Global Dynamics of an HIV Infection Model with Two Classes of Target Cells and Distributed Delays

Abstract

We investigate the global dynamics of an HIV infection model with two classes of target cells and multiple distributed intracellular delays. The model is a 5-dimensional nonlinear delay ODEs that describes the interaction of the HIV with two classes of target cells, CD4⁺ T cells and macrophages. The incidence rate of infection is given by saturation functional response. The model has two types of distributed time delays describing time needed for infection of target cell and virus replication. This model can be seen as a generalization of several models given in the literature describing the interaction of the HIV with one class of target cells, CD4⁺ T cells. Lyapunov functionals are constructed to establish the global asymptotic stability of the uninfected and infected steady states of the model. We have proven that if the basic reproduction number 𝑅₀ is less than unity then the uninfected steady state is globally asymptotically stable, and if 𝑅₀>1 then the infected steady state exists and it is globally asymptotically stable.

1. Introduction

In the last decade, several mathematical models have been developed to describe the interaction of the human immunodeficiency virus (HIV) with target cells [1]. HIV is responsible for acquired immunodeficiency syndrome (AIDS). Mathematical modeling and model analysis of the HIV dynamics are important for exploring possible mechanisms and dynamical behaviors of the viral infection process, estimating key parameter values, and guiding development efficient antiviral drug therapies. Some of the existing HIV infection models are given by nonlinear ODEs by assuming that the infection could occur and the viruses are produced from infected target cells instantaneously, once the uninfected target cells are contacted by the virus particles (see e.g., [2–4]). Other accurate models incorporate the delay between the time, the viral entry into the target cell, and the time the production of new virus particles, modeled with discrete time delay or distributed time delay using functional differential equations (see e.g., [5–9]). The basic virus dynamics model with distributed intracellular time delay has been proposed in [9] and given by 𝑥̇(𝑡)=𝜆−𝑑𝑥(𝑡)−1−𝑢_𝑟𝑡(𝑡)𝑣(𝑡),(𝑡)=𝛽𝑥𝑦̇1−𝑢_𝑟𝑡𝛽^∞₀𝑓(𝜏)𝑒^−𝑚𝜏𝑥()𝑣()(𝑡),(𝑡)=𝑡−𝜏𝑡−𝜏𝑑𝜏−𝑎𝑦𝑣̇1−𝑢_𝑝𝑝^∞₀𝑔(𝜏)𝑦(𝑡−𝜏)𝑑𝜏−𝑐𝑣(𝑡), (1.1)(1.2)(1.3) where 𝑥(𝑡),  𝑦(𝑡) and 𝑣(𝑡) represent the populations of uninfected CD4⁺ T cells, infected cells, and free virus particles at time 𝑡, respectively. Here, 𝜆 represents the rate of which new CD4⁺ T cells are generated from sources within the body, 𝑑 is the death rate constant, and 𝛽 is the constant rate at which a target cell becomes infected via contacting with virus. Equation (1.2) describes the population dynamics of the infected cells and shows that they die with rate constant 𝑎. The virus particles are produced by the infected cells with rate constant 𝑝 and are removed from the system with rate constant 𝑐. The model includes two kinds of antiretroviral drugs, reverse transcriptase inhibitors (RTI) to prevent the virus from infecting cells and protease inhibitors (PI) drugs to prevent already infected host cells from producing infectious virus particles. The parameters 𝑢_𝑟𝑡∈[0,1] and 𝑢_𝑝∈[0,1] are the efficacies of RTI and PI, respectively. To account for the time lag between viral contacting a target cell and the production of new virus particles, two distributed intracellular time delays are introduced. It is assumed that the target cells are contacted by the virus particles at time 𝑡−𝜏 become infected cells at time 𝑡, where 𝜏 is a random variable with a probability distribution 𝑓(𝜏). The factor 𝑒^−𝑚𝜏 accounts for the loss of target cells during time period [𝑡−𝜏,𝑡]. On the other hand, it is assumed that a cell infected at time 𝑡−𝜏 starts to yield new infectious virus at time 𝑡, where 𝜏 is distributed according to a probability distribution 𝑔(𝜏).

A tremendous effort has been made in developing various mathematical models of HIV infection with discrete or distributed delays and studying their basic and global properties, such as positive invariance properties, boundedness of the model solutions, and stability analysis [5–20]. Most of the existing delayed HIV infection models are based on the assumption that the virus attacks one class of target cells, CD4⁺ T cells. In 1997, it was observed by Perelson et al. [21] that the HIV attacks two classes of target cells, CD4⁺ T cells and macrophages. In [3, 4], an HIV model with two target cells has been proposed. Also, in very recent works [22–25], we have proposed several HIV models with two target cells and investigated the global asymptotic stability of their steady states. In [26], we have studied a class of virus infection models assuming that the virus attacks multiple classes of target cells. In very recent works, [27, 28], discrete-time delays have been incorporated into the HIV models.

The purpose of this paper is to propose a delayed HIV infection model with two target cells and establish the global stability of its steady states. We assume that the infection rate is given by saturation functional response. We incorporate two types of distributed delays into this model to account the time delay between the time the target cells are contacted by the virus particle and the time the emission of infectious (matures) virus particles. The global stability of this model is established using Lyapunov functionals, which are similar in nature to those used in [29]. We prove that the global dynamics of these models are determined by the basic reproduction number 𝑅₀. If 𝑅₀≤1, then the uninfected steady state is globally asymptotically stable (GAS) and if 𝑅₀>1, then the infected steady state exists and it is GAS.

2. HIV Infection Model with Two Classes of Target Cells and Distributed Delays

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 saturation infection rate and multiple distributed intracellular delays. This model can be considered as an extension of HIV infection models given in [3, 4, 22].

