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Computational and Mathematical Methods in Medicine
Volume 2015, Article ID 206205, 14 pages
http://dx.doi.org/10.1155/2015/206205
Research Article

Stability and Hopf Bifurcation in a Delayed HIV Infection Model with General Incidence Rate and Immune Impairment

1Department of Applied Mathematics, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China
2Fundamental Science Department, North China Institute of Aerospace Engineering, Langfang, Hebei 065000, China

Received 16 May 2015; Accepted 29 June 2015

Academic Editor: Chung-Min Liao

Copyright © 2015 Fuxiang Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We investigate the dynamical behavior of a delayed HIV infection model with general incidence rate and immune impairment. We derive two threshold parameters, the basic reproduction number and the immune response reproduction number . By using Lyapunov functional and LaSalle invariance principle, we prove the global stability of the infection-free equilibrium and the infected equilibrium without immunity. Furthermore, the existence of Hopf bifurcations at the infected equilibrium with CTL response is also studied. By theoretical analysis and numerical simulations, the effect of the immune impairment rate on the stability of the infected equilibrium with CTL response has been studied.

1. Introduction

In recent years, mathematical models have been proved to be valuable in understanding the dynamics of viral infection (see, e.g., [18]). In most virus infections, cytotoxic T lymphocyte (CTL) cells play a significant role in antiviral defense by attacking virus-infected cells. In order to study the role of the population dynamics of the viral infection with CTL response, Nowak and Bangham et al. proposed a basic viral infection model describing the interactions between a replicating virus population and a specific antiviral CTL response, which takes into account four populations: uninfected cells, actively infected cells, free virus, and CTL cells (see, e.g., [14, 9, 10]). Now, the population dynamics of viral infection with CTL response has been paid much attention and many properties have been investigated (see, e.g., [1116]).

Furthermore, the state of latent infection cannot be ignored in many biological models. The infected cells are separated into two distinct compartments, latently infected and actively infected. These latently infected cells do not produce virus and can evade from viral cytopathic effects and host immune mechanisms (see, e.g., [1720]). Recently, the following model with latent infection and CTL response has been proposed (see, e.g., [11]):where , , , , and represent the numbers of uninfected cells, latently infected cells, actively infected cells, free virus, and CTLs at time , respectively. Uninfected cells are produced at the rate , die at the rate , and become infected at the rate . The constant is the rate of latently infected cells translating to actively infected cells and is the death rate of actively infected cells. The constant represents the death rate of latently infected cells. The constant is the rate of CTL-mediated lysis and is the rate of CTL proliferation. The constant is the rate of production of virus by infected cells and is the clearance rate of free virus. The removal rate of CTLs is .

However, in plenty of previous papers, many models are constructed under the assumption that the presence of antigen can stimulate immunity and ignore the immune impairment (see, e.g., [8, 11, 16, 17]). In fact, some pathogens can also suppress immune response or even destroy immunity especially when the load of pathogens is too high such as HIV, HBV (see, e.g., [15, 2125]). Regoes et al. consider an ordinary differential equation (ODE) model with an immune impairment term (see, e.g., [12, 26, 27]), where denotes the immune impairment rate. Time delay should be considered in models for CTL response. It is shown that time delay plays an important role to the dynamic properties in models for CTL response (see, e.g., [1, 5, 6, 8, 15]). In fact, antigenic stimulation generating CTLs may need a period of time ; that is, the CTL response at time may depend on the numbers of CTLs and infected cells at time , for a time lag (see, e.g., [1, 5, 13]).

Motivated by the above works, in this paper, we will study a delay differential equation (DDE) model of HIV infection with immune impairment and delayed CTL response. Furthermore, we know that the actual incidence rate is probably not linear over the entire range of and . Based on the works mentioned above (see, e.g., [21, 2831]), we propose the following system with general incidence function:where the state variables , , , , and and the parameters , , , , , , , , , and have the same biological meaning as in system (1). is the immune impairment rate. Suppose all the parameters are nonnegative. We assume the incidence rate is the general incidence function , where satisfies the following hypotheses:(H1), for all and ; if and only if ;(H2), for all and ;(H3), for all and ;(H4), for all and .

