, 146 quinine, 158

R radial distance, 236 radiation, 134, 234, 245, 250 radiation damage, 245 radiation therapy, 245 radical, 244 radioactive isotopes, 245 radiotherapy, 230, 234, 245, 248, 249, 250 radius, 105, 208, 210, 236, 238, 240, 241, 243, 246, 250 radon, 233 rainfall, 152 random, 135, 136 range, 93, 151, 152, 156, 183, 184, 185, 186, 244 real numbers, 11 reality, 65 reasoning, 255 recall, 19 recovery, 16, 39, 72, 160, 215 red blood cell, 155, 158, 160, 161, 172 red blood cells, 158, 160, 161

270

Index

reduction, 34, 39, 98, 125, 159, 162, 164, 177, 186, 189, 192, 193, 223 refugees, 152 regression, 161 regular, vii, 133, 157 regulation, 161, 169, 170 relapse, 86, 154 relapses, 154 relationship, 203, 256 relationships, 35, 168, 255 relatives, 233 repair, 243 replication, 87, 171, 211, 230, 234, 259, 260 reproduction, 31, 32, 34, 35, 43, 44, 49, 53, 71, 76, 82, 85, 91, 92, 93, 98, 99, 104, 105, 107, 110, 111, 112, 117, 119, 120, 121, 122, 123, 124, 129, 155, 178, 208, 210, 211, 225, 257, 258 Republic of the Congo, 171 research, vii, 36, 156, 157, 158, 178, 193, 225, 229, 233, 234, 242, 254 researchers, 163, 178, 234, 239 resilience, 167 resistance, vii, 36, 83, 85, 86, 87, 89, 99, 100, 101, 102, 113, 114, 122, 123, 124, 125, 129, 130, 131, 153, 154, 156, 158, 159, 161, 164, 166, 167, 169, 170, 171, 173, 174, 197, 198, 199, 200, 202, 203, 211, 216, 220, 222, 223, 224, 227, 243 resolution, 5, 255 resources, 144, 146, 152, 157, 170 responsiveness, 171 retinoblastoma, 233 retrovirus, 234 rice, 171, 172 rice field, 171 risk, 36, 37, 85, 125, 162, 165, 177, 198, 225, 233 risk factors, 233 risks, 167 RNA, 32, 199, 234 robustness, 70 RTS, 159, 165, 166 RTT, 91, 92, 100, 117, 118, 119 rural, 48, 165, 195 Rwanda, 152, 170

S saliva, 134, 155, 178, 179 salivary glands, 156, 178, 179, 180, 182, 183, 185, 186, 187, 188, 189, 190, 193 saturation, 219, 256, 257 savannah, 168 scalar, 11, 219 schistosomiasis, 167 school, 31, 33, 152, 165, 171 scientists, 33, 239, 250 search, 163 searches, 154 seasonality, 76, 165

secretion, 207 seeds, 193 selecting, 250 Senegal, 167, 173, 184 sensitivity, 35, 37, 38, 70, 71, 73, 77, 82, 87, 130, 180, 185, 188, 192, 200, 215, 216, 219, 220, 221, 223, 258 Sensitivity Analysis, 70, 188 series, 6, 7, 159 serum, 33, 198 severity, 158, 163 sex, 171, 183, 195, 196 sex ratio, 195, 196 sexual contact, 102, 113 sexual development, 156, 195 sexual reproduction, 178 shape, 161, 244 side effects, 87 sign, 45, 52, 67, 69, 80, 81, 127, 128 signaling, 251 signs, 13, 35, 69, 81, 134, 162 similarity, 33, 57 simulation, 37, 111, 184, 186, 244 simulations, 31, 36, 38, 77, 99, 110, 111, 197, 214, 215, 216, 217, 218, 220, 221, 222, 223, 257, 258 Singapore, 150 SIR, 128, 135, 160, 253, 254, 255 SIS, vii, 1, 14, 15, 27, 128 sites, 157 skin, 155 solid tumors, 232, 250 solutions, 21, 22, 25, 31, 35, 41, 44, 50, 60, 67, 68, 76, 84, 90, 104, 116, 133, 147, 148, 150, 153, 203, 232 South Africa, 85, 128, 130, 169 Spanish influenza, 32 spatial, 230, 235, 236, 242, 245, 250 species, 62, 154, 155, 157, 158, 159, 165, 177, 184 speech, 134 spiritual, 133 sputum, 226 stability, 1, 2, 3, 4, 5, 11, 14, 21, 22, 31, 32, 35, 37, 44, 45, 49, 55, 56, 57, 59, 67, 76, 83, 85, 91, 93, 96, 98, 105, 106, 109, 117, 118, 129, 130, 133, 137, 141, 175, 211, 225, 232, 257, 258 stabilization, 13 stabilize, 10, 14, 28 stages, 86, 142, 154, 155, 158, 159, 161, 168, 171, 172, 175, 178, 179, 180, 181, 191, 194, 232, 237, 244 steady state, 1, 3, 4, 5, 10, 14, 16, 17, 18, 20, 21, 22, 23, 24, 25, 27, 28, 29, 30, 37, 55, 62, 69, 72, 92, 161, 205, 257, 258 sterile, 159 stimulus, 229 stings, 134 stochastic, 34, 36, 37, 128 stochastic model, 37 stomach, 155

