Global Dynamics of HIV Infection of CD4⁺ T Cells and Macrophages

Abstract

We study the global dynamics of an HIV infection model describing the interaction of the HIV with CD4⁺ T cells and macrophages. The incidence rate of virus infection and the growth rate of the uninfected CD4⁺ T cells and macrophages are given by general functions. We have incorporated two types of distributed delays into the model to account for the time delay between the time the uninfected cells are contacted by the virus particle and the time for the emission of infectious (matures) virus particles. We have established a set of conditions which are sufficient for the global stability of the steady states of the model. Using Lyapunov functionals and LaSalle's invariant principle, we have proven that if the basic reproduction number 𝑅₀ is less than or equal to unity, then the uninfected steady state is globally asymptotically stable (GAS), and if the infected steady state exists, then it is GAS.

1. Introduction

Recently, many mathematical models have been developed to describe the interaction of human immunodeficiency virus (HIV) with the immune system [1]. Those mathematical models can provide some insights into the dynamics of HIV viral load in vivo and may play a significant role in the development of a better understanding of HIV/AIDS and drug therapies. For example, they provided a quantitative understanding of the level of virus production during the long asymptomatic stage of HIV infection [2].

The basic mathematical model describing the dynamics of HIV infection of CD4⁺ T cells is given by [3] ̇̇̇𝑥=𝜆−𝑑𝑥−𝛽𝑥𝑣,𝑦=𝛽𝑥𝑣−𝑎𝑦,𝑣=𝑘𝑦−𝑐𝑣, (1)(2)(3) where 𝑥, 𝑦, and 𝑣 represent the populations of the uninfected CD4⁺ T cells, infected cells, and free virus particles, respectively. The uninfected cells are generated from sources within the body at a rate 𝜆. The parameter 𝑑 is the death rate constant of the uninfected cells. The incidence rate is given by the bilinear form 𝛽𝑥𝑣, where 𝛽 is the rate constant characterizing infections of the cells. Equation (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 a rate constant 𝑘 and are cleared from plasma with a rate constant 𝑐.

Many researchers suggested that the bilinear incidence rate is insufficient to describe the infection process in detail [4–8]. Therefore, different forms of the incidence rate of infection have been proposed such as saturated incidence rate 𝛽𝑥𝑣/(1+𝛾𝑣) [4], Holling type II functional response 𝛽𝑥𝑣/(1+𝛼𝑥) [5], Beddington-DeAngelis infection rate 𝛽𝑥𝑣/(1+𝛼𝑥+𝛾𝑣) [6], Crowley-Martin functional response 𝛽𝑥𝑣/((1+𝛼𝑥)(1+𝛾𝑣)) [7, 8], where 𝛼, 𝛾≥0, and a general function 𝜓(𝑥,𝑣) [9].

In model (1)–(3), it is assumed that the infection could occur and the viruses are produced from infected cells instantaneously, once the uninfected cells are contacted by the virus particles. However, this assumption is unrealistic. Therefore, more realistic HIV dynamics models incorporate the delay between the time of viral entry into the uninfected cell and the time for the production of new virus particles, modeled with discrete time delay or distributed time delay using functional differential equations (see, e.g., [2, 5, 10–18]). As pointed by Li and Shu [18], 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 [18].

All of the afore mentioned delayed HIV infection models are mainly modeled with the interaction of the HIV with one class of target cells, CD4⁺ T cells. More accurate modeling was developed in 1997 when Perelson et al. [19] observed that, after the rapid first phase of decay during the initial 1-2 weeks of antiretroviral treatment, plasma virus levels declined at a considerably slower rate. This second phase of viral decay was attributed to the turnover of a longer-lived virus reservoir of infected cells. These cells are called macrophages and considered as the second target cell for the HIV. To model the second class of target cells, two additional equations describing the population dynamics of the uninfected and infected macrophages have to be added to the basic model (1)–(3) (see [20, 21]). In [20–27], the HIV models have been proposed to describe the HIV dynamics with the CD4⁺ T cells and macrophages. The global stability analysis of these models has been investigated in [24–27]. Elaiw [24] studied the global properties of HIV infection model with bilinear and nonlinear incidence rates. In [25], Beddington-DeAngelis functional response has been considered. In [24, 25], the effect of time delay is neglected. Elaiw et al. [26] studied the global stability of HIV model with Beddington-DeAngelis functional response and one kind of discrete time delay. Elaiw [27] studied the global dynamics of a delay HIV model with saturated functional response.

