A bacteria-immunity system with delayed quorum sensing

J Appl Math Comput DOI 10.1007/s12190-008-0167-8

JAMC

A bacteria-immunity system with delayed quorum sensing Zhang Zhonghua · Suo Yaohong · Zhang Juan · Peng Jigeng

Received: 12 June 2007 / Revised: 17 June 2008 © Korean Society for Computational and Applied Mathematics 2008

Abstract Considering the mechanism of quorum sensing, we formulate a bacteriaimmunity model to describe the competition between bacteria and immune cells on the basis of Zhang’s model (see Zhang et al. 2008, to appear, for more details). A distributed delay is introduced to characterize the time in which bacteria receive signal molecules and then combat with immune cells. We analyze the stability of the equilibrium points and discuss the existence of Hopf bifurcation near the positive equilibrium point. Finally, numerical simulation is carried out to illustrate our qualitative results. Keywords Quorum sensing · Hopf bifurcation · Stability · Immunity Mathematics Subject Classification (2000) 34D20 · 34C23 · 92D30 1 Introduction Most of living organisms are continuously exposed to the substances that are capable of causing them harm. Most organisms protect themselves against such substances This work was supported by the Natural Science Foundation of China under the contract 10531030. Z. Zhonghua () · S. Yaohong School of Sciences, Xi’an University of Science and Technology, Xi’an 710054, China e-mail: [email protected] S. Yaohong e-mail: [email protected] Z. Juan Department of Mathematics, North China Electric Power University, Beijing 102206, China P. Jigeng Research Center for Applied Mathematics, Xi’an Jiaotong University, Xi’an 710049, China

Z. Zhonghua et al.

in more than one way (e.g. physical barriers, or chemicals), repel or kill invaders. Animals with backbones, called vertebrates, have these types of general protective mechanisms, but they also have a more advanced protective system called the immune system. The immune system is a complex network of organs containing cells that recognize foreign substances in the body and destroy them. It protects vertebrates against pathogens, or infectious agents, such as viruses, bacteria, fungi, and other parasites. There are two basic kinds of immunity [2, 3]: the innate immunity and the adaptive one. The innate immunity is the first line of defence, it is nonspecific i.e., it is not directed against specific invaders but against any pathogens that enter the body, and it can suffice to clear the pathogens in most cases, but sometimes it is insufficient. In fact, some pathogens may possess ways to overcome the innate immunity and successfully colonize and infect the host. When the innate immunity fails, a completely different cascade of events ensues leading to adaptive immunity. Unlike the innate immunity, the adaptive immunity is specific i.e., it can recognize and destroy specific pathogen. The defensive reaction of the adaptive immune system is called the immune response. Any substance capable of generating such a response is called an antigen, or immunogen. Quorum sensing is a process that enables bacteria to communicate using secreted signaling molecules called autoinducers [4]. This process enables a population of bacteria to regulate gene expression collectively and, therefore, control behavior on a community-wide scale. The quorum sensing mechanism was initial observed in the marine bacterium Vibrio fisheri around 30 years ago [5, 6]. More recently, many other species have been discovered to exhibit quorum sensing behavior [4, 7, 8], including importantly, major human pathogens such as Staphylococcus aureus and Pseudomonas aeruginosa [4, 8, 9]. Recently, quorum sensing mechanism has been paid more and more attention by biomathematics scholars, and quite a few mathematical models have been formulated (see, for example, [1, 9–11] and references therein). Obviously, new models which characterizes this mechanism from different views are extremely needed. In the present paper, considering bacterial quorum sensing mechanism in the competition between bacteria and immune system and introducing a distributed delay to describe the time in which bacteria receive signal molecules and then combat with immune cells, we formulate a bacteria-immunity model with delayed quorum sensing. Subsequently, we analyze the stability of the equilibrium points, and discuss the existence of Hopf bifurcation near the positive equilibrium point by using the inverse of the average delay as a bifurcation parameter. Finally, we illustrate our qualitative results by numerical simulation. This paper is arranged as follows: Sect. 2 formulates the model, Sect. 3 analyzes the local stability of the equilibrium points and the existence of Hopf bifurcation, Sect. 4 presents the global stability of the positive equilibrium point, Sect. 5 describes the numerical results, Sect. 6 provides the conclusion and discussion.

