Computational and Mathematical Methods in Medicine
Volume 2015 (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

Fuxiang Li,¹ Wanbiao Ma,¹ Zhichao Jiang,² and Dan Li¹

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., [1–8]). 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., [1–4, 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., [11–16]).

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., [17–20]). 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, 21–25]). 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, 28–31]), 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 have

Since , 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 have

Therefore, 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 is

Clearly, 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) is

Note 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 form

where 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 by

where

When , (40) becomes

where

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 have

Squaring and adding the two equations of (53), it follows that

where

Letting , (54) can be written as

Then we have

Denote

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, let

where . Then are a pair of purely imaginary roots of (54) with .

Define

Let 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

Then

Therefore, it follows that

Since , we can know that and have the same sign.

From the above analysis, we have the following results.

Theorem 9. Let , and be defined by (62) and (63). If and are satisfied, then the following results hold:

(i)If the conditions of Lemma 8 are all not satisfied, then the infected equilibrium with CTL response is locally asymptotically stable for all time delay (ii)If one of the conditions of Lemma 8 is satisfied, then the infected equilibrium with CTL response is locally asymptotically stable for and unstable for .(iii)If all the conditions as stated in (ii) hold and , then system (2) undergoes a Hopf bifurcation at when .

5. Conclusion and Numerical Simulations

In this paper, we proposed a class of delayed HIV infection model (2) with general incidence rate and immune impairment. This general incidence only satisfies some general hypotheses and includes many types of special incidence functions as special cases. First, we discussed the nonnegativity and boundedness of the solutions and the existence of equilibria of system (2). Then, by constructing suitable Lyapunov functionals and using Lyapunov-LaSalle invariance principle and Hopf bifurcation theorem, we proved the following results.

If , the infection-free equilibrium is globally asymptotically stable for any time delay ; that is, any solution . In biology, this means that the virus can be finally cleared from the body and the disease dies out. At the same time, as the time increases, the numbers of latently infected cells, actively infected cells, and CTLs trends to zero and the number of uninfected cells trends to a constant .

If and , the infected equilibrium without immunity is globally asymptotically stable for any time delay ; that is, any solution . In biology, this indicates that the HIV infection will finally become chronic with no persistent CTL response.

If and , there exists a unique infected equilibrium with CTL response . The result of Theorem 9 implies that the time delay can destabilize the stability of the infected equilibrium with CTL response and leads to the occurrence of Hopf bifurcations.

If the time delay , the infected equilibrium with CTL response is locally asymptotically stable. In biology, this implies that the HIV infection may become chronic and the CTL immune response may be persistent. When the time delay passes through the critical value , the infected equilibrium with CTL response will become unstable and a Hopf bifurcation occurs under some conditions. In biology, this suggests that as the time delay increases, the numbers of the uninfected cells, latently infected cells, actively infected cells, free virus, and CTLs will first attend constant values and then become oscillated.

We now give numerical simulations to illustrate the main results in Sections 3 and 4.

Let us choose . Then we have that and . Based on the numerical simulations in [12, 19, 21, 22, 27], let us take the following data:

Direct calculations show that and ; system (2) has the infection-free equilibrium . By Theorem 4, the infection-free equilibrium is globally asymptotically stable for any time delay . Figure 1 gives the phase trajectories of system (2) with suitable initial condition.

Figure 1: Phase trajectories of system (2) with .

Next, let us choose the following data:

Direct computations show that , , and ; system (2) has the infected equilibrium without immunity . Therefore, by Theorem 5, the infected equilibrium without immunity is globally asymptotically stable for any time delay . Figure 2 gives the phase trajectories of system (2) with suitable initial condition.

Figure 2: Phase trajectories of system (2) with .

Furthermore, let us choose the following data:

Then we have that , , and and (56) has no positive root. System (2) has the infected equilibrium with CTL response . From Theorem 9(i), the infected equilibrium with CTL response is locally asymptotically stable for any time delay . Figure 3 gives the phase trajectories of system (2) with suitable initial condition.

Figure 3: Phase trajectories of system (2) with . The initial condition is (7970, 1000, 2.6, 0.2, 17).

Finally, let us choose the following data:

Then we have that , , and , (56) has two positive roots, and . By simple computations, we have and . From Theorem 9(ii), the infected equilibrium with CTL response is asymptotically stable if and unstable if . Figure 4 gives the phase trajectories of system (2) with and suitable initial condition. Figure 5 gives the phase trajectories of system (2) with and suitable initial condition and shows the occurrence of the Hopf bifurcations.

Figure 4: Phase trajectories of system (2) with . The initial condition is (7970, 1000, 2.6, 0.2, 17).

Figure 5: Phase trajectories of system (2) with . The initial condition is (7970, 1000, 2.6, 0.2, 17).

Since and , it is easy to see that the number of free viruses is increased and the number of CTLs is decreased with respect to the immune impairment rate . For example, if we choose , then and . If we choose , then and . If we choose , then and . Figure 6 shows that, for any initial conditions, when , the numbers of free viruses trend to , , and , respectively. Figure 7 shows that, for any initial conditions, when , the numbers of CTLs trend to , , and , respectively. In Figures 6 and 7, all the data are chosen as in Figure 4 except the immune impairment rate .

Figure 6: The curves of the free virus of system (2) with , , , and . The initial conditions are chosen as , , and , respectively.

Figure 7: The curves of the CTLs of system (2) with , , , and . The initial conditions are chosen as , , and , respectively.

As immune impairment rate increases, the CTL response gradually becomes weak and the individuals eventually develop AIDS. Thus, in order to control the HIV infection, we should decrease the value of . Numerical simulations show the similar known results (see, e.g., [22]).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors thank the editor and anonymous reviewers of the journal for their helpful and valuable comments. The research is supported by NNSF of China (11471034).

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