Index storage, 84 strain, vii, 33, 36, 37, 85, 87, 88, 89, 93, 99, 100, 101, 102, 110, 111, 112, 113, 120, 121, 124, 171, 197, 199, 200, 202, 203, 206, 210, 211, 214, 215, 216, 220, 223, 224, 225 strains, 32, 33, 36, 76, 87, 99, 100, 101, 106, 109, 112, 113, 119, 121, 122, 130, 152, 158, 159, 197, 199, 200, 203, 205, 208, 210, 211, 214, 215, 216, 220, 223, 225 strategic, 34 strategies, 33, 34, 35, 37, 70, 75, 77, 121, 122, 129, 151, 153, 156, 157, 158, 159, 163, 164, 165, 166, 167, 180, 193, 250, 255 strength, 192 students, vii subgroups, 114 Sub-Saharan Africa, 34, 86, 87, 101, 151, 152, 165, 174 substances, 155 Sudan, 172 suffering, 114, 157 supervision, 198 supply, 34, 37, 87, 124, 146, 205, 244 suppression, 86, 162, 175 Surgery, 234 surveillance, 129 survival, 62, 153, 244 surviving, 18, 146 susceptibility, 37, 154, 172, 173 sustainability, 156 Switzerland, 82 symmetry, 238, 246 symptoms, 32, 88, 101, 102, 113, 114, 152, 161, 162, 163, 215, 234 synchronization, 174 synergistic, 164, 245 synergistic effect, 164 systems, 152, 167, 253, 257

T T cell, 199, 200, 203, 206, 207, 211, 212, 216, 217, 218, 220, 221, 222, 255, 256, 257, 260 T cells, 199, 200, 203, 206, 207, 211, 212, 216, 217, 218, 220, 221, 222, 255, 256, 257, 260 T lymphocyte, 251 T lymphocytes, 160, 251 tactics, 146 Tanzania, 31, 33, 36, 84, 151 targets, 193, 245 TBI, 204, 207, 215, 217, 218, 219, 221, 222, 224 TBIs, 200, 201, 203, 205, 212, 215 TCP, 245, 246 temperature, 152, 154, 155, 156, 166, 169, 176, 180 temporal, 251 testimony, 130 Texas, 251