A bacteria-immunity system with delayed quorum sensing

2 Model formulation Denote the concentrations at time t of uninfected target cells, infected target cells, bacteria, innate immune cells and adaptive immune cells as XU (t), XI (t), B(t), IR (t) and IA (t), respectively. Suppose the dynamic relations among them are as the following: Uninfected target cells have a natural turnover SU and a half-life μXU and can be infected (mass-action term α1 XU B). Infected target cells can be cleared by adaptive immune cells (mass action term α2 XI IA ) or half-life μXI . Both innate and adaptive immune cells have a source term and a half-life term. For innate immunity, the source term SIR includes a wide range of cells involved in the first wave of defense of the host (e.g. natural killer cells, polymorphonuclear cells, etc.). For adaptive immunity, the source term SIA represent memory cells that are present, derived from a previous infection or vaccination. A zero source means the first infection with this pathogen (i.e. no memory cells). Both the numbers of the two kinds of immune cells are increased by the signals that we have captured by means of bacteria load. Finally, the bacteria population has a net growth term, represented by a logistic function α20 B(1 − Bσ ) and is also cleared by innate immunity (mass action term α3 BIR ) and adaptive immunity (mass action term α4 BIA ). We consider a mechanism named quorum sensing for bacteria, by which the bacteria control their growth rate or the expression of their genes in response to their own or the density of other microorganisms (e.g. bacteria, immune cells) in the environment. Further, a distributed delay to characterize the time in which bacteria receive single molecules and combat with immune cells is introduced. Consequently, the vital dynamics are governed by ⎧ dB(t) t B(t) = α20 (1 + −∞ βe−β(t−u) B(u) ⎪ dt B0 du − σ )B(t) − α3 B(t)IR (t) ⎪ ⎪ ⎪ ⎪ ⎪ − α4 B(t)IA (t), ⎪ ⎪ ⎪ ⎪ ⎪ dX (t) ⎪ ⎨ dtU = SU − α1 XU (t)B(t) − μXU XU (t), dXI (t) ⎪ = α1 XU (t)B(t) − α2 IA (t)XI (t) − μXI XI (t), ⎪ dt ⎪ ⎪ ⎪ ⎪ dIR (t) ⎪ ⎪ ⎪ dt = SIR + β1 B(t) − μIR IR (t), ⎪ ⎪ ⎪ ⎩ dIA (t) dt = SIA + β2 B(t) − μIA IA (t),

(1)

where α20 is the effective reproductive rate of bacteria (the reproduction rate minus the death rate), σ the effective carrying capacity the environment, α20 B(t)(1 −  t ofβ(t−u) B(t) ) the logistic growth of bacteria, α B(t) βe B(u)du the concentration 20 −∞ σ of bacteria competing with immune cells at time t, and which receive the signal molecules in quorum sensing phenomena u time units ago, and B0 a positive constant. Suppose all of parameters in system (1) are positive. According to MacDonald [12], βe−βs > 0 is called the weak delay kernel and the average delay τ is defined by  τ= 0

∞

βse−βs ds =

1 . β

Z. Zhonghua et al.

The initial values of system (1) are B(s) = ψ(s)ψ(0) > 0, XU (0) = XU0 > 0, IR (0) = IR0 > 0,

ψ(s) ≥ 0, s ∈ (−∞, 0), eβ· ψ(·) ∈ L(−∞, 0];

XI (0) = XI0 > 0,

(2)

IA (0) = IA0 > 0.

Remark 1 The first equation of (1) suggests that the growth rate of bacteria can be controlled and increased by quorum sensing mechanism expect for its net growth, and also can be cleared by two kinds of immune cells. The second and third equations of (1) indicate the dynamics of the uninfected target cells and the infected target cells, respectively. The uninfected target cells are infected by bacteria in mass action law and have a constant input flow. In the same time, infected target cells can be killed by adaptive immune cells. The last two equations of (1) show that each kind of the immune cells has a special source term and its responses can be enhanced by bacteria load. Furthermore, target cells and immune cells have their own half-life terms. Similar to Lemma 1 in [10], we easily prove that the solution of system (1) remains positive whenever it exists.

3 Local stability of equilibrium points and existence of Hopf bifurcation It is clear that the equations related to B(t), IR (t) and IA (t) are independent to the other equations in system (1). Here, we mainly focus on the dynamical properties of bacteria B, innate immune cells IR and adaptive immune cells IA in the positive cone 3 . Therefore, only the following subsystem is focused on. R+ t ⎧ dB(t) = α20 (1 + −∞ βe−β(t−u) B(u) ⎪ dt B0 du − ⎪ ⎪ ⎪ ⎪ ⎨ − α4 B(t)IA (t),

B(t) σ )B(t) − α3 B(t)IR (t)

dIR (t) ⎪ ⎪ dt = SIR + β1 B(t) − μIR IR (t), ⎪ ⎪ ⎪ ⎩ dIA (t) dt = SIA + β2 B(t) − μIA IA (t),

(3)

with initial values B(s) = ψ(s), IR (0) = IR0 > 0,

ψ(0) > 0, ψ(s) ≥ 0, s ∈ (−∞, 0), eβ· ψ(·) ∈ L(−∞, 0]; IA (0) = IA0 > 0.

Let

 X(t) =

t

−∞

βe−β(t−u) B(u)du.