Consider the following: 𝑥̇₁(𝑡)=𝜆₁−𝑑₁𝑥₁(𝑡)−𝛽₁𝑥₁(𝑡)𝑣(𝑡)1+𝛼₁𝑣(𝑡),𝑦₁̇(𝑡)=𝛽₁^∞₀𝑓₁(𝜏)𝑒^−𝑚1𝜏𝑥₁(𝑡−𝜏)𝑣(𝑡−𝜏)1+𝛼₁𝑣(𝑡−𝜏)𝑑𝜏−𝑎₁𝑦₁(𝑡),𝑥̇₂(𝑡)=𝜆₂−𝑑₂𝑥₂(𝑡)−𝛽₂𝑥₂(𝑡)𝑣(𝑡)1+𝛼₂𝑣(𝑡),𝑦₂̇(𝑡)=𝛽₂^∞₀𝑓₂(𝜏)𝑒^−𝑚2𝜏𝑥₂(𝑡−𝜏)𝑣(𝑡−𝜏)1+𝛼₂𝑣(𝑡−𝜏)𝑑𝜏−𝑎₂𝑦₂(𝑡),𝑣̇(𝑡)=𝑝₁^∞₀𝑔₁(𝜏)𝑒^−𝑛1𝜏𝑦₁(𝑡−𝜏)𝑑𝜏+𝑝₂^∞₀𝑔₂(𝜏)𝑒^−𝑛2𝜏𝑦₂(𝑡−𝜏)𝑑𝜏−𝑐𝑣(𝑡). (2.1)(2.2)(2.3)(2.4)(2.5) The state variables describes the plasma concentrations of: 𝑥₁, the uninfected CD4⁺ T cells; 𝑦₁, the infected CD4⁺ T cells; 𝑥₂, the uninfected macrophages; 𝑦₂, the infected macrophages; 𝑣, the free virus particles. Here, 𝛼_𝑖,  𝑖=1,2 are positive constants, 𝛽_𝑖=(1−𝑢_𝑟𝑡)𝛽_𝑖, and 𝑝_𝑖=(1−𝑢_𝑝)𝑝_𝑖,  𝑖=1,2. The factors 𝑒^{−𝑛𝑖𝜏},  𝑖=1,2 account for the cells loss during the delay period. All the other parameters of the model have the same meanings as given in (1.1)–(1.3).

The probability distribution functions 𝑓_𝑖(𝜏) and 𝑔_𝑖(𝜏) are assumed to satisfy 𝑓_𝑖(𝜏)>0 and 𝑔_𝑖(𝜏)>0,  𝑖=1,2 and ^∞₀𝑓_𝑖(𝜏)𝑑𝜏=^∞₀𝑔_𝑖(𝜏)𝑑𝜏=1,𝑖=1,2,_∞0𝑓_𝑖(𝑟)𝑒^𝑠𝑟𝑑𝑟<∞,^∞₀𝑔_𝑖(𝑟)𝑒^𝑠𝑟𝑑𝑟<∞,𝑖=1,2, (2.6) where 𝑠 is a positive number. Then 0<^∞₀𝑓_𝑖(𝜏)𝑒^{−𝑚𝑖𝜏}𝑑𝜏≤1,for𝑚_𝑖≥0,𝑖=1,2,0<^∞₀𝑔_𝑖(𝜏)𝑒^{−𝑛𝑖𝜏}𝑑𝜏≤1,for𝑛_𝑖≥0,𝑖=1,2. (2.7) The initial conditions for system (2.1)–(2.5) take the form 𝑥₁(𝜃)=𝜑₁(𝜃),𝑦₁(𝜃)=𝜑₂(𝜃),𝑥₂(𝜃)=𝜑₃(𝜃),𝑦₂(𝜃)=𝜑₄(𝜃),𝑣(𝜃)=𝜑₅(𝜃),𝜑_𝑗(𝜃)()𝜑≥0,𝜃∈−∞,0,𝑗=1,…,5,_𝑗(0)>0,𝑗=1,…,5, (2.8) where (𝜑₁(𝜃),𝜑₂(𝜃),…,𝜑₅(𝜃))∈𝑈𝐶((−∞,0],ℝ⁵₊), and 𝑈𝐶 is the Banach space of fading memory type defined as [30] 𝑈𝐶(−∞,0],ℝ⁵₊=𝜑∈𝐶(−∞,0],ℝ⁵₊∶𝜑(𝑟)𝑒^𝑠𝑟isuniformlycontinuouson(−∞,0],𝜑=sup_𝑟≤0𝜑(𝑟)𝑒^𝑠𝑟<∞, (2.9) where 𝐶((−∞,0],ℝ⁵₊) is the Banach space of continuous functions mapping the interval (−∞,0]into ℝ⁵₊. By the fundamental theory of functional differential equations [31], system (2.1)–(2.5) has a unique solution satisfying the initial conditions (2.8).

2.1. Nonnegativity and Boundedness of Solutions

In the following, we establish the nonnegativity and boundedness of solutions of (2.1)–(2.5) with initial conditions (2.8).

Proposition 2.1. Let (𝑥₁(𝑡),𝑦₁(𝑡),𝑥₂(𝑡),𝑦₂(𝑡),𝑣(𝑡)) be any solution of (2.1)–(2.5) satisfying the initial conditions (2.8), then 𝑥₁(𝑡),  𝑦₁(𝑡),  𝑥₂(𝑡),  𝑦₂(𝑡) and 𝑣(𝑡) are all nonnegative for 𝑡≥0 and ultimately bounded.