In this paper, we study the global dynamics of HIV infection of CD4⁺ T cells and macrophages. We assume that the incidence rate and the growth rate of the uninfected cells are given by general functions. We incorporate two types of distributed delays into the model to account for the time delay between the time the uninfected cells are contacted by the virus particle and the time for the emission of infectious (mature) virus particles. We established a set of conditions which are sufficient for the global stability of the steady states of the model. Using Lyapunov functionals and LaSalle’s invariant principle, we prove that if the basic reproduction number 𝑅₀ is less than or equal to unity, then the uninfected steady state is globally asymptotically stable (GAS), and if the infected steady state exists, then it is GAS.

2. The Model

In this sections we propose an HIV dynamics model which describes the interaction of the HIV with the CD4⁺ T cells and macrophages. Two types of distributed time delays are incorporated into the model 𝑥̇_𝑖=𝑟_𝑖𝑥_𝑖−𝜓_𝑖𝑥_𝑖𝑦̇,𝑣,𝑖=1,2,_𝑖=𝛽^′_𝑖^ℎ𝑖₀𝑓_𝑖(𝜏)𝜓_𝑖𝑥_𝑖(𝑡−𝜏),𝑣(𝑡−𝜏)𝑑𝜏−𝑎_𝑖𝑦_𝑖̇,𝑖=1,2,𝑣=²_𝑖=1𝑝_𝑖^𝑙𝑖₀𝑔_𝑖(𝜔)𝑦_𝑖(𝑡−𝜔)𝑑𝜔−𝑐𝑣, (4)(5)(6) where 𝑥_𝑖 and 𝑦_𝑖 represent the populations of the uninfected and infected cells, respectively, where 𝑖=1 and 2 correspond to CD4⁺ T cells and macrophages. Parameters 𝛽^′_𝑖 and 𝑝_𝑖 satisfy 𝛽^′_𝑖<1 and 𝑝_𝑖<𝑘_𝑖, 𝑖=1,2. All the variables and other parameters of the model have the same meanings as given in model (1)–(3). To take into account the delay between viral infection of an uninfected target cell and the production of an actively infected target cell, we let 𝜏 be the random variable that describes the time between viral entry and the transcription of viral RNA (stages (i) and (ii)) with a probability distribution 𝑓_𝑖(𝜏) over the interval [0,ℎ_𝑖] and ℎ_𝑖 is limit superior of this delay. On the other hand, to consider the delay between viral RNA transcription and viral release and maturation, we let 𝜔 be the random variable that is the time between these two events with a probability distribution 𝑔_𝑖(𝜔) over the interval [0,𝑙_𝑖] and 𝑙_𝑖 is limit superior of this delay [18]. The probability distribution functions 𝑓_𝑖(𝜏)∶[0,ℎ_𝑖]→ℝ₊ and 𝑔_𝑖(𝜔)∶[0,𝑙_𝑖]→ℝ₊ are integral functions with ^ℎ𝑖₀𝑓_𝑖(𝜏)𝑑𝜏=^𝑙𝑖₀𝑔_𝑖(𝜔)𝑑𝜔=1,𝑖=1,2. (7) Function 𝜓_𝑖(𝑥_𝑖,𝑣) represents the rate for the uninfected target cell of class 𝑖 to be infected by the mature viruses. Special forms of function 𝜓_𝑖(𝑥_𝑖,𝑣) have been presented in the literature as follows:(i)bilinear incidence rate [3]: 𝜓(𝑥,𝑣)=𝛽𝑥𝑣,(ii)saturated incidence rate [4, 28, 29]: 𝜓(𝑥,𝑣)=𝛽𝑥𝑣/(1+𝛾𝑣),(iii)Holling type II functional response [5]: 𝜓(𝑥,𝑣)=𝛽𝑥𝑣/(1+𝛼𝑥),(iv)Beddington-DeAngelis infection rate [6, 25]: 𝜓(𝑥,𝑣)=𝛽𝑥𝑣/(1+𝛼𝑥+𝛾𝑣),(v)Crowley-Martin functional response [7, 8, 30]: 𝜓(𝑥,𝑣)=𝛽𝑥𝑣/((1+𝛼𝑥)(1+𝛾𝑣)).

The growth rate of the uninfected cells is given by general function 𝑟_𝑖(𝑥_𝑖). The following particular forms of function 𝑟_𝑖(𝑥_𝑖) have widely been used in the literature of HIV dynamics: 𝑟_𝑖𝑥_𝑖=𝜆_𝑖−𝑑𝑥_𝑖,𝑟_𝑖̃𝑥_𝑖=𝜆_𝑖−𝑑𝑥_𝑖+𝑏_𝑖𝑥_𝑖𝑥1−^𝑖𝑥_𝑖,max, (8) where 𝑏_𝑖 is the maximum proliferation rate of the target cells of class 𝑖 and 𝑥_𝑖,max is the maximum level of uninfected cells population in the body [13, 16, 31].

Initial Conditions. The initial conditions for system (4)–(6) take the form 𝑥₁(𝜃)=𝜑₁(𝜃),𝑥₂(𝜃)=𝜑₂(𝜃),𝑦₁(𝜃)=𝜑₃(𝜃),𝑦₂(𝜃)=𝜑₄(𝜃),𝑣(𝜃)=𝜑₅(𝜃),𝜑_𝑗(𝜃)[)≥0,𝜃∈−ℓ,0,𝜑_𝑗(0)>0,𝑗=1,…,5,ℓ=maxℎ₁,ℎ₂,𝑙₁,𝑙₂, (9) where (𝜑₁(𝜃),𝜑₂(𝜃),…,𝜑₅(𝜃))∈𝐶 and 𝐶=𝐶([−ℓ,0],ℝ⁵₊) is the Banach space of continuous functions mapping the interval [−ℓ,0] into ℝ⁵₊. By the fundamental theory of functional differential equations [32], system (4)–(6) has a unique solution (𝑥₁(𝑡),𝑥₂(𝑡),𝑦₁(𝑡),𝑦₂(𝑡),𝑣(𝑡)) satisfying initial conditions (9).

We assume that functions 𝑟_𝑖 and 𝜓_𝑖 satisfy the following assumptions.

Assumption 1. Function 𝑟_𝑖∶[0,∞)→ℝ satisfies the following:(i)𝑟_𝑖(𝑥_𝑖) is continuous and differentiable and 𝑟_𝑖(0)>0,(ii)there exits an 𝑥⁰_𝑖>0 such that 𝑟_𝑖𝑥⁰_𝑖=0,𝑟^′_𝑖𝑥⁰_𝑖𝑥<0,_𝑖−𝑥⁰_𝑖𝑟_𝑖𝑥_𝑖≤0,𝑥_𝑖≠𝑥⁰_𝑖. (10)

Assumption 2. (i) 𝜓_𝑖(𝑥_𝑖,𝑣) is positive, continuous, and differentiable for all 𝑥_𝑖>0, and 𝑣>0,
(ii) 𝜓_𝑖(0,𝑣)=𝜓_𝑖(𝑥_𝑖,0)=0, 𝜕𝜓_𝑖(𝑥_𝑖,0)/𝜕𝑣>0 for any 𝑥>0, 𝑣>0, 𝑖=1,2,
(iii) 𝜕𝜓_𝑖(𝑥_𝑖,𝑣)/𝜕𝑥_𝑖>0, 𝜕𝜓_𝑖(𝑥_𝑖,𝑣)/𝜕𝑣>0 for all 𝑥_𝑖>0, 𝑣>0, 𝑖=1,2.

2.1. Nonnegativity and Boundedness of Solutions

In the following, we establish the nonnegativity and boundedness of solutions of (4)–(6) with initial conditions (9).

Proposition 3. Assume that Assumptions 1 and 2 are satisfied. Let 𝑋(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),𝑦₁(𝑡),𝑦₂(𝑡),𝑣(𝑡))^𝑇 be any solution of (4)–(6) satisfying the initial conditions (9); then 𝑋(𝑡) is nonnegative for 𝑡≥0 and ultimately bounded.

Proof. First, we prove that 𝑥_𝑖(𝑡)>0, 𝑖=1,2, for all 𝑡≥0. Assume that 𝑥_𝑖(𝑡) loses its nonnegativity on some local existence interval [0,𝜌] for some constant 𝜌 and let 𝑡^∗_𝑖∈[0,𝜌] be such that 𝑥_𝑖(𝑡^∗_𝑖)=0, 𝑖=1,2. From (4), we have 𝑥̇_𝑖𝑡^∗_𝑖=𝑟_𝑖𝑥_𝑖𝑡^∗_𝑖−𝜓_𝑖𝑥_𝑖𝑡^∗_𝑖,𝑣𝑡^∗_𝑖=𝑟_𝑖(0)−𝜓_𝑖0,𝑣𝑡^∗_𝑖=𝑟_𝑖(0)>0,𝑖=1,2. (11) Hence, 𝑥_𝑖(𝑡)>0 for some 𝑡∈(𝑡^∗_𝑖,𝑡^∗_𝑖+𝜀), where 𝜀>0 is sufficiently small. This leads to a contradiction and hence 𝑥_𝑖(𝑡)>0 for all 𝑡≥0. Further, from (5) and (6), we have 𝑦_𝑖(𝑡)=𝑦_𝑖(0)𝑒^{−𝑎𝑖𝑡}+𝛽^′_𝑖^𝑡₀𝑒^{−𝑎𝑖(𝑡−𝜂)}×^ℎ𝑖₀𝑓_𝑖(𝜏)𝜓_𝑖𝑥_𝑖𝑣(𝑡)(0)𝑒𝜂−𝜏,𝑣𝜂−𝜏𝑑𝜏𝑑𝜂,𝑖=1,2,=𝑣^−𝑐𝑡+²_𝑖=1𝑝_𝑖^𝑡₀𝑒^{−𝑐(𝑡−𝜂)𝑙𝑖}₀𝑔_𝑖(𝜔)𝑦_𝑖𝜂−𝜔𝑑𝜔𝑑𝜂, (12) confirming that 𝑦_𝑖(𝑡)≥0, 𝑖=1,2, and 𝑣(𝑡)≥0 for all 𝑡∈[0,ℓ]. By a recursive argument, we obtain 𝑦_𝑖(𝑡)≥0, 𝑖=1,2, and 𝑣(𝑡)≥0 for all 𝑡≥0.
Now, we show the boundedness of the solutions of (4)–(6). Assumption 1 and (4) imply that limsup_𝑡→∞𝑥_𝑖(𝑡)≤𝑥⁰_𝑖. It follows that ^ℎ𝑖₀𝑓_𝑖(𝜏)𝑥_𝑖(𝑡−𝜏)𝑑𝜏≤𝑥⁰_𝑖. (13) Let 𝑇_𝑖(𝑡)=𝛽^′_𝑖∫^ℎ𝑖₀𝑓_𝑖(𝜏)𝑥_𝑖(𝑡−𝜏)𝑑𝜏+𝑦_𝑖(𝑡), 𝑆_𝑖=sup_{𝑥𝑖∈[0,𝑥0𝑖]}𝑟_𝑖(𝑥_𝑖) and 𝑎_𝑖≤min{𝑎_𝑖,𝑆_𝑖/𝑥⁰_𝑖}, 𝑖=1,2; then 𝑇̇_𝑖(𝑡)=^ℎ𝑖₀𝛽^′_𝑖𝑓_𝑖(𝜏)𝑟_𝑖𝑥_𝑖(𝑡−𝜏)−𝜓_𝑖𝑥_𝑖(𝑡−𝜏),𝑣(𝑡−𝜏)𝑑𝜏+𝛽^′ℎ𝑖₀𝑓_𝑖(𝜏)𝜓_𝑖𝑥_𝑖(𝑡−𝜏),𝑣(𝑡−𝜏)𝑑𝜏−𝑎_𝑖𝑦_𝑖(𝑡)≤𝛽^′_𝑖𝑆_𝑖−𝑎_𝑖𝑦_𝑖(𝑡)≤𝛽^′_𝑖𝑆_𝑖−𝑎_𝑖𝑦_𝑖(𝑡)+𝛽^′_𝑖𝑆_𝑖−𝑎_𝑖𝛽^′_𝑖^ℎ𝑖₀𝑓_𝑖(𝜏)𝑥_𝑖(𝑡−𝜏)𝑑𝜏≤2𝛽^′_𝑖𝑆_𝑖−𝑎_𝑖𝑇_𝑖(𝑡). (14) Hence, limsup_𝑡→∞𝑇_𝑖(𝑡)≤𝐿_𝑖, where 𝐿_𝑖=2𝛽^′_𝑖𝑆_𝑖/𝑎_𝑖. Since ∫^ℎ𝑖₀𝑓_𝑖(𝜏)𝑥_𝑖(𝑡−𝜏)𝑑𝜏>0, we get limsup_𝑡→∞𝑦_𝑖(𝑡)≤𝐿_𝑖. On the other hand, 𝑣̇(𝑡)≤²_𝑖=1𝑝_𝑖𝐿_𝑖^𝑙𝑖₀𝑔_𝑖(𝜏)=𝑑𝜏−𝑐𝑣_2𝑖=1𝑝_𝑖𝐿_𝑖−𝑐𝑣. (15) Then limsup_𝑡→∞𝑣(𝑡)≤𝐿^∗, where 𝐿^∗=∑²_𝑖=1(𝑝_𝑖𝐿_𝑖/𝑐). Therefore, 𝑋(𝑡) is ultimately bounded.

2.2. Steady States

Let Assumptions 1 and 2 be satisfied; then system (4)–(6) has an uninfected steady state 𝐸₀(𝑥⁰₁,𝑥⁰₂,𝑦⁰₁,𝑦⁰₂,𝑣⁰), where 𝑥⁰_𝑖 is defined in Assumption 1, 𝑦⁰_𝑖=0, 𝑖=1,2, and 𝑣⁰=0. The system can also have another steady state 𝐸_∗(𝑥^∗₁,𝑥^∗₂,𝑦^∗₁,𝑦^∗₂,𝑣^∗) which is the infected steady state; the coordinates of the infected steady state, if they exist, satisfy the equalities 𝑟_𝑖𝑥^∗_𝑖=𝜓_𝑖𝑥^∗_𝑖,𝑣^∗𝑎,𝑖=1,2,_𝑖𝑦^∗_𝑖=𝛽^′_𝑖𝜓_𝑖𝑥^∗_𝑖,𝑣^∗,𝑖=1,2,𝑐𝑣^∗=²_𝑖=1𝑝_𝑖𝑦^∗_𝑖. (16) We define the basic reproduction number for system (4)–(6) as 𝑅₀=²_𝑖=1𝑅_𝑖=²_𝑖=1𝑝_𝑖𝛽^′_𝑖𝑎_𝑖𝑐𝜕𝜓_𝑖𝑥⁰_𝑖,0.𝜕𝑣 (17) The term 𝜕𝜓_𝑖(𝑥⁰_𝑖,0)/𝜕𝑣 denotes the maximal average number of target cells 𝑖 that each virus infects, and 𝑅_𝑖 is the basic reproduction number for the dynamics of the virus and the uninfected cell of class 𝑖.

2.3. Global Stability Analysis

In this section, we establish a set of conditions which are sufficient for the global stability of the uninfected and infected steady states of system (4)–(6). The strategy of the proofs is to use suitable Lyapunov functionals which are similar in nature to those used in [33, 34]. We will use the following notation: 𝑧=𝑧(𝑡) for any 𝑧∈{𝑥_𝑖,𝑦_𝑖,𝑣,𝑖=1,2}. We also define a function 𝐻∶(0,∞)→[0,∞) as 𝐻(𝑧)=𝑧−1−ln𝑧. (18) It is clear that 𝐻(𝑧)≥0 for any 𝑧>0 and 𝐻 has the global minimum 𝐻(1)=0.

Assumption 4. 𝜓_𝑖(𝑥_𝑖,𝑣)≤𝑣(𝜕𝜓_𝑖(𝑥_𝑖,0)/𝜕𝑣), for all 𝑣>0.

Assumption 5. (1−(𝜕𝜓_𝑖(𝑥⁰_𝑖,0)/𝜕𝑣)/(𝜕𝜓_𝑖(𝑥_𝑖,0)/𝜕𝑣))𝑟_𝑖(𝑥_𝑖)≤0, 𝑥_𝑖>0.

The global stability of the uninfected steady state will be established in the next theorem.

Theorem 6. If Assumptions 1–5 hold true and 𝑅₀≤1, then 𝐸₀ is GAS.

Proof. Define a Lyapunov functional 𝑊₀ as follows: 𝑊₀=²_𝑖=1𝑝_𝑖𝑎_𝑖𝛽^′_𝑖𝑥_𝑖−𝑥⁰_𝑖−^𝑥𝑖_𝑥0𝑖lim_𝑣→0+𝜓_𝑖𝑥⁰_𝑖,𝑣𝜓_𝑖(𝜃,𝑣)𝑑𝜃+𝑦_𝑖+𝛽^′_𝑖^ℎ𝑖₀𝛿_{𝑓𝑖,ℎ𝑖}(𝜏)𝜓_𝑖𝑥_𝑖(𝑡−𝜏),𝑣(𝑡−𝜏)𝑑𝜏+𝑎_𝑖^𝑙𝑖₀𝛿_{𝑔𝑖,𝑙𝑖}(𝜔)𝑦_𝑖(𝑡−𝜔)𝑑𝜔+𝑣, (19) where 𝛿_{𝑓𝑖,ℎ𝑖}(𝜏)=^ℎ𝑖_𝜏𝑓_𝑖(𝜎)𝛿𝑑𝜎,_{𝑔𝑖,𝑙𝑖}(𝜔)=^𝑙𝑖_𝜔𝑔_𝑖(𝜎)𝑑𝜎,𝑖=1,2. (20) Functions 𝛿_{𝑓𝑖,ℎ𝑖} and 𝛿_{𝑔𝑖,𝑙𝑖} satisfy 𝛿_{𝑓𝑖,ℎ𝑖}(0)=1,𝛿_{𝑓𝑖,ℎ𝑖}ℎ_𝑖=0,𝑑𝛿_{𝑓𝑖,ℎ𝑖}(𝜏)𝑑𝜏=−𝑓_𝑖(𝜏),𝛿_{𝑔𝑖,𝑙𝑖}(0)=1,𝛿_{𝑔𝑖,𝑙𝑖}𝑙_𝑖=0,𝑑𝛿_{𝑔𝑖,𝑙𝑖}(𝜔)𝑑𝜔=−𝑔_𝑖(𝜔). (21) We note that 𝑊₀ is defined and continuous for all (𝑥₁,𝑥₂,𝑦₁,𝑦₂,𝑣)>0. Also, the global minimum 𝑊₀=0 occurs at the uninfected steady state 𝐸₀. The time derivative of 𝑊₀ along the solution of (4)–(6) is given by 𝑑𝑊₀=𝑑𝑡_2𝑖=1𝑝_𝑖𝑎_𝑖𝛽^′_𝑖1−lim_𝑣→0+𝜓_𝑖𝑥⁰_𝑖,𝑣𝜓_𝑖𝑥_𝑖𝑟,𝑣^𝑖𝑥_𝑖−𝜓_𝑖𝑥_𝑖,𝑣+𝛽^′_𝑖^ℎ𝑖₀𝑓_𝑖(𝜏)𝜓_𝑖𝑥_𝑖(𝑡−𝜏),𝑣(𝑡−𝜏)𝑑𝜏−𝑎_𝑖𝑦_𝑖+𝛽^′_𝑖^ℎ𝑖₀𝑓_𝑖(𝜏)𝜓_𝑖𝑥_𝑖,𝑣−𝜓_𝑖𝑥_𝑖(𝑡−𝜏),𝑣(𝑡−𝜏)𝑑𝜏+𝑎_𝑖^𝑙𝑖₀𝑔_𝑖(𝜔)𝑦_𝑖−𝑦_𝑖()+𝑡−𝜔𝑑𝜔_2𝑖=1𝑝_𝑖^𝑙𝑖₀𝑔_𝑖(𝜔)𝑦_𝑖()=𝑡−𝜔𝑑𝜔−𝑐𝑣_2𝑖=1𝑝_𝑖𝑎_𝑖𝛽^′_𝑖1−lim_𝑣→0+𝜓_𝑖𝑥⁰_𝑖,𝑣𝜓_𝑖𝑥_𝑖𝑟,𝑣^𝑖𝑥_𝑖+𝛽^′_𝑖𝜓_𝑖𝑥_𝑖,𝑣lim_𝑣→0+𝜓_𝑖𝑥⁰_𝑖,𝑣𝜓_𝑖𝑥_𝑖=,𝑣−𝑐𝑣_2𝑖=1𝑝_𝑖𝑎_𝑖𝛽^′_𝑖1−𝜕𝜓_𝑖𝑥⁰_𝑖,0/𝜕𝑣𝜕𝜓_𝑖𝑥_𝑖𝑟,0/𝜕𝑣^𝑖𝑥_𝑖+𝛽^′_𝑖𝜓_𝑖𝑥_𝑖,𝑣𝜕𝜓_𝑖𝑥⁰_𝑖,0/𝜕𝑣𝜕𝜓_𝑖𝑥_𝑖≤,0/𝜕𝑣−𝑐𝑣_2𝑖=1𝑝_𝑖𝑎_𝑖𝛽^′_𝑖1−𝜕𝜓_𝑖𝑥⁰_𝑖,0/𝜕𝑣𝜕𝜓_𝑖𝑥_𝑖𝑟,0/𝜕𝑣^𝑖𝑥_𝑖+𝛽^′_𝑖𝑣𝜕𝜓_𝑖𝑥_𝑖,0𝜕𝑣𝜕𝜓_𝑖𝑥⁰_𝑖,0/𝜕𝑣𝜕𝜓_𝑖𝑥_𝑖=,0/𝜕𝑣−𝑐𝑣_2𝑖=1𝑝_𝑖𝑎_𝑖𝛽^′_𝑖1−𝜕𝜓_𝑖𝑥⁰_𝑖,0/𝜕𝑣𝜕𝜓_𝑖𝑥_𝑖𝑟,0/𝜕𝑣^𝑖𝑥_𝑖+²_𝑖=1𝑝_𝑖𝛽^′_𝑖𝑎_𝑖𝑐𝜕𝜓_𝑖𝑥⁰_𝑖,0=𝜕𝑣−1𝑐𝑣_2𝑖=1𝑝_𝑖𝑎_𝑖𝛽^′_𝑖1−𝜕𝜓_𝑖𝑥⁰_𝑖,0/𝜕𝑣𝜕𝜓_𝑖𝑥_𝑖𝑟,0/𝜕𝑣^𝑖𝑥_𝑖+𝑅₀−1𝑐𝑣. (22) The first term of (22) is less than or equal to zero according to Assumption 5, so it can be seen that if 𝑅₀≤1, then 𝑑𝑊₀/𝑑𝑡≤0 for all 𝑥_𝑖,𝑣>0, 𝑖=1,2. By Theorem 5.3.1 in [32], the solutions of system (4)–(6) limit to 𝑀, the largest invariant subset of {𝑑𝑊₀/𝑑𝑡=0}. Clearly, it follows from (22) that 𝑑𝑊₀/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, and then 𝑣̇=0. From (6), we derive that 0=𝑣̇=²_𝑖=1𝑝_𝑖^𝑙𝑖₀𝑔(𝜔)𝑦_𝑖(𝑡−𝜔)𝑑𝜔. (23) Since 𝑦_𝑖(𝑡−𝜃)≥0 for all 𝜃∈[0,ℓ], then ∑²_𝑖=1𝑝_𝑖∫^𝑙𝑖₀𝑔(𝜔)𝑦_𝑖(𝑡−𝜔)𝑑𝜔=0 if and only if 𝑦_𝑖(𝑡−𝜔)=0, 𝑖=1,2. Hence, 𝑑𝑊₀/𝑑𝑡=0 if and only if 𝑥_𝑖=𝑥⁰_𝑖, 𝑦_𝑖=0, 𝑖=1,2, and 𝑣=0. From LaSalle’s invariance principle, 𝐸₀ is GAS.

To establish the global stability of the infected steady state, we need the following conditions.

Assumption 7. (1−(𝜓_𝑖(𝑥^∗_𝑖,𝑣^∗)/𝜓_𝑖(𝑥_𝑖,𝑣^∗)))((𝑟_𝑖(𝑥_𝑖)/𝑟_𝑖(𝑥^∗_𝑖))−1)≤0.

Assumption 8. ((𝜓_𝑖(𝑥_𝑖,𝑣)/𝜓_𝑖(𝑥_𝑖,𝑣^∗))−(𝑣/𝑣^∗))(1−(𝜓_𝑖(𝑥_𝑖,𝑣^∗)/𝜓_𝑖(𝑥_𝑖,𝑣)))≤0.

Theorem 9. Assume that Assumptions 1, 2, 7, and 8 hold true and 𝐸_∗ exists; then 𝐸_∗ is GAS.

Proof. Define Lyapunov functional 𝑊₁ as follows: 𝑊₁=²_𝑖=1𝑝_𝑖𝑎_𝑖𝛽^′_𝑖𝑥_𝑖−𝑥^∗_𝑖−^𝑥𝑖_𝑥∗𝑖𝜓_𝑖𝑥^∗_𝑖,𝑣^∗𝜓_𝑖(𝜃,𝑣^∗)𝑑𝜃+𝑦^∗_𝑖𝑦_𝑖𝑦^∗_𝑖𝑦−1−ln^𝑖𝑦^∗_𝑖+𝛽^′_𝑖𝜓_𝑖𝑥^∗_𝑖,𝑣^∗×^ℎ𝑖₀𝛿_{𝑓𝑖,ℎ𝑖}(𝜏)𝐻𝜓_𝑖𝑥_𝑖(𝑡−𝜏),𝑣(𝑡−𝜏)𝜓_𝑖𝑥^∗_𝑖,𝑣^∗𝑑𝜏+𝑎_𝑖𝑦^∗_𝑖^𝑙𝑖₀𝛿_{𝑔𝑖,𝑙𝑖}(𝜔)𝐻𝑦_𝑖(𝑡−𝜔)𝑦^∗_𝑖𝑑𝜔+𝑣^∗𝑣𝑣^∗𝑣−1−ln𝑣^∗. (24) By calculating the time derivative along (4)–(6), we get 𝑑𝑊₁=𝑑𝑡_2𝑖=1𝑝_𝑖𝑎_𝑖𝛽^′_𝑖𝜓1−^𝑖𝑥^∗_𝑖,𝑣^∗𝜓_𝑖𝑥_𝑖,𝑣^∗𝑟_𝑖𝑥_𝑖−𝜓_𝑖𝑥_𝑖+𝑦,𝑣1−^∗𝑖𝑦_𝑖×𝛽^′_𝑖^ℎ𝑖₀𝑓_𝑖(𝜏)𝜓_𝑖𝑥_𝑖(𝑡−𝜏),𝑣(𝑡−𝜏)𝑑𝜏−𝑎_𝑖𝑦_𝑖+𝛽^′_𝑖𝜓_𝑖𝑥^∗_𝑖,𝑣^∗×^ℎ𝑖₀𝑓_𝑖(𝜏)𝜓_𝑖𝑥_𝑖,𝑣𝜓_𝑖𝑥^∗_𝑖,𝑣^∗−𝜓_𝑖𝑥_𝑖(𝑡−𝜏),𝑣(𝑡−𝜏)𝜓_𝑖𝑥^∗_𝑖,𝑣^∗𝜓+ln^𝑖𝑥_𝑖(𝑡−𝜏),𝑣(𝑡−𝜏)𝜓_𝑖