(4)

(5)

A bacteria-immunity system with delayed quorum sensing

By linear chain trick technology [12–14, 16], the three dimensional system (3) is equivalently transformed into the following four dimensional system ⎧ dB(t) B(t) = α20 (1 + X(t) ⎪ B0 − σ )B(t) − α3 B(t)IR (t) − α4 B(t)IA (t), ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎨ dIR (t) = SI + β1 B(t) − μI IR (t), R R dt (6) ⎪ dI (t) A ⎪ = S + β B(t) − μ I (t), ⎪ dt IA 2 IA A ⎪ ⎪ ⎪ ⎩ dX(t) dt = βB(t) − βX(t), with initial values B(s) = ψ(s), IR (0) = IR0

ψ(0) > 0, ψ(s) ≥ 0, s ∈ (−∞, 0), eβ· ψ(·) ∈ L(−∞, 0];  0 (7) > 0, IA (0) = IA0 > 0, X(0) = βeβs ψ(s)ds. −∞

Remark 2 If (B(t), IR (t), IA (t)) is a solution of system (3–4), then (B(t), IR (t), IA (t), X(t)) obviously solves system (6–7), where X(t) is defined by (5). Conversely, if (B(t), IR (t), IA (t), X(t)) is a solution of system (6–7), then X(t) satisfies (5), and further then (B(t), IR (t), IA (t)) solves system (3–4). For the convenience of description, we introduce the following notation: R0 =

α3 SIR α4 SIA + , α20μIR α20μIA

a=

1 , B0

=

1 − a. σ

The biological meaning of R0 will be given in Sect. 6. Clearly, we have the following results related to the existence of the equilibrium points of system (6). Here, the proof is omitted. α3 β1 α4 β2 α3 β1 α20 μIR + α20 μIA ) > 0, i.e., R0 < 1 and  > − α20 μIR − α4 β2 α3 β1 α4 β2 α20 μIA , or R0 > 1 and  < − α20 μIR − α20 μIA , there are a unique positive equilibrium point E ∗ = (B ∗ , IR∗ , IA∗ , X ∗ ) and a bacteria free equilibrium point E0 = (0, IR0 , IA0 , 0), otherwise, there only exists the bacteria free equilibrium point E0 , SI SI 1−R0 where B ∗ = X ∗ = , II∗R = μIR + μβI1 B ∗ , II∗A = μIA + μβI2 B ∗ , and α β α β + 3 1 + 4 2 R R A A

Theorem 1 If (1 − R0 )( +

α20 μI

II0R

=

SIR μ IR

,

II0A

=

SIA μ IA

R

α20 μI

A

.

Next, we investigate the locally stability of the equilibrium points of (6). Theorem 2 (1) If R0 < 1, E0 is unstable, while if R0 > 1, E0 is locally asymptotically stable; (2) If R0 < 1 and one of the following conditions hold, E ∗ is locally asymptotically stable: (1) β is small enough, further  < 0, (2)  ≥ 0; (3) If R0 > 1, E ∗ is unstable when it exists.

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Proof (1) The Jacobian Matrix at bacteria free equilibrium point has the form ⎞ ⎛ α20 (1 − R0 ) 0 0 0 −μIR 0 0 ⎟ β1 ⎜ ⎠. ⎝ 0 −μIA 0 β2 β 0 0 −β It is easy to see that if R0 < 1, the bacteria free equilibrium point is unstable, while if R0 > 1, the bacteria free equilibrium point is locally asymptotically stable. (2) The Jacobian Matrix at positive equilibrium point has the form ∗ ∗ ⎞ ⎛ −α20 Bσ −α3 B ∗ −α4 B ∗ α20 BB0 −μIR 0 0 ⎟ ⎜ β1 M := ⎝ ⎠. 0 −μIA 0 β2 β 0 0 −β Then, the characteristic equation of M has the form λ4 + a1 λ3 + a2 λ2 + a3 λ + a4 = 0,

(8)

where λ is a complex number and a1 = α20 B ∗ ( + a) + μIA + μIR + β > 0, a2 = α20 B ∗ [(μIR + μIA )(a + ) + β] + (μIR + μIA )β + μIR μIA + (β1 α3 + β2 α4 )B ∗ , a3 = α20 B ∗ [β(μIR + μIA ) + μIR μIA (a + )] + β(μIR μIA + β2 α4 B ∗ + β1 α3 B ∗ ) + B ∗ (β2 α4 μIR + β1 α3 μIA ),   β2 α4 β1 α3 a4 = βμIR μIA B ∗ α20  + + = βα20 μIR μIA (1 − R0 ). μI A μI R After tedious computation, we get 2 2 ∗2 a1 a2 − a3 = (α20 B ∗ 2 μIA + α20 B μIR )(a + )2 + (α20 B ∗ μ2IR + α20 B ∗ μ2IA 2 ∗2 + α20 B β + α20 B ∗ 2 β2 α4 + α20 B ∗ 2 β1 α3 + 2μIR α20 B ∗ μIA