Proof. From (2.1) and (2.3) we have 𝑥_𝑖(𝑡)=𝑥_𝑖(0)𝑒^{−∫𝑡0[𝑑𝑖+𝛽𝑖𝑣(𝜉)/(1+𝛼𝑖𝑣(𝜉))]𝑑𝜉}+𝜆_𝑖^𝑡₀𝑒^{−∫𝑡𝜂[𝑑𝑖+𝛽𝑖𝑣(𝜉)/(1+𝛼𝑖𝑣(𝜉))]𝑑𝜉}𝑑𝜂,𝑖=1,2, (2.10) which indicates that 𝑥_𝑖(𝑡)≥0, for all 𝑡≥0. Now from (2.2), (2.4), and (2.5) we have 𝑦_𝑖(𝑡)=𝑦_𝑖(0)𝑒^{−𝑎𝑖𝑡}+𝛽_𝑖^𝑡₀𝑒^{−𝑎𝑖(𝑡−𝜂)∞}₀𝑓_𝑖(𝜏)𝑒^{−𝑚𝑖𝜏}𝑥_𝑖𝜂−𝜏𝑣𝜂−𝜏1+𝛼_𝑖𝑣𝑣(𝑡)(0)𝑒𝜂−𝜏𝑑𝜏𝑑𝜂,𝑖=1,2,=𝑣^−𝑐𝑡+𝑝₁^𝑡₀𝑒^{−𝑐(𝑡−𝜂)∞}₀𝑔₁(𝜏)𝑒^−𝑛1𝜏𝑦₁𝜂−𝜏𝑑𝜏𝑑𝜂+𝑝₂^𝑡₀𝑒^{−𝑐(𝑡−𝜂)∞}₀𝑔₂(𝜏)𝑒^−𝑛2𝜏𝑦₂𝜂−𝜏𝑑𝜏𝑑𝜂, (2.11) confirming that 𝑦₁(𝑡),  𝑦₂(𝑡)≥0, and 𝑣(𝑡)≥0 for all 𝑡≥0.
Next we show the boundedness of the solutions. From (2.1) and (2.3) we have 𝑥̇_𝑖(𝑡)≤𝜆_𝑖−𝑑_𝑖𝑥_𝑖(𝑡), 𝑖=1,2. This implies limsup_𝑡→∞𝑥_𝑖(𝑡)≤𝜆_𝑖/𝑑_𝑖,  𝑖=1,2.
Let 𝑋_𝑖(𝑡)=∫^∞₀𝑓_𝑖(𝜏)𝑒^{−𝑚𝑖𝜏}𝑥_𝑖(𝑡−𝜏)𝑑𝜏+𝑦_𝑖(𝑡),  𝑖=1,2, then 𝑋̇_𝑖(𝑡)=^∞₀𝑓_𝑖(𝜏)𝑒^{−𝑚𝑖𝜏}𝜆_𝑖−𝑑_𝑖𝑥_𝑖(𝑡−𝜏)−𝛽_𝑖𝑥_𝑖(𝑡−𝜏)𝑣(𝑡−𝜏)1+𝛼_𝑖𝑣()+𝑡−𝜏𝑑𝜏_∞0𝑓_𝑖(𝜏)𝑒^{−𝑚𝑖𝜏}𝛽_𝑖𝑥_𝑖(𝑡−𝜏)𝑣(𝑡−𝜏)1+𝛼_𝑖𝑣(𝑡−𝜏)𝑑𝜏−𝑎_𝑖𝑦_𝑖(𝑡)=𝜆_𝑖^∞₀𝑓_𝑖(𝜏)𝑒^{−𝑚𝑖𝜏}𝑑𝜏−𝑑_𝑖^∞₀𝑓_𝑖(𝜏)𝑒^{−𝑚𝑖𝜏}𝑥_𝑖(𝑡−𝜏)𝑑𝜏−𝑎_𝑖𝑦_𝑖(𝑡)≤𝜆_𝑖^∞₀𝑓_𝑖(𝜏)𝑒^{−𝑚𝑖𝜏}𝑑𝜏−𝜎_𝑖^∞₀𝑓_𝑖(𝜏)𝑒^{−𝑚𝑖𝜏}𝑥_𝑖(𝑡−𝜏)𝑑𝜏+𝑦_𝑖(𝑡)=𝜆_𝑖^∞₀𝑓_𝑖(𝜏)𝑒^{−𝑚𝑖𝜏}𝑑𝜏−𝜎_𝑖𝑋_𝑖(𝑡)≤𝜆_𝑖−𝜎_𝑖𝑋_𝑖(𝑡), (2.12) where 𝜎_𝑖=min{𝑑_𝑖,𝑎_𝑖}. Hence limsup_𝑡→∞𝑋_𝑖(𝑡)≤𝐿_𝑖, where 𝐿_𝑖=𝜆_𝑖/𝜎_𝑖, 𝑖=1,2. On the other hand, 𝑣̇(𝑡)≤𝑝₁𝐿₁^∞₀𝑔₁(𝜏)𝑒^−𝑛1𝜏𝑑𝜏+𝑝₂𝐿₂^∞₀𝑔₂(𝜏)𝑒^−𝑛2𝜏𝑑𝜏−𝑐𝑣≤𝑝₁𝐿₁+𝑝₂𝐿₂−𝑐𝑣, (2.13) then limsup_𝑡→∞𝑣(𝑡)≤(𝑝₁𝐿₁+𝑝₂𝐿₂)/𝑐. Therefore, 𝑥₁(𝑡),  𝑦₁(𝑡),  𝑥₂(𝑡),  𝑦₂(𝑡), and 𝑣(𝑡) are ultimately bounded.