271

theory, 85, 93, 94, 98, 106, 107, 109, 110, 118, 129, 174, 251 therapy, 37, 86, 87, 101, 119, 120, 124, 129, 130, 131, 158, 161, 198, 199, 210, 215, 229, 230, 234, 239, 245, 260 threat, 31, 34 three-dimensional, 251 threshold, 35, 43, 44, 47, 48, 49, 52, 77, 85, 257 threshold level, 47 thresholds, 78, 260 throat, 32, 233 time, 2, 15, 18, 35, 39, 40, 41, 49, 63, 72, 73, 74, 75, 76, 77, 81, 88, 99, 100, 102, 111, 112, 135, 136, 137, 138, 142, 146, 147, 148, 159, 160, 162, 163, 168, 177, 180, 181, 182, 185, 187, 191, 197, 203, 204, 205, 215, 216, 220, 223, 235, 236, 237, 238, 239, 240, 241, 243, 244, 246, 254 time frame, 185 tissue, 229, 245 tobacco, 233 total energy, 242 tourism, 152 trade, 152 training, 152 traits, 161 trajectory, 12 transcription, 199 transfer, 39, 104, 208 transformation, 5, 155, 181, 182, 185, 191 transgenic, 158, 171, 172 transition, 2, 168, 191, 194 transition rate, 2 translation, 29 transmission, vii, 1, 14, 27, 31, 33, 34, 35, 36, 37, 47, 50, 70, 73, 75, 76, 77, 79, 81, 82, 84, 120, 122, 125, 130, 131, 135, 150, 151, 152, 154, 156, 157, 158, 159, 160, 162, 163, 164, 166, 167, 168, 169, 170, 171, 173, 175, 176, 178, 184, 192, 193, 194, 195, 196, 216, 225, 254 transplantation, 153, 250 transportation, 152 travel, 155 trend, 73 trial, 131, 159, 165, 166 triggers, 155, 166, 256 Tuberculosis, vii, 82, 83, 85, 86, 88, 124, 129, 130, 131, 197, 198, 199, 201, 203, 205, 207, 209, 211, 213, 215, 217, 219, 221, 223, 225, 226, 227, 254, 256, 259, 260 tumor, vii, 229, 230, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251 tumor cells, 229, 230, 235, 236, 237, 238, 239, 241, 242, 243, 244, 245, 249, 250, 251 tumor growth, vii, 229, 232, 235, 239, 240, 241, 250, 251 tumor necrosis factor, 229, 239, 240, 241 tumors, 229, 230, 232, 234, 235, 244, 245, 250, 251 two-dimensional, 68

272

Index

U uncertainty, 37, 258 undergraduate, vii UNICEF, 156 uniform, 250 United Kingdom, 84, 169 United Nations, 156 United Nations Development Program, 156

V vaccination, vii, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 43, 44, 46, 45, 47, 48, 49, 53, 71, 72, 73, 75, 76, 77, 78, 79, 81, 82, 83, 130, 158, 159, 164, 167 vaccine, 31, 33, 34, 35, 36, 38, 39, 47, 48, 70, 71, 72, 73, 76, 84, 114, 153, 158, 159, 160, 163, 164, 165, 166, 170, 172, 193, 195, 196, 226 vacuole, 159 values, 3, 12, 17, 35, 48, 59, 68, 70, 71, 72, 73, 75, 79, 85, 89, 93, 94, 99, 101, 102, 110, 111, 112, 123, 124, 142, 145, 179, 182, 183, 184, 185, 186, 188, 189, 192, 207, 211, 213, 214, 215, 216, 220, 223, 224, 236, 244, 247, 249, 258 variability, 160, 167, 176 variable, 11, 18, 20, 25, 50, 69, 70, 78, 130, 152, 154, 162, 173, 180, 182, 185, 259 variables, 19, 41, 46, 50, 62, 65, 66, 67, 69, 78, 79, 93, 103, 106, 180, 203, 204, 211, 255 variation, 160, 242 vascular system, 233 vasculature, 232, 250 vector, 5, 7, 11, 41, 42, 59, 90, 93, 106, 147, 148, 152, 153, 154, 155, 157, 159, 164, 165, 166, 174, 176, 178, 192, 234 vehicles, 254 vessels, 232, 244 video games, 134 viral infection, 128 virulence, 32, 82, 130

virus, 32, 33, 87, 153, 230, 234, 251, 260 virus replication, 87 viruses, 32, 33, 134, 230, 234 vomiting, 242 voodoo, 134 vulnerability, 32

W walking, 134 war, 156 warrants, 34 watches, 198 water, 154, 170, 171 water resources, 170 weakness, 32 wear, 163 West Africa, 172 white blood cells, 242 wild type, 199, 208, 216, 223 women, 157, 162, 165, 169, 242 workers, 152 World Bank, 156 World Health Organization (WHO), 33, 34, 84, 129, 131, 152, 156, 157, 168, 176, 177, 196, 198 World War, 150 worms, 176 writing, 20, 26, 34, 118

Y yield, 67, 155, 193, 259

Z Zimbabwe, 85, 151, 167, 197, 253 zygote, 156, 179, 181, 185, 186, 187, 188, 191, 192 zygotes, 180, 181, 183, 184, 185, 186, 191