+ 2βα20 B ∗ μIA + 2βα20 B ∗ μIR )(a + ) + μIA β2 α4 B ∗ + μ2IR β + 2μIR μIA β + α20 B ∗ β 2 + μ2IR μIA + μ2IA β + μIR β1 α3 B ∗ + μIR μ2IA + μIA β 2 + μIR β 2 and a1 a2 a3 − a32 − a12 a4 = A1 β 3 + A2 β 2 + A3 β + A4 := Ψ (β),

(9)

A bacteria-immunity system with delayed quorum sensing

where A1 = μ2IR μIA + 2α20 B ∗ μIR μIA + α20 B ∗ μ2IA + μIR β1 α3 B ∗ + α20 B ∗ μ2IR 2 + μIR μ2IA + α20 B ∗ 2  2 μIA + μIA β2 α4 B ∗ + α20 B ∗ 2 β2 α4 + α20 B ∗ 2 β1 α3 2 ∗2 2 + α20 B  μI R , 2 2 ∗2 A2 = 2(3μ2IA μIR α20 B ∗ + 3μ2IR α20 B ∗ μIA + α20 B ∗ 3 β1 α3 + 4α20 B μIR μIA 2 ∗3 3 + 2μ2IR μ2IA + α20 B β2 α4 + 2α20 B ∗ 2 μIA β2 α4 + α20 B ∗ 3  2 μI A 3 ∗3 2 2 ∗2 2 2 ∗2 2 + α20 B  μIR + μ2IA β2 α4 B ∗ + 2α20 B μIA  + 2α20 B μI R 

+ 2α20 B ∗ 2 μIR β1 α3 )(a + ) + α20 B ∗ 2 μIA β1 α3 + μ2IR β1 α3 B ∗ + μ3IA μIR + μ2IR μIA α20 B ∗  + μ3IA α20 B ∗  + α20 B ∗ 2 μIR β2 α4 + μIR μ2IA α20 B ∗  + μIR μIA β1 α3 B ∗ + μ3IR μIA + μ3IR α20 B ∗  + μIR μIA β2 α4 B ∗ , 3 3 2 ∗2 2 2 ∗2 2 A3 = (α20 B ∗ 3 μ2IR  + 2α20 B ∗ 3 μIR μIA + 3α20 B μIR μIA + 3α20 B μI A μI R 3 ∗3 2 2 2 + α20 B μIA  + α20 B ∗ 3 μIR β1 α3 + α20 B ∗ 3 μIA β2 α4 )(a + )2 2 2 ∗2 + (α20 B ∗ 2 μ3IR  + 4μ2IR μ2IA α20 B ∗ + α20 B μ2IR μIA + 2α20 B ∗ μ3IR μIA

+ α20 B ∗ 2 μ2IA β2 α4 + 3β2 α4 B ∗ 2 μIR α20 μIA + 3β1 α3 B ∗ 2 μIR α20 μIA 2 + 2α20 B ∗ μ3IA μIR + α20 B ∗ 2 μ2IA μIR + α20 B ∗ 2 μ2IA β1 α3 + α20 B ∗ 2 μ2IR β2 α4 2 + α20 B ∗ 2 μ2IR β1 α3 + 2α20 B ∗ 3 β1 α3 β2 α4 + α20 B ∗ 3 β12 α32 + α20 B ∗ 3 β2 α4 μIA 2 2 ∗3 2 ∗2 3 + α20 B 3 β22 α42 + α20 B ∗ 3 β1 α3 μIR + 2α20 B β2 α4 μIR + α20 B μI A  2 ∗3 B β1 α3 μIA )(a + ) + μIR β12 α32 B ∗ 2 + μ2IR β1 α3 B ∗ 2 α20  + μ3IR μ2IA + 2α20

+ μ2IR μ3IA + μ2IR μIA β2 α4 B ∗ + μ2IA β2 α4 B ∗ 2 α3  + α20 B ∗ 2 μIA β2 α4 μIR + βIR α3 B ∗ 2 β2 α4 μIR + μIR μ2IA β1 α3 B ∗ + 2μ2IR β1 α3 B ∗ μIA + α20 B ∗ 2 μIR βIR α3 μIA + β1 α3 B ∗ 2 μIA β2 α4 + 2μ2IA β2 α4 B ∗ μIR + μIA β22 α42 B ∗ 2 and A4 = 2β1 α3 B ∗ 2 μ2IA μIR α20 (a + ) + 2β2 α4 B ∗ 2 μ2IR α20 (a + )μIA 2 + α20 B ∗ 2 (a + )2 μ3IR μIA + α20 B ∗ 2 (a + )μ3IR β2 α4 2 + 2α20 B ∗ 2 (a + )μ2IR β1 α3 μIA + α20 B ∗ 2 (a + )2 μ3IA μIR