2.2. Steady States

It is clear that system (2.1)–(2.5) has an uninfected steady state 𝐸₀=(𝑥⁰₁,0,𝑥⁰₂,0,0), where 𝑥⁰_𝑖=𝜆_𝑖/𝑑_𝑖, 𝑖=1,2. In addition to 𝐸₀, the system can also have a positive infected steady state 𝐸₁(𝑥^∗₁,𝑦^∗₁,𝑥^∗₂,𝑦^∗₂,𝑣^∗). The coordinates of the infected steady state, if they exist, satisfy the following equalities: 𝜆_𝑖=𝑑_𝑖𝑥^∗_𝑖+𝛽_𝑖𝑥^∗_𝑖𝑣^∗1+𝛼_𝑖𝑣^∗𝑎,𝑖=1,2,_𝑖𝑦^∗_𝑖=𝐹_𝑖𝛽_𝑖𝑥^∗_𝑖𝑣^∗1+𝛼_𝑖𝑣^∗,𝑖=1,2,𝑐𝑣^∗=𝐺₁𝑝₁𝑦^∗₁+𝐺₂𝑝₂𝑦^∗₂, (2.14)(2.15)(2.16) where 𝐹_𝑖=^∞₀𝑓_𝑖(𝜏)𝑒^{−𝑚𝑖𝜏}𝑑𝜏,𝐺_𝑖=^∞₀𝑔_𝑖(𝜏)𝑒^{−𝑛𝑖𝜏}𝑑𝜏,𝑖=1,2. (2.17) Following van den Driessche and Watmough [32], we define the basic reproduction number for system (2.1)–(2.5) as 𝑅₀=²_𝑖=1𝑅_𝑖=²_𝑖=1𝐹_𝑖𝐺_𝑖𝛽_𝑖𝑝_𝑖𝜆_𝑖𝑎_𝑖𝑑_𝑖𝑐, (2.18) where 𝑅₁ and 𝑅₂ are the basic reproduction numbers of the HIV dynamics with CD4⁺ T cells (in the absence of macrophages) and the HIV dynamics with macrophages (in the absence of CD4⁺ T cells), respectively.

Lemma 2.2. If 𝑅₀>1, then there exists a positive steady state 𝐸₁.

Proof. From (2.14) and (2.15) we have 𝑥^∗_𝑖=𝑥⁰_𝑖1+𝛼_𝑖𝑣^∗1+𝛿_𝑖𝑣^∗𝑦,𝑖=1,2,_∗𝑖=𝐹_𝑖𝛽_𝑖𝑥⁰_𝑖𝑣^∗𝑎_𝑖1+𝛿_𝑖𝑣^∗,𝑖=1,2, (2.19)(2.20) where 𝛿_𝑖=𝛼_𝑖+𝛽_𝑖/𝑑_𝑖. From (2.20) into (2.16) we get 𝐹1=¹𝐺₁𝑝₁𝛽₁𝑥⁰₁𝑎₁𝑐1+𝛿₁𝑣^∗+𝐹₂𝐺₂𝑝₂𝛽₂𝑥⁰₂𝑎₂𝑐1+𝛿₂𝑣^∗=𝑅₁1+𝛿₁𝑣^∗+𝑅₂1+𝛿₂𝑣^∗. (2.21) Equation (2.21) can be written as 𝛿₁𝛿₂𝑣^∗2+𝛿₁𝑅₁+𝛿₂𝑅₂+1−𝑅₀𝛿₁+𝛿₂𝑣^∗+1−𝑅₀=0. (2.22) If 𝑅₀>1, then the positive solution of (2.21) is given by: 𝑣^∗=−𝛿₁𝑅₁+𝛿₂𝑅₂+1−𝑅₀𝛿₁+𝛿₂+𝛿₁𝑅₁+𝛿₂𝑅₂+(1−𝑅₀)(𝛿₁+𝛿₂)²−4𝛿₁𝛿₂1−𝑅₀2𝛿₁𝛿₂. (2.23) It follows that, if 𝑅₀>1 then 𝑥^∗₁,  𝑦^∗₁,  𝑥^∗₂,  𝑦^∗₂ and 𝑣^∗ are all positive.

2.3. Global Stability

In this section, we prove the global stability of the uninfected and infected steady states of system (2.1)–(2.3) employing the method of Lyapunov functional which is used in [29] for SIR epidemic model with distributed delay. Next we shall use the following notation: 𝑧=𝑧(𝑡), for any 𝑧∈{𝑥₁,𝑦₁,𝑥₂,𝑦₂,𝑣}. We also define a function 𝐻∶(0,∞)→[0,∞) as 𝐻(𝑧)=𝑧−1−ln𝑧. (2.24) It is clear that 𝐻(𝑧)≥0 for any 𝑧>0 and 𝐻 has the global minimum 𝐻(1)=0.

Theorem 2.3. If 𝑅₀≤1, then 𝐸₀ is GAS.

