#### Abstract

We study the global stability of a human immunodeficiency virus (HIV) infection model with Cytotoxic T Lymphocytes (CTL) immune response. The model describes
the interaction of the HIV with two classes of target cells, CD4^{+} T cells and macrophages. Two types of distributed time delays are incorporated into the model to describe the time needed for infection of target cell and virus replication. Using the method of Lyapunov functional, we have established that the global stability of the model is determined by two threshold numbers, the basic reproduction number
and the immune response reproduction number . We have proven that, if , then the uninfected steady state is globally asymptotically stable (GAS), if , then the infected steady state without CTL immune response is GAS, and, if , then the infected steady state with CTL immune response is GAS.

#### 1. Introduction

One of the most diseases that have attracted the attention of many mathematicians is the acquired immunodeficiency syndrome (AIDS) caused by human immunodeficiency virus (HIV). HIV infects the CD4^{+} T cell which plays the central role in the immune system. Mathematical modeling and model analysis of HIV dynamics are important to discover the dynamical behaviors of the viral infection process and estimating key parameter values which leads to development of efficient antiviral drug therapies. Several mathematical models have been proposed to describe the HIV dynamics with CD4^{+} T cells [1–15]. In these papers, the Cytotoxic T Lymphocytes (CTL) immune response was not taken into account. The role of CTL is universal and necessary to eliminate or control the disease during viral infections. In particular, as a part of innate response, CTL plays a particularly important rate in antiviral defense by attacking infected cells. The basic HIV infection model which takes into consideration the CTL immune response has been proposed in [16] as
The state variables describe the plasma concentrations of, the uninfected CD4^{+} T cells;, the infected CD4^{+} T cells;, the free virus particles; and, the CTL cells at time. Here, (1) describes the population dynamics of the uninfected CD4^{+} T cells, whererepresents the rate of new uninfected cells that are generated from sources within the body,is the death rate constant, andis the infection rate constant at which a target cell becomes infected via contacting with virus. Equation (2) describes the population dynamics of the infected CD4^{+} T cells and shows that they die with rate constantand are killed by the CTL immune response with rate constant. Equation (3) describes the population dynamics of the free virus particles and shows that they are produced by the infected cells with rate constantand removed from the body with rate constant. Equation (4) describes the population dynamics of the CTL cells which are produced with rate constantand die with rate constant. Model (1)–(4) is based on the assumption that, once the virus contacts a target cell, the cell begins producing new virus particles. However, as pointed by Li and Shu [17], the period between the time for HIV to enter the target cell and the time for new virions to be produced from the infected cell needs the following stages: (i) the period between the viral entry of a target cell and integration of viral DNA into the host genome, (ii) the period from the integration of viral DNA to the transcription of viral RNA and translation of viral proteins such as reverse transcriptase, integrase, and protease, and (iii) the period between the transcription of viral RNA and the release and maturation of virus [17]. More realistic models incorporate the delay between the time of viral entry into the target cell and the time of the production of new virus particles, modeled with discrete time delay or distributed time delay (see, e.g., [3–7]). In [3–7], the HIV infection models did not take into account the impact of the immune response. The time delay has been incorporated into the HIV infection models with CTL immune response in [18–22]. It was assumed that the HIV attacks one class of target cells, CD4^{+} T cells. In 1997, Perelson et al. [23], observed that the HIV attacks two classes of target cells, CD4^{+} T cells and macrophages. HIV infection models with two classes of target cells, CD4^{+} T cells and macrophages, have been proposed in [1, 2, 8, 9, 11]; however, the effect of CTL immune response was neglected. In [24, 25], HIV infection models with two classes of target cells and with CTL immune response have been proposed. In [24], one type of discrete delay (stages (i) and (ii)) has been incorporated into the model. However, it is more realistic to consider the second type of delays between viral RNA transcription and viral release and maturation.

The purpose of the present paper is to propose an HIV infection model with two classes of target cells and two types of distributed delays taking into consideration the CTL immune response. The global stability of the steady state of the model are established using Lyapunov functional. We have proven that the global dynamics of this model is determined by the basic reproduction numberand the immune response reproduction number. We have shown that, if, then the uninfected steady state is globally asymptotically stable (GAS), if, then the infected steady state without CTL immune response is GAS, and, if, then the infected steady state with CTL immune response is GAS.

##### 1.1. The Model

In this section, we propose a mathematical model of HIV infection which describes two cocirculation populations of target cells, potentially representing CD4^{+} T cells and macrophages, taking into account the CTL immune response and multiple distributed intracellular delays
Herecorresponds to the CD4^{+} T cells and macrophages, respectively. All the variables and parameters of the model have the same meanings as given in (1)–(4). To take into account the delay between viral infection of an uninfected target cell and the production of an actively infected target cell, we letbe the random variable that describes the time between viral entry and the transcription of viral RNA (stages (i) and (ii)) with a probability distributionover the interval, andis limit superior to this delay. The factoraccounts for the loss of target cells during this delay period, whereis constant. On the other hand, to consider the delay between viral RNA transcription and viral release and maturation, we letbe the random variable; that is, the time between these two events with a probability distributionover the interval, andis limit superior to this delay [17]. The factoraccounts for the loss of infected cells during this delay period, whereis constant.