Z. Zhonghua et al.

+ 2α20 B ∗ 2 (a + )μ2IA β2 α4 μIR + α20 B ∗ 2 (a + )μ3IA β1 α3 + μ3IR μ2IA α20 B ∗ (a + ) + μ3IR μIA β2 α4 B ∗ + μ2IR μ2IA β1 α3 B ∗ 2 + μ2IR μ3IA α20 B ∗ (a + ) + μ2IR μ2IA β2 α4 B ∗ + 2μ2IR α20 B ∗ 2 (a + )2 μ2IA 3 2 + α20 B ∗ 3 (a + )3 μ2IR μIA + α20 B ∗ 3 (a + )2 μ2IR β2 α4 2 3 + 2α20 B ∗ 3 (a + )2 μIR β1 α3 μIA + α20 B ∗ 3 (a + )3 μ2IA μIR 2 + μ2IR β1 α3 B ∗ 2 β2 α4 + 2α20 B ∗ 3 (a + )2 μIA β2 α4 μIR 2 + α20 B ∗ 3 (a + )2 μ2IA β1 α3 + μIR β12 α32 B ∗ 2 μIA + μIA β22 α42 B ∗ 2 μIR

+ μ2IA β2 α4 B ∗ 2 β1 α3 + α20 B ∗ 3 (a + )β1 α3 β2 α4 μIR + α20 B ∗ 3 (a + )β12 α32 μIA + α20 B ∗ 3 (a + )β22 α42 μIR + μIR μ3IA β1 α3 B ∗ + α20 B ∗ 3 (a + )β2 α4 β1 α3 μIA > 0. It is easy to get that a2 , a3 , a4 > 0 and a1 a2 − a3 > 0 and if either β is small enough, further  < 0 and R0 < 1, or  ≥ 0 and R0 < 1, the positive equilibrium point is locally asymptotically stable when it exists. (3) If R0 > 1, it is easy to get a4 < 0. By Routh-Hurwitz criterion, the positive equilibrium point is unstable when it exists.  Next, under the condition of  < 0 and adopting the notations ai , Ai , i = 1, 2, 3, 4 and the function Ψ which are defined in the proof of Theorem 2, we further analyze the qualitative properties of system (6). For convenience, we respectively rewrite ai as ai = ai1  + ai2 , i = 2, 3, 4 , i = 2, 3, 4}, M2 = min {− aai2 , i = 2, 3, 4}. Obviously, and denote M1 = max {− aai2 i1 i1 M1 , M2 < 0. It is easy to verify that ai > 0 if M1 <  < 0 ai < 0 if  < M2 , i = 2, 3, 4. For last case, the positive equilibrium point is unstable. Throughout the remainder of this section, we suppose M1 <  < 0,

A3 > 0,

a1 a2 − a3 > 0.

(10)

Therefore, by Routh-Hurwitz criterion, (8) has a pair of purely imaginary roots iff there exists a positive number β ∗ such that Ψ (β ∗ ) = 0. Clearly, if β ∗ exists, Hopf bifurcation maybe occurs near the positive equilibrium point as β passes through β ∗ , otherwise, the positive equilibrium point remains stable for any positive β under the condition of (10). In order to study the existence of β ∗ , we introduce the Sturm sequence [15].  Suppose that a polynomial f has no repeated roots. Then f and its derivative f  are relatively prime. Let f = f0 and f = f1 . We obtain the following sequence of equations by the division algorithm: f0 = q0 f1 − f 2 , f1 = q1 f2 − f 3 ,

A bacteria-immunity system with delayed quorum sensing

.. . fs−2 = qs−2 fs−1 − K, where K is a constant. The sequence of Sturm functions f0 , f1 , . . . , fs−1 , fs ≡ K is called a Sturm chain. We may determine the number of real roots of the polynomial f in any interval in the following manner: plug in each endpoint of the interval, and obtain a sequence of signs (suppose the interval to be (a, b), then at left and right endpoints the sequences of signs are sgn(fi (a)) and sgn(fi (b)), i = 0, 1, 2, 3, respectively). The number of real roots in the interval is the difference between the number of sign changes in the sequence at each endpoint. Now, take the sturm chain of the polynomial (9), denote f0 , f1 , f2 , f3 . Then f0 = A1 β 3 + A2 β 2 + A3 β + A4 , f1 = 3A1 β 2 + 2A2 β + A3 ,     A2 2 A2 A3 f2 = − , A3 − 2 β + A4 − 3 3A1 9A1 f3 =

A2 A3 9A1 ) A22 3A1

3A2 (A4 − A3 −

−

A2 A3 2 4A1 ) A22 2 3A1 )

27A1 (A4 − 4(A3 −

− A3 .