Proof. Define a Lyapunov functional 𝑊₁ as follows: 𝑊₁=²_𝑖=1𝛾_𝑖𝑥⁰_𝑖𝐻𝑥_𝑖𝑥⁰_𝑖+1𝐹_𝑖𝑦_𝑖+𝛽_𝑖𝐹_𝑖^∞₀𝑓_𝑖(𝜏)𝑒^{−𝑚𝑖𝜏𝜏}₀𝑥_𝑖(𝑡−𝜃)𝑣(𝑡−𝜃)1+𝛼_𝑖𝑣()+𝑎𝑡−𝜃𝑑𝜃𝑑𝜏_𝑖𝐹_𝑖𝐺_𝑖^∞₀𝑔_𝑖(𝜏)𝑒^{−𝑛𝑖𝜏𝜏}₀𝑦_𝑖(𝑡−𝜃)𝑑𝜃𝑑𝜏+𝑣, (2.25) where 𝛾_𝑖=𝑝_𝑖𝐹_𝑖𝐺_𝑖/𝑎_𝑖, 𝑖=1,2.
The time derivative of 𝑊₁ along the trajectories of (2.1)–(2.5) satisfies 𝑑𝑊₁=𝑑𝑡²_𝑖=1𝛾_𝑖𝑥1−^0𝑖𝑥_𝑖𝜆_𝑖−𝑑_𝑖𝑥_𝑖−𝛽_𝑖𝑥_𝑖𝑣1+𝛼_𝑖𝑣+𝛽_𝑖𝐹_𝑖^∞₀𝑓_𝑖(𝜏)𝑒^{−𝑚𝑖𝜏}𝑥_𝑖(𝑡−𝜏)𝑣(𝑡−𝜏)1+𝛼_𝑖𝑣()−𝑎𝑡−𝜏𝑑𝜏_𝑖𝐹_𝑖𝑦_𝑖+𝛽_𝑖𝐹_𝑖^∞₀𝑓_𝑖(𝜏)𝑒^{−𝑚𝑖𝜏}𝑥_𝑖𝑣1+𝛼_𝑖𝑣−𝑥_𝑖(𝑡−𝜏)𝑣(𝑡−𝜏)1+𝛼_𝑖𝑣()+𝑎𝑡−𝜏𝑑𝜏_𝑖𝐹_𝑖𝐺_𝑖^∞₀𝑔_𝑖(𝜏)𝑒^{−𝑛𝑖𝜏}𝑦_𝑖−𝑦_𝑖()+𝑡−𝜏𝑑𝜏_2𝑖=1𝑝_𝑖^∞₀𝑔_𝑖(𝜏)𝑒^{−𝑛𝑖𝜏}𝑦_𝑖(𝑡−𝜏)𝑑𝜏−𝑐𝑣. (2.26) Collecting terms of (2.26) we get 𝑑𝑊₁=𝑑𝑡²_𝑖=1𝛾_𝑖𝜆_𝑖−𝑑_𝑖𝑥_𝑖−𝜆_𝑖𝑥⁰_𝑖𝑥_𝑖+𝑑_𝑖𝑥⁰_𝑖+𝛽_𝑖𝑥⁰_𝑖𝑣1+𝛼_𝑖𝑣=−𝑐𝑣_2𝑖=1𝛾_𝑖𝜆_𝑖𝑥2−^𝑖𝑥⁰_𝑖−𝑥⁰_𝑖𝑥_𝑖−𝑐𝑣+𝑐𝑣²_𝑖=1𝐹_𝑖𝐺_𝑖𝑝_𝑖𝛽_𝑖𝑥⁰_𝑖𝑎_𝑖𝑐1+𝛼_𝑖𝑣=−²_𝑖=1𝛾_𝑖𝑑_𝑖𝑥_𝑖−𝑥⁰_𝑖²𝑥_𝑖−𝑐𝑣+𝑐𝑣²_𝑖=1𝑅_𝑖1+𝛼_𝑖𝑣=−²_𝑖=1𝛾_𝑖𝑑_𝑖𝑥_𝑖−𝑥⁰_𝑖²𝑥_𝑖−²_𝑖=1𝑅_𝑖𝛼_𝑖𝑐𝑣²1+𝛼_𝑖𝑣+𝑅₀−1𝑐𝑣. (2.27) If 𝑅₀≤1 then 𝑑𝑊₁/𝑑𝑡≤0 for all 𝑥₁,𝑥₂,𝑣>0. By Theorem 5.3.1 in [31], the solutions of system (2.1)–(2.5) limit to 𝑀, the largest invariant subset of {𝑑𝑊₁/𝑑𝑡=0}. Clearly, it follows from (2.27) that 𝑑𝑊₁/𝑑𝑡=0 if and only if 𝑥_𝑖=𝑥⁰_𝑖,  𝑖=1,2, and  𝑣=0. Noting that 𝑀 is invariant, for each element of 𝑀 we have 𝑣=0, then 𝑣̇=0. From (2.5) we drive that 0=𝑣̇=𝑝₁^∞₀𝑔₁(𝜏)𝑒^−𝑛1𝜏𝑦₁(𝑡−𝜏)𝑑𝜏+𝑝₂^∞₀𝑔₂(𝜏)𝑒^−𝑛2𝜏𝑦₂(𝑡−𝜏)𝑑𝜏. (2.28) This yields 𝑦₁=𝑦₂=0. Hence 𝑑𝑊₁/𝑑𝑡=0 if and only if 𝑥_𝑖=𝑥⁰_𝑖,  𝑦_𝑖=0,  𝑖=1,2, and 𝑣=0. From La Salle's Invariance Principle, 𝐸₀ is GAS.

Theorem 2.4. If 𝑅₀>1, then 𝐸₁ is GAS.