Clearly, the hypotheses can be satisfied by different types of the incidence rate including the mass action, the Holling type II function, the saturation incidence, Beddington-DeAngelis incidence function, Crowley-Martin incidence function, and the more generalized incidence functions (see, e.g., [4, 6, 17, 32, 33]). Further, in order to study the global stability of the equilibria of system (2) by the method of Lyapunov functionals, we assume the following hypotheses hold (see, e.g., [28]):(H5), as or ;(H6), as or ;(H7), as or .

The main purpose of this paper is to carry out a complete theoretical analysis on the global stability of the equilibria of system (2). The organization of this paper is as follows. In Section 2, we consider the nonnegativity and boundedness of the solutions and the existence of the equilibria of system (2). In Section 3, we consider the global stability of the infection-free equilibrium and the infected equilibrium without immunity by constructing suitable Lyapunov functionals and using LaSalle invariance principle. In Section 4, we discuss the local stability of the infected equilibrium with CTL response and the existence of Hopf bifurcations. Finally, in Section 5, the brief conclusions are given and some numerical simulations are carried out to illustrate the main results.

2. Basic Results

2.1. The Nonnegativity and Boundedness of the Solutions

According to biological meanings, the initial condition of system (2) is given as follows:where and and is the Banach space of the continuous functions mapping the interval into .

Under the initial condition (3), it easily shows that the solution of system (2) is unique and nonnegative for all and ultimately bounded. It has the following result.

Proposition 1. Under the initial condition (3), the solution of system (2) is unique and nonnegative for all and also ultimately bounded, when (H1)–(H7) are satisfied.

Proof. The uniqueness and nonnegativity of the solution can be easily proved by using the theorems in [34, 35].
Next, for , define By the nonnegativity of the solutions, it follows that, for , where . Thus, it has that , from which it has that the solution is ultimately bounded.

2.2. The Existence of the Equilibria

Next, we consider the existence of the equilibria. The equilibrium of system (2) satisfies

If , and , system (2) has only one equilibrium, that is, the infection-free equilibrium , where .

If , and , we haveSince , we have that . Hence, we only need to consider the case of .

Consider the following function defined on the interval by Under hypotheses (H2) and (H3), we have We know that the function is strictly monotonically increasing with respect to . Denote the basic reproduction number of system (2) by Clearly, we have It has that there exists a unique such that , if . Then we can compute and by (8). Hence, we get the unique infected equilibrium without immunity .

If and , we get the following equations:

Since , we have , where Hence, the existence of the equilibrium requires and (13) has a solution on the interval .

Denote Hence, if , it has . Denote Under hypothesis (H2), we know that the function is strictly monotonically increasing with respect to . Clearly, we have where Hence, we have that there exists such that , if and . Then we can compute , , , and by (14) and (15).

Denote the immune response reproduction number of system (2) as . Therefore, we have that there exists a unique infected equilibrium with CTL response , if and . This proves the following theorem.

Theorem 2. Suppose that hypotheses (H1)(H4) are satisfied; the following conclusions hold.(i)System (2) always has an infection-free equilibrium .(ii)System (2) has an infected equilibrium without immunity if .(iii)System (2) has an infected equilibrium with immunity if and .

From hypotheses (H1)(H3), it is clear that . In order to study the global stability of the infected equilibrium in the next section, we give the following remark.

Remark 3. Suppose that is satisfied; then the following results hold:(i)If , then .(ii)If , then .

Let us give the proof of Remark 3. Firstly, for Case (i), since , then Since the function is strictly monotonically increasing with respect to and , we have . Therefore Then

Secondly, for Case (ii), since , then We have . Therefore Then

3. The Global Stability of the Equilibria

In this section, we study the global stability of the equilibria of system (2). Firstly, we analyze the global stability of the infection-free equilibrium .

Theorem 4. Suppose that hypotheses (H1)(H7) are satisfied. If , then the infection-free equilibrium is globally asymptotically stable for any time delay . If , then the infection-free equilibrium is unstable for any time delay .

Proof. Let , , , , be a positive solution of system (2) with the initial condition (3) for . Motivated by the works in [14, 28, 31, 36, 37], we consider the following Lyapunov functional:where . By (H1)(H5), it is obvious that is positive definite with respect to . For , the time derivative of along the solutions of system (2) is Since hypotheses and , we haveTherefore, if . Then it follows from stability theorems in [34, 35] that the infection-free equilibrium is stable for any time delay if .
Furthermore, note that, for each , implies that , . Let be the largest invariant set in the set We have from the first four equations of system (2) and the invariance of that . Since any solution of system (2) is bounded, it follows from LaSalle invariance principle (see, e.g., [34, 35]) that the infection-free equilibrium is also globally attractive for any time delay if .
The characteristic equation of system (2) at the infection-free equilibrium isClearly, if , (31) has at least a positive real root. Thus, the infection-free equilibrium is unstable.

Next we study the global stability of the infected equilibrium without immunity .

Theorem 5. Suppose that hypotheses (H1)(H7) and are satisfied. If , then the infected equilibrium without immunity is globally asymptotically stable for any time delay . If , then the infected equilibrium without immunity is unstable for any time delay .

Proof. Let , , , , be a positive solution of system (2) with the initial condition (3) for . Consider the following Lyapunov functional: Let . Then, has the global minimum at and . Furthermore, for . Hence, is positive definite with respect to . For , the time derivative of along the solutions of system (2) isNote that , , and ; we have Since the arithmetic mean is greater than or equal to the geometric mean, it has From hypotheses (H3)(H4), we have Note Remark 3, we have . Therefore, if . Then it follows from stability theorems in [34, 35] that the infected equilibrium without immunity is stable for any time delay if .
Furthermore, note that, for each , implies that , , , and . Let be the largest invariant set in the set We have from system (2) and the invariance of that . Since any solution of system (2) is bounded, it follows from LaSalle invariance principle (see, e.g., [34, 35]) that the infected equilibrium without immunity is also globally attractive for any time delay if .
The characteristic equation of system (2) at takes the formwhere is a polynomial with respect to . Let Thus we have and . From Remark 3, we have that if . Thus, if . Hence, if , then has at least a positive real root; that is, (38) has at least a positive real root. Therefore, the infected equilibrium without immunity is unstable.

4. The Local Stability of the Infected Equilibrium and Hopf Bifurcation

The characteristic equation of system (2) at the infected equilibrium with CTL response is given bywhereWhen , (40) becomeswhere Denote Since and , we have , , and

Thus, where

Assume further that(H8); that is, and

We have Therefore, , By Routh-Hurwitz criterion, all the roots of (42) have negative real parts. Hence we have the following result.

Proposition 6. When , if , , and (H8) hold, then the infected equilibrium with CTL response is locally asymptotically stable.

In fact, when , we can show that if and hold, the infected equilibrium with CTL response is globally asymptotically stable by constructing suitable Lyapunov function.

Proposition 7. Suppose that hypotheses (H1)(H7) and are satisfied. If , then the infected equilibrium with CTL response is globally asymptotically stable when .

Proof. By the following Lyapunov function, is positive definite with respect to . For , the time derivative of along the solutions of system (2) is Note that , , and ; we have Since the arithmetic mean is greater than or equal to the geometric mean, it has From hypotheses (H3), we have Therefore, if . Then it follows from stability theorems in [34, 35] that the infected equilibrium CTL response is stable for if . Similarly, by LaSalle invariance principle, we can show that the infected equilibrium CTL response is also globally attractive for if .

Next, we consider the case when . Since , is not a root of (40). We suppose (40) has a purely imaginary root for some . Substituting into (40) and separating the real and imaginary parts, we haveSquaring and adding the two equations of (53), it follows thatwhere Letting , (54) can be written asThen we haveDenote

By a similar argument as that in [38], we have the following results.

Lemma 8. For the polynomial equation (56), the following results hold.(i)Equation (56) has at least one positive root, if one of the following conditions holds:(a)(b), and or and there exists such that and , where , and (c), or and there exists such that and , where , and (d), and , where .(ii)If the conditions of are all not satisfied, then (56) has no positive real root.

Suppose that has positive real roots. Without loss of generality, we may assume that (56) has positive real roots, denoted, respectively, as . Then, (54) has positive real roots . From (40), we get Therefore, letwhere . Then are a pair of purely imaginary roots of (54) with .

DefineLet be a root of (40) satisfying and Differentiating the two sides of (40) with respect to and noticing that is a function of , it follows that Thus, we get From (40), we attain ThenTherefore, it follows that