We evaluate the nonnegative real line, i.e., from 0 to ∞, and construct a table of the signs at these endpoints (Table 1). Denote u and v as the number of sign changes of f0 , f1 , f2 , f3 at 0 and ∞, respectively. Then, we have the following results about the stability of the positive equilibrium point E ∗ . Here, the proof is also omitted. Theorem 3 A2

2 A3 (1) If f3 > 0, A4 > A9A , A3 > 3A21 and A1 > 0, we have u − v = 0. That is, E ∗ is 1 locally asymptotically stable;

A2

2 A3 , A3 > 3A21 and A1 < 0, we have u − v = 1. That is, there (2) If f3 > 0, A4 > A9A 1 exists a positive constant β ∗ such that Ψ (β ∗ ) = 0. If the transversality condition dλ1,2 ∗ ∗ dβ |β=β = 0 holds, then Hopf bifurcation occurs at E when β passes through ∗ the critical values β ;

Table 1

0

∞

f0

+

+ if A1 > 0 or − if A1 < 0

f1

+

+ if A1 > 0 or − if A1 < 0

f2

A2 A3 − sgn(A4 − 9A ) 1

− sgn(A3 − 3A2 ) 1

f3

A2

Z. Zhonghua et al. A2

2 A3 (3) If f3 > 0, A4 > A9A , and A3 < 3A21 , we have u − v = 2. That is, there exist two 1 ∗ positive constants β1 and β2∗ such that Ψ (βi∗ ) = 0, i = 1, 2. If the transversaldλ ity condition dβ1,2 |β=βi∗ = 0 holds, then Hopf bifurcation occurs at E ∗ when β passes through the critical values βi∗ , i = 1, 2;

A2

2 A3 , A3 > 3A21 and A1 > 0, we have u − v = −1. That is, there (4) If f3 > 0, A4 < A9A 1 exists a positive constant β ∗ such that Ψ (β ∗ ) = 0. If the transversality condition dλ1,2 ∗ ∗ dβ |β=β = 0 holds, then Hopf bifurcation occurs at E when β passes through ∗ the critical values β ;

A2

2 A3 , A3 > 3A21 and A1 < 0, we have u − v = 0. That is, E ∗ is (5) If f3 > 0, A4 < A9A 1 locally asymptotically stable;

A2

2 A3 and A3 < 3A21 , we have u − v = 1. That is, there ex(6) If f3 > 0, A4 < A9A 1 ists a positive constant β ∗ such that Ψ (β ∗ ) = 0. If the transversality condition dλ1,2 ∗ ∗ dβ |β=β = 0 holds, then Hopf bifurcation occurs at E when β passes through ∗ the critical values β ;

A2

2 A3 , A3 > 3A21 , and A1 > 0, we have u − v = 0. That is, E ∗ is (7) If f3 < 0, A4 > A9A 1 locally asymptotically stable;

A2

2 A3 , A3 > 3A21 and A1 < 0, we have u − v = 1. That is, there (8) If f3 < 0, A4 > A9A 1 exists a positive constant β ∗ such that Ψ (β ∗ ) = 0. If the transversality condition dλ1,2 ∗ dβ |β=β = 0 holds, then Hopf bifurcation occurs at the positive equilibrium point when β passes through the critical values β ∗ ; 2 A3 , and A3 < (9) If f3 < 0, A4 > A9A 1 asymptotically stable;

A22 3A1 ,

we have u − v = 0. That is, E ∗ is locally

A2

2 A3 (10) If f3 < 0, A4 < A9A , A3 > 3A21 and A1 > 0, we have u − v = 0. That is, E ∗ is 1 locally asymptotically stable;

A2

2 A3 , A3 > 3A21 and A1 < 0, we have u − v = 1. That is, there (11) If f3 < 0, A4 < A9A 1 exists a positive constant β ∗ such that Ψ (β ∗ ) = 0. If the transversality condition dλ1,2 ∗ ∗ dβ |β=β = 0 holds, then Hopf bifurcation occurs at E when β passes through ∗ the critical values β ; 2 A3 , and A3 < (12) If f3 < 0, A4 < A9A 1 asymptotically stable.

A22 3A1 ,

we have u − v = 0. That is, E ∗ is locally

4 The global stability of the positive equilibrium point By Lyapunov’s second method, we study the global stability of the positive equilib∗ rium point. For the end, we define three variables: x(t) = ln B(t) B ∗ , y(t) = IR (t) − IR ∗ and z(t) = IA − IA . Then, system (3) is transformed into ⎧ dx(t) t −β(t−u) B ∗ (ex(u) − 1)du − ex(t) −1 ) − α y(t) − α z(t), ⎪ 3 4 ⎪ dt = α20 ( −∞ βe B0 σ ⎪ ⎨ dy(t) x(t) − 1)B ∗ − μ y(t), (11) IR dt = β1 (e ⎪ ⎪ ⎪ ⎩ dz(t) x(t) − 1)B ∗ − μ z(t). IA dt = β2 (e