Proof. We construct the following Lyapunov functional: 𝑊₂=²_𝑖=1𝛾_𝑖𝑥^∗_𝑖𝐻𝑥_𝑖𝑥^∗_𝑖+1𝐹_𝑖𝑦^∗_𝑖𝐻𝑦_𝑖𝑦^∗_𝑖+1𝐹_𝑖𝛽_𝑖𝑥^∗_𝑖𝑣^∗1+𝛼_𝑖𝑣^∗∞₀𝑓_𝑖(𝜏)𝑒^{−𝑚𝑖𝜏𝜏}₀𝐻𝑥_𝑖(𝑡−𝜃)𝑣(𝑡−𝜃)1+𝛼_𝑖𝑣^∗𝑥^∗_𝑖𝑣^∗1+𝛼_𝑖𝑣()+𝑎𝑡−𝜃𝑑𝜃𝑑𝜏_𝑖𝑦^∗_𝑖𝐹_𝑖𝐺_𝑖^∞₀𝑔_𝑖(𝜏)𝑒^{−𝑛𝑖𝜏𝜏}₀𝐻𝑦_𝑖(𝑡−𝜃)𝑦^∗_𝑖𝑑𝜃𝑑𝜏+𝑣^∗𝐻𝑣𝑣^∗. (2.29) Differentiating with respect to time yields 𝑑𝑊₂=𝑑𝑡²_𝑖=1𝛾_𝑖𝑥1−^∗𝑖𝑥_𝑖𝜆_𝑖−𝑑_𝑖𝑥_𝑖−𝛽_𝑖𝑥_𝑖𝑣1+𝛼_𝑖𝑣+1𝐹_𝑖𝑦1−^∗𝑖𝑦_𝑖𝛽_𝑖^∞₀𝑓_𝑖(𝜏)𝑒^{−𝑚𝑖𝜏}𝑥_𝑖(𝑡−𝜏)𝑣(𝑡−𝜏)1+𝛼_𝑖𝑣(𝑡−𝜏)𝑑𝜏−𝑎_𝑖𝑦_𝑖+𝛽_𝑖𝐹_𝑖^∞₀𝑓_𝑖(𝜏)𝑒^{−𝑚𝑖𝜏}×𝑥_𝑖𝑣1+𝛼_𝑖𝑣−𝑥_𝑖(𝑡−𝜏)𝑣(𝑡−𝜏)1+𝛼_𝑖𝑣()+𝑥𝑡−𝜏^∗𝑖𝑣^∗1+𝛼_𝑖𝑣^∗𝑥ln^𝑖(𝑡−𝜏)𝑣(𝑡−𝜏)1+𝛼_𝑖𝑣𝑥_𝑖𝑣1+𝛼_𝑖𝑣(+𝑎𝑡−𝜏)𝑑𝜏_𝑖𝐹_𝑖𝐺_𝑖^∞₀𝑔_𝑖(𝜏)𝑒^{−𝑛𝑖𝜏}𝑦_𝑖−𝑦_𝑖(𝑡−𝜏)+𝑦^∗_𝑖𝑦ln^𝑖(𝑡−𝜏)𝑦_𝑖+𝑣𝑑𝜏1−^∗𝑣²_𝑖=1𝑝_𝑖^∞₀𝑔_𝑖(𝜏)𝑒^{−𝑛𝑖𝜏}𝑦_𝑖(𝑡−𝜏)𝑑𝜏−𝑐𝑣. (2.30) Collecting terms we obtain 𝑑𝑊₂=𝑑𝑡²_𝑖=1𝛾_𝑖𝜆_𝑖−𝑑_𝑖𝑥_𝑖−𝜆_𝑖𝑥^∗_𝑖𝑥_𝑖+𝑑_𝑖𝑥^∗_𝑖+𝛽_𝑖𝑥^∗_𝑖𝑣1+𝛼_𝑖𝑣−𝛽_𝑖𝑦^∗_𝑖𝐹_𝑖𝑦_𝑖^∞₀𝑓_𝑖(𝜏)𝑒^{−𝑚𝑖𝜏}𝑥_𝑖(𝑡−𝜏)𝑣(𝑡−𝜏)1+𝛼_𝑖𝑣()+𝑎𝑡−𝜏𝑑𝜏_𝑖𝐹_𝑖𝑦^∗_𝑖+1𝐹_𝑖𝛽_𝑖𝑥^∗_𝑖𝑣^∗1+𝛼_𝑖𝑣^∗∞₀𝑓_𝑖(𝜏)𝑒^{−𝑚𝑖𝜏}𝑥ln^𝑖(𝑡−𝜏)𝑣(𝑡−𝜏)1+𝛼_𝑖𝑣𝑥_𝑖𝑣1+𝛼_𝑖𝑣()+𝑎𝑡−𝜏𝑑𝜏_𝑖𝑦^∗_𝑖𝐹_𝑖𝐺_𝑖^∞₀𝑔_𝑖(𝜏)𝑒^{−𝑛𝑖𝜏}𝑦ln^𝑖(𝑡−𝜏)𝑦_𝑖−𝑣𝑑𝜏−𝑐𝑣_∗𝑣²_𝑖=1𝑝_𝑖^∞₀𝑔_𝑖(𝜏)𝑒^{−𝑛𝑖𝜏}𝑦_𝑖(𝑡−𝜏)𝑑𝜏+𝑐𝑣^∗. (2.31) Using the infected steady state conditions (2.14)–(2.16), and the following equality: 𝑐𝑣=𝑐𝑣^∗𝑣𝑣^∗=𝑣𝑣^∗2_𝑖=1𝐺_𝑖𝑝_𝑖𝑦^∗_𝑖=𝑣𝑣^∗2_𝑖=1𝛾_𝑖𝑎_𝑖𝐹_𝑖𝑦^∗_𝑖, (2.32) we obtain 𝑑𝑊₂=𝑑𝑡²_𝑖=1𝛾_𝑖𝑑_𝑖𝑥^∗_𝑖+𝑎_𝑖𝐹_𝑖𝑦^∗_𝑖−𝑑_𝑖𝑥_𝑖−𝑥^∗_𝑖𝑥_𝑖𝑑_𝑖𝑥^∗_𝑖+𝑎_𝑖𝐹_𝑖𝑦^∗_𝑖+𝑑_𝑖𝑥^∗_𝑖+𝑎_𝑖𝐹_𝑖𝑦^∗_𝑖𝑣1+𝛼_𝑖𝑣^∗𝑣^∗1+𝛼_𝑖𝑣−𝑎_𝑖𝐹²_𝑖𝑦^∗_𝑖^∞₀𝑓_𝑖(𝜏)𝑒^{−𝑚𝑖𝜏}𝑦^∗_𝑖𝑥_𝑖(𝑡−𝜏)𝑣(𝑡−𝜏)1+𝛼_𝑖𝑣^∗𝑦_𝑖𝑥^∗_𝑖𝑣^∗1+𝛼_𝑖𝑣()𝑎𝑡−𝜏𝑑𝜏+^𝑖𝐹_𝑖𝑦^∗_𝑖+𝑎_𝑖𝐹²_𝑖𝑦^∗_𝑖^∞₀𝑓_𝑖(𝜏)𝑒^{−𝑚𝑖𝜏}𝑥ln^𝑖(𝑡−𝜏)𝑣(𝑡−𝜏)1+𝛼_𝑖𝑣𝑥_i𝑣1+𝛼_𝑖𝑣()+𝑎𝑡−𝜏𝑑𝜏_𝑖𝐹_𝑖𝐺_𝑖𝑦^∗_𝑖^∞₀𝑔_𝑖(𝜏)𝑒^{−𝑛𝑖𝜏}𝑦ln^𝑖(𝑡−𝜏)𝑦_𝑖−𝑎𝑑𝜏_𝑖𝐹_𝑖𝑦^∗_𝑖𝑣𝑣^∗−𝑎_𝑖𝐹_𝑖𝐺_𝑖𝑦^∗_𝑖^∞₀𝑔_𝑖(𝜏)𝑒^{−𝑛𝑖𝜏}𝑣^∗𝑦_𝑖(𝑡−𝜏)𝑣𝑦^∗_𝑖𝑎𝑑𝜏+^𝑖𝐹_𝑖𝑦^∗_𝑖. (2.33) Then