The probability distribution functions, , andare assumed to satisfy, and, and whereis a positive constant. Let Then The initial conditions for system (5)–(8) take the form whereandwhereis the Banach space of continuous functions mapping the intervalinto. By the fundamental theory of functional differential equations [26], system (5)–(8) have a unique solution satisfying the initial conditions (12).

##### 1.2. Nonnegativity and Boundedness of Solutions

In the following, we establish the nonnegativity and boundedness of solutions of (5)–(8) with initial conditions (12).

Proposition 1. *Letbe any solution of (5)–(8) satisfying the initial conditions (12); then, andare all non-negative forand ultimately bounded.*

* Proof . *First, we prove that, for allAssume thatloses its nonnegativity on some local existence intervalfor some constantand letbe such thatFrom (5) we haveHencefor somewhereis sufficiently small. This leads to a contradiction and hencefor allFurther, from (6) and (7) we have
confirming that, andfor all. By a recursive argument, we obtain, , andfor allNow from (8) we have
Thenfor all.

Next we show the boundedness of the solutions of system (5)–(8). From (5) we have, . This implies, .

Let. Then
where. Hence, where. Since,and then,, and. On the other hand,
Then. Therefore,, andare ultimately bounded.

##### 1.3. Steady States

First we define the basic reproduction numberand immune response reproduction numberof system (5)–(8) as
We can rewriteas
whereandare the immune strengths of CD4^{+} T cells and macrophages, respectively. Clearly.

Lemma 2. * If, then there exists only one uninfected steady state.** If, then there existand an infected steady state without CTL immune response.** If, then there exist,, and an infected steady state with CTL immune response.*

* Proof . *The steady states of (5)–(8) satisfy the following equations:
From (22) we have
Equation (23) has two possible solutions,or
Ifthen from (19) and (20) we obtainandas
and inserting them into (21) we obtain
Equation (26) has two possible solutionsor.

If, then substituting it in (25) leads to an uninfected steady state, where,. Ifthen we have
Now we can write (27) as
where, The solution of (28) is given by
Clearly if, thenand: then we chooseTherefore, if, then system (5)–(8) has an infected steady state without CTL immune responsewhereand
If, then from (24) we have
and inserting it into (21) we obtain
From (32) to (19) and (20) we get
and inserting (31) into (20) we get
We have, and if, thenand. It follows that, if, then there exists an infected steady state with CTL immune response.

Hence, if, then there exists only one steady state, if, then there exist two steady statesand, and, if, then there exist three steady states,, and.

##### 1.4. Global Stability

In this section, we establish the global stability of the three steady states of system (5)–(8) employing the method of Lyapunov functional which is used in [27] for SIR epidemic model with distributed delay. Next, we will use the following notation:for any. We also define a functionas. It is clear thatfor anyandhas the global minimum.

Theorem 3. *If, thenis GAS.*

* Proof . *Define a Lyapunov functionalas follows:
where,.

The time derivative ofalong the trajectories of (5)–(8) satisfies
Collecting terms of (36) we get
If, thenfor all. By Theoremin [26], the solutions of system (5)–(8) are limited to, the largest invariant subset of. Clearly, it follows from (37) thatif and only if,,, and. Noting thatis invariant, for each element of, we have; then. From (7) we drive that
Hence, this yields. Sincefor, then. Henceif and only if,,, , and. From LaSalle's Invariance Principle,is GAS.

Theorem 4. *If, thenis GAS.*

* Proof. *We construct the following Lyapunov functional:
Differentiating with respect to time yields
Collecting terms we obtain
From (19)–(21) we have
Using (42) and the following equality:
we obtain
Then collecting terms of (44) and using the following equalities:
we obtain
Equation (46) can be rewritten as
We can rewriteas:
Now we show that, if, then. Assume that; then
From (27) we have
Comparing (49) and (50) we get
Then
Using (42) we have
Then
Now if, thenIt follows that, if, then. By Theoremin [26], the solutions of system (5)–(8) are limited to, the largest invariant subset of. It can be seen thatif and only if,, and: that is,
If, then from (19) we haveand from (55) we have. It follows thatat. LaSalle's Invariance Principle implies the global stability of.

Theorem 5. *If, thenis GAS.*

* Proof . *We construct the following Lyapunov functional:
Differentiating with respect to time yields
Collecting terms we obtain
Using (19)–(21), we have equalities
We obtain
Then collecting terms of (60)
Using the following equalities:
we obtain
Equation (63) can be rewritten as
It can be easily seen that, if, , then. By Theoremin [26], the solutions of system (5)–(8) are limited to, the largest invariant subset of. It can be seen thatif and only ifand; that is,
If, then from (19) we haveand from (65) we haveMoreover, from (20) we have
Henceat. LaSalle's Invariance Principle implies the global stability of.

#### 2. Conclusion

In this paper, we have proposed an HIV infection model describing the interaction of the HIV with two classes of target cells, CD4^{+} T cells and macrophages, taking into account the CTL immune response. Two types of distributed time delays have been incorporated into the model to describe the time needed for infection of target cell and virus replication. The global stability of the three steady states of the model has been established by constructing suitable Lyapunov functionals and using LaSalle’s Invariant Principle. We have proven that(i)if