A bacteria-immunity system with delayed quorum sensing

It is easy to see that (0, 0, 0) is an equilibrium point of system (11) and its stability is equivalent to that of the positive equilibrium point of system (6). Theorem 4 If  > αα203μβI1 + αα204μβI2 , the positive equilibrium point is globally asympR A totically stable when it exists. Proof Let V1 (t) = |x(t)|, V2 (t) = |y(t)| and V3 (t) = |z(t)|. Calculating the respective upper-right derivative of V1 (t), V2 (t) and V3 (t) along the solution of (11), it follows that    t ∗ ex(t) − 1 ∗ + −β(t−u) B x(u) B D V1 (t) = sgn x(t) α20 βe (e − 1)du − B0 σ −∞  (12) − α3 y(t) − α4 z(t) ,   D + V2 (t) = sgn y(t) ex(t) − 1 B ∗ β1 − μIR |y(t)| and

  D + V3 (t) = sgn z(t) ex(t) − 1 B ∗ β2 − μIA |z(t)|.

(13)

(14)

Let V4 = V1 +

α3 α4 V2 + V3 . μI R μI A

(15)

By the equalities (12–14), the upper-right derivative of V4 satisfies  B ∗ x(u) |ex(t) − 1| ∗ B − sgn x(t) α3 y(t) |e − 1)|du − B0 σ −∞   α3 |ex(t) − 1|B ∗ β1 − μIR |y(t)| − sgn x(t) α4 z(t) + μI R   α4 x(t) ∗ |e + − 1|B β2 − μIA |z(t)| μI A  t  B∗ |ex(t) − 1| ∗ B βe−β(t−u) |ex(u) − 1)|du − ≤ α20 B0 σ −∞ α3 x(t) α 4 + |e − 1|B ∗ β1 + |ex(t) − 1|B ∗ β2 . (16) μI R μI A

D + V4 ≤ α20



t

βe−β(t−u)

Define

 V5 (t) =

t −∞

e−β(t−u) |ex(u) − 1|du.

Clearly, the upper-right derivative of V5 (t) has the form  t + x(t) D V5 (t) = |e − 1| − βe−β(t−u) |ex(u) − 1|du. −∞

(17)

Z. Zhonghua et al.

Let V (t) = V4 (t) +

α20 B ∗ V5 (t). B0

By (16–17), the upper-right derivative of V satisfies 

+

D V ≤ α20

t

−∞

βe

−β(t−u) B

∗

B0

|e

x(u)

|ex(t) − 1| ∗ B − 1)|du − σ



α3 x(t) α4 x(t) |e − 1|B ∗ β1 + |e − 1|B ∗ β2 μI R μI A    t α20 B ∗ x(t) −β(t−u) x(u) |e + − 1| − βe |e − 1|du B0 −∞   α3 β1 α4 β2 ∗ + −  |ex(t) − 1|. = B α20 α20 μIR α20 μIA +

Then the positive equilibrium point is globally stable if  >

α3 β1 α20 μIR

+

α4 β2 α20 μIA

.



5 Numerical results on the Hopf bifurcation at the positive equilibrium point In this section, using β as bifurcation parameter, we illustrate the above results by numerical simulation. Take α20 = 0.8, B10 = 0.00999, σ1 = 0.001991, α3 = 0.15, α4 = 0.2, β1 = 0.001, β2 = 0.002, μIA = 0.05, μIR = 0.05, SIR = 0.1907984 and SIA = 0.0476996. Then, system (6) has a positive equilibrium point (B ∗ , IR∗ , IA∗ ) = (8, 3.975968, 1.273992), and there exists a critical number β ∗ ≈ 2.713133605 such that, if β < β ∗ , every root of (8) has negative real part, while if β > β ∗ , there at least exists a root with positive real part. Let λ(β) = x(β) + y(β)i solve (8), where i is the unit of complex. ∗ It is easily obtained that dx(β) dβ |β=β ≈ 0.0008289301495 > 0, i.e., the transversality condition holds at β = β ∗ . Clearly, a1 , a2 , a3 , a4 , a1 a2 − a3 > 0 and A1 = −0.00000510937190 < 0, A2 = 0.00001272673705, A3 = 0.000003019058924, A4 = 0.0000001686748731. Then, f3 < 0, 9A1 A4 − A2 A3 < 0 and 3A3 A1 − A22 < 0. By the 8th case of Theorem 3, Hopf bifurcation occurs near (B ∗ , IR∗ , IA∗ ) as β passes through β ∗ . Under the condition β > β ∗ or β < β ∗ , the solution of system (6) are respectively showed in Figs. 1–3. With β = 1.2 < β ∗ , Fig. 1 shows the positive equilibrium point is locally asymptotically stable, i.e., the concentrations of the bacteria and the adaptive immune cells are convergent to 8 and 3.975968, respectively. Figure 2 shows the positive equilibrium point loses its stability, and a periodic solution is bifurcated. Under the condition of β = 6 > β ∗ , Fig. 3 exhibits the positive equilibrium point is unstable, i.e., the concentrations of bacteria and the adaptive immune cells oscillate around 8 and 1.273992, respectively. Moreover, the amplitudes increase with the increasing of time t, which means the positive equilibrium point is unstable.