collecting terms of (2.33) and using the following equalities: 𝑥ln^𝑖(𝑡−𝜏)𝑣(𝑡−𝜏)1+𝛼_𝑖𝑣𝑥_𝑖𝑣1+𝛼_𝑖𝑣(𝑦𝑡−𝜏)=ln^∗𝑖𝑥_𝑖(𝑡−𝜏)𝑣(𝑡−𝜏)1+𝛼_𝑖𝑣^∗𝑦_𝑖𝑥^∗_𝑖𝑣^∗1+𝛼_𝑖𝑣(𝑥𝑡−𝜏)+ln^∗𝑖𝑥_𝑖𝑣+ln^∗𝑦_𝑖𝑣𝑦^∗_𝑖+ln1+𝛼_𝑖𝑣1+𝛼_𝑖𝑣^∗𝑦,𝑖=1,2,ln^𝑖(𝑡−𝜏)𝑦_𝑖=ln𝑣𝑦^∗_𝑖𝑣^∗𝑦_𝑖𝑣+ln^∗𝑦_𝑖(𝑡−𝜏)𝑣𝑦^∗_𝑖𝑣,𝑖=1,2ln^∗𝑦_𝑖𝑣𝑦^∗_𝑖+ln𝑣𝑦^∗_𝑖𝑣^∗𝑦_𝑖=ln(1)=0,𝑖=1,2 (2.34) we obtain 𝑑𝑊₂=𝑑𝑡²_𝑖=1𝛾_𝑖𝑑_𝑖𝑥^∗_𝑖𝑥2−^∗𝑖𝑥_𝑖−𝑥_𝑖𝑥^∗_𝑖+𝑎_𝑖𝐹_𝑖𝑦^∗_𝑖𝑥1−^∗𝑖𝑥_𝑖+2𝑎_𝑖𝐹_𝑖𝑦^∗_𝑖+𝑎_𝑖𝐹_𝑖𝑦^∗_𝑖𝑣1+𝛼_𝑖𝑣^∗𝑣^∗1+𝛼_𝑖𝑣−𝑣𝑣^∗−𝑎_𝑖𝐹²_𝑖𝑦^∗_𝑖^∞₀𝑓_𝑖(𝜏)𝑒^{−𝑚𝑖𝜏}𝑦^∗_𝑖𝑥_𝑖(𝑡−𝜏)𝑣(𝑡−𝜏)1+𝛼_𝑖𝑣^∗𝑦_𝑖𝑥^∗_𝑖𝑣^∗1+𝛼_𝑖𝑣()+𝑎𝑡−𝜏𝑑𝜏_𝑖𝐹²_𝑖𝑦^∗_𝑖^∞₀𝑓_𝑖(𝜏)𝑒^{−𝑚𝑖𝜏}×𝑦ln^∗𝑖𝑥_𝑖(𝑡−𝜏)𝑣(𝑡−𝜏)1+𝛼_𝑖𝑣^∗𝑦_𝑖𝑥^∗_𝑖𝑣^∗1+𝛼_𝑖𝑣(𝑥𝑡−𝜏)+ln^∗𝑖𝑥_𝑖𝑣+ln^∗𝑦_𝑖𝑣𝑦^∗_𝑖+ln1+𝛼_𝑖𝑣1+𝛼_𝑖𝑣^∗+𝑎𝑑𝜏_𝑖𝐹_𝑖𝐺_𝑖𝑦^∗_𝑖^∞₀𝑔_𝑖(𝜏)𝑒^{−𝑛𝑖𝜏}ln𝑣𝑦^∗_𝑖𝑣^∗𝑦_𝑖𝑣+ln^∗𝑦_𝑖(𝑡−𝜏)𝑣𝑦^∗_𝑖−𝑎𝑑𝜏_𝑖𝐹_𝑖𝐺_𝑖𝑦^∗_𝑖^∞₀𝑔_𝑖(𝜏)𝑒^{−𝑛𝑖𝜏}𝑣^∗𝑦_𝑖(𝑡−𝜏)𝑣𝑦^∗_𝑖𝑑𝜏. (2.35) Equation (2.35) can be rewritten as 𝑑𝑊₂=𝑑𝑡²_𝑖=1𝛾_𝑖𝑑_𝑖𝑥^∗_𝑖𝑥2−^∗𝑖𝑥_𝑖−𝑥_𝑖𝑥^∗_𝑖−𝑎_𝑖𝐹_𝑖𝑦^∗_𝑖𝑥^∗_𝑖𝑥_𝑖𝑥−1−ln^∗𝑖𝑥_𝑖+𝑎_𝑖𝐹_𝑖𝑦^∗_𝑖𝑣−1+1+𝛼_𝑖𝑣^∗𝑣^∗1+𝛼_𝑖𝑣−𝑣𝑣^∗+1+𝛼_𝑖𝑣1+𝛼_𝑖𝑣^∗−𝑎_𝑖𝐹_𝑖𝑦^∗_𝑖1+𝛼_𝑖𝑣1+𝛼_𝑖𝑣^∗−1−ln1+𝛼_𝑖𝑣1+𝛼_𝑖𝑣^∗−𝑎_𝑖𝐹²_𝑖𝑦^∗_𝑖^∞₀𝑓_𝑖(𝜏)𝑒^{−𝑚𝑖𝜏}×𝑦^∗_𝑖𝑥_𝑖(𝑡−𝜏)𝑣(𝑡−𝜏)1+𝛼_𝑖𝑣^∗𝑦_𝑖𝑥^∗_𝑖𝑣^∗1+𝛼_𝑖𝑣()𝑦𝑡−𝜏−1−ln^∗𝑖𝑥_𝑖(𝑡−𝜏)𝑣(𝑡−𝜏)1+𝛼_𝑖𝑣^∗𝑦_𝑖𝑥^∗_𝑖𝑣^∗1+𝛼_𝑖𝑣()−𝑎𝑡−𝜏𝑑𝜏_𝑖𝐹_𝑖𝐺_𝑖𝑦^∗_𝑖^∞₀𝑔_𝑖(𝜏)𝑒^{−𝑛𝑖𝜏}𝑣^∗𝑦_𝑖(𝑡−𝜏)𝑣𝑦^∗_𝑖𝑣−1−ln^∗𝑦_𝑖(𝑡−𝜏)𝑣𝑦^∗_𝑖𝑑𝜏. (2.36) Using the following equality: 𝑣−1+1+𝛼_𝑖𝑣^∗𝑣^∗1+𝛼_𝑖𝑣−𝑣𝑣^∗+1+𝛼_𝑖𝑣1+𝛼_𝑖𝑣^∗=−𝛼_𝑖𝑣−𝑣^∗2𝑣^∗1+𝛼_𝑖𝑣^∗1+𝛼_𝑖𝑣,𝑖=1,2, (2.37) we can rewrite 𝑑𝑊₂/𝑑𝑡 as 𝑑𝑊₂𝑑𝑡=−²_𝑖=1𝛾_𝑖𝑑_𝑖𝑥_𝑖−𝑥^∗_𝑖²𝑥_𝑖+𝑎_𝑖𝐹_𝑖𝑦^∗_𝑖𝛼_𝑖𝑣−𝑣^∗2𝑣^∗1+𝛼_𝑖𝑣^∗1+𝛼_𝑖𝑣+𝑎_𝑖𝐹_𝑖𝑦^∗_𝑖𝐻𝑥^∗_𝑖𝑥_𝑖+𝑎_𝑖𝐹_𝑖𝑦^∗_𝑖𝐻1+𝛼_𝑖𝑣1+𝛼_𝑖𝑣^∗+𝑎_𝑖𝑦^∗_𝑖𝐹²_𝑖^∞₀𝑓_𝑖(𝜏)𝑒^{−𝑚𝑖𝜏}𝐻𝑦^∗_𝑖𝑥_𝑖(𝑡−𝜏)𝑣(𝑡−𝜏)1+𝛼_𝑖𝑣^∗𝑦_𝑖𝑥^∗_𝑖𝑣^∗1+𝛼_𝑖𝑣()+𝑎𝑡−𝜏𝑑𝜏_𝑖𝑦^∗_𝑖𝐹_𝑖𝐺_𝑖^∞₀𝑔_𝑖(𝜏)𝑒^{−𝑛𝑖𝜏}𝐻𝑣^∗𝑦_𝑖(𝑡−𝜏)𝑣𝑦^∗_𝑖𝑑𝜏. (2.38) It is easy to see that if 𝑥^∗_𝑖,𝑦^∗_𝑖,𝑣^∗>0,  𝑖=1,2, then 𝑑𝑊₂/𝑑𝑡≤0. By Theorem 5.3.1 in [31], the solutions of system (2.1)–(2.5) limit to 𝑀, the largest invariant subset of {𝑑𝑊₂/𝑑𝑡=0}. It can be seen that 𝑑𝑊₂/𝑑𝑡=0 if and only if 𝑥_𝑖=𝑥^∗_𝑖,  𝑣=𝑣^∗, and 𝐻=0, that is, 𝑦^∗_𝑖𝑥_𝑖(𝑡−𝜏)𝑣(𝑡−𝜏)1+𝛼_𝑖𝑣^∗𝑦_𝑖𝑥^∗_𝑖𝑣^∗1+𝛼_𝑖𝑣(=𝑣𝑡−𝜏)^∗𝑦_𝑖(𝑡−𝜏)𝑣𝑦^∗_𝑖=1foralmostall𝜏∈(0,∞). (2.39) If 𝑣=𝑣^∗ then from (2.39) we have 𝑦_𝑖=𝑦^∗_𝑖, and hence 𝑑𝑊₂/𝑑𝑡 equal to zero at 𝐸₁. LaSalle's Invariance Principle implies global stability of 𝐸₁.

3. 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 saturation infection rate. Two types of distributed time delays describing time needed for infection of target cell and virus replication have been incorporated into the model. The global stability of the uninfected and infected steady states of the model has been established by using suitable Lyapunov functionals and LaSalle invariant principle. We have proven that, if the basic reproduction number 𝑅₀ is less than unity, then the uninfected steady state is GAS and if 𝑅₀>1, then the infected steady state exists and it is GAS.

Acknowledgment

This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The author, therefore, acknowledges with thanks DSR technical and financial support.