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

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 R 0 and the immune response reproduction number R 1. 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.


Introduction
In recent years, mathematical models have been proved to be valuable in understanding the dynamics of viral infection (see, e.g., [1][2][3][4][5][6][7][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][12][13][14][15][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][18][19][20]). Recently, the following model with latent infection and CTL response has been proposed (see, e.g., [11]): where ( ), ( ), ( ), V( ), 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 1 , and become infected at the rate . The constant is the rate of latently infected cells translating to actively infected cells and 3 is the death rate of actively infected cells. The constant 2 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 4 is the clearance rate of free virus. The removal rate of CTLs is 5 .

Computational and Mathematical Methods in Medicine
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][22][23][24][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 > 0 (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 V. Based on the works mentioned above (see, e.g., [21,[28][29][30][31]), we propose the following system with general incidence function: where the state variables ( ), ( ), ( ), V( ), and ( ) and the parameters , , , , , 1 , 2 , 3 , 4 , and 5 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 ( , V)V, where ∈ 1 ([0, +∞] × [0, +∞], ) satisfies the following hypotheses: 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]): 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 infectionfree equilibrium 0 and the infected equilibrium without immunity 1 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.

The Nonnegativity and Boundedness of the Solutions.
According to biological meanings, the initial condition of system (2) is given as follows: Proof. The uniqueness and nonnegativity of the solution ( ( ), ( ), ( ), V( ), ( )) can be easily proved by using the theorems in [34,35].

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Denote the immune response reproduction number of system (2) as 1 . Therefore, we have that there exists a unique infected equilibrium with CTL response * = ( * , * , * , V * , * ), if > 1 and 1 > 1. This proves the following theorem.
From hypotheses (H1)-(H3), it is clear that 1 < 0 . In order to study the global stability of the infected equilibrium 1 in the next section, we give the following remark.
Remark 3. Suppose that > 1 is satisfied; then the following results hold: Let us give the proof of Remark 3. Firstly, for Case (i), since 1 > 1, then Since the function ( ) is strictly monotonically increasing with respect to and ( 1 ) = 0, we have 1 < . Therefore Then Secondly, for Case (ii), since 1 ≤ 1, then We have 1 ≥ . Therefore Then

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 0 .
The characteristic equation of system (2) at the infectionfree equilibrium 0 is Clearly, if 0 > 1, (31) has at least a positive real root. Thus, the infection-free equilibrium 0 is unstable.
Next we study the global stability of the infected equilibrium without immunity 1 .
The characteristic equation of system (2) at 1 takes the form where 0 ( ) is a polynomial with respect to . Let Thus we have lim → +∞ 1 ( ) > 0 and 1 (0 Hence, if 1 > 1, then 1 ( ) = 0 has at least a positive real root; that is, (38) has at least a positive real root. Therefore, the infected equilibrium without immunity 1 is unstable.
In fact, when = 0, we can show that if > 1 and 1 > 1 hold, the infected equilibrium with CTL response * is globally asymptotically stable by constructing suitable Lyapunov function.

(ii) If the conditions (a)-(d) of ( ) are all not satisfied, then
(56) has no positive real root.
Let ( ) = ( ) + ( ) be a root of (40) satisfying ( ( ) ) = 0 and ( ( ) ) = . Differentiating the two sides of (40) with respect to and noticing that is a function of , it follows that Thus, we get Computational and Mathematical Methods in Medicine 11 From (40), we attain Then Therefore, it follows that Since ] > 0, we can know that Re[ ( )/ | = ( ) ] and ℎ (] ) have the same sign.
From the above analysis, we have the following results.

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 0 ≤ 1, the infection-free equilibrium 0 is globally asymptotically stable for any time delay ≥ 0; that is, any solution ( ( ), ( ), ( ), V( ), ( )) → 0 = ( 0 , 0, 0, 0, 0). 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 0 .
If 1 > 1 and > 1, 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 ∈ [0, 0 ), 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 0 , 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.
Finally, let us choose the following data: Then we have that 0 = 3.2452 > 1, = 3.8942 > 1, and 1 = 2.4119 > 1, (56) has two positive roots, and ℎ (] ) ̸ = 0. By simple computations, we have 0 ≈ 0.0394 and    Figure 4 gives the phase trajectories of system (2) with < 0 and suitable initial condition. Figure 5 gives the phase trajectories of system (2) with > 0 and suitable initial condition and shows the occurrence of the Hopf bifurcations.
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]).