A bacteria-immunity system with delayed quorum sensing

(a)

(b) Fig. 1 β = 1.2 < β ∗ . With the initial value (8.1, 3.975968, 1.29, 0.080919), (a) is B versus t and (b) is the projected B − IA phase plane

Z. Zhonghua et al.

(a)

(b) Fig. 2 β = 2.8 > β ∗ . (a) is B versus t with the initial value (8.1, 3.975968, 1.29, 0.080919). (b) is the projected B − IA phase plane, further the inner trajectory and the outer have the initial values (8.001, 3.9, 1.27, 0.87912999) and (5, 3.88, 1.8, 0.04995), respectively

A bacteria-immunity system with delayed quorum sensing

(a)

(b) Fig. 3 β = 6 > β ∗ . (a) and (b) are B and IA versus t with the initial value (8.1, 3.975968, 1.29, 0.080919), respectively

Z. Zhonghua et al.

6 Conclusion and discussion In this paper, a kind of delayed quorum sensing model describing the competition between the bacteria and the immune system is formulated. By Theorem 2, the bacteria free equilibrium point E0 is locally asymptotically stable if R0 > 1, and unstable if R0 < 1, while the positive equilibrium point E ∗ is locally asymptotically stable if R0 < 1 and either B0 < σ , β is small enough, or B0 ≥ σ , and unstable if R0 > 1. Subsequently, under the condition of B0 < σ and considering β as bifurcation parameter, we discuss the existence of Hopf bifurcation near positive equilibrium point. In Sect. 4, under additional condition, the global stability of the positive equilibrium point is discussed. Finally, in Sect. 5, a numeric example is provided to illustrate the qualitative behavior of the positive equilibrium point. SI From the biological viewpoint, μIR means the concentration of the initial immune R

SI

cells at bacteria free equilibrium point. So the product of μIR and the bacteria clearR ance factor α3 measures the strength of the innate immune system defense against the bacteria challenge; while the bacteria productivity factor α20 measures the bacteria’s SI offensive strength. Similarly, μIA means the concentration of the adaptive immune A

S

cells at bacteria free equilibrium point, and the product of μIIA and the bacteria clearA ance factor α4 measures the strength of the adaptive immune system defense against SI IR

α3 μ R

α4 SI A μI A α20

the bacteria challenge. So with R0 = α20 + , we can compare the strength of the immune system against the bacterial offensive. Thus, Theorem 2 has the biological explication: in the domain of attraction of E0 , bacteria will be cleared if R0 > 1, i.e., the strength of the immune system defense against the bacteria challenge is not weaker the bacteria’s offensive strength; in the domain of attraction of E ∗ , bacteria co-exist with immune cells if R0 < 1, i.e., the strength of the immune system defence against the bacterial challenge is weaker than bacteria’s offensive strength. Unfortunately, the global stability of bacteria free equilibrium is not obtained. It will be solved in the future.

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A bacteria-immunity system with delayed quorum sensing 8. Chun, C.K., Ozer, E.A., Welsh, M.J., Zabner, J., Greenberg, E.P.: Inactivation of a Pseudomonas aeruginosa quorum-sensing signal by human airway epithelia. Proc. Natl. Acad. Sci. 101, 3587–3590 (2004) 9. Anguige, K., King, J.R., Ward, J.P., Williams, P.: Mathematical modelling of therapies targeted at bacterial quorum sensing. Math. Biosci. 92, 39–83 (2004) 10. Fergola, P., Beretta, E., Cerasuolo, M.: Some new results on an allelopathic competition model with quorum sensing and delayed toxicant production. Nonlinear Anal. Real World Appl. 7, 1081–1095 (2006) 11. Fergola, P., Cerasuolo, M., Beretta, E.: An allelopathic competition model with quorum sensing and delayed toxicant production. Math. Biosci. Eng. 3, 37–50 (2006) 12. MacDonald, N.: Time Delays in Biological Models. Springer, Heidelberg (1978) 13. He, X.Z., Ruan, S.G.: Global stability in chemostat-type plankton models with delayed nutrient recycling. J. Math. Biol. 37, 253–271 (1998) 14. Gail, S.K., Wolk, O., Xia, H.X., Wu, J.H.: Global dynamics of a chemostat competition model with distributed delay. J. Math. Biol. 38, 285–316 (1999) 15. Shmerling, E., Hochberg, K.J.: Sturm’s method in counting roots of randoms polynomial equations. Methodol. Comput. Appl. Probab. 6, 203–218 (2004) 16. Kuang, Y.: Delay differential equations with applications in population dynamics. Academic Press, San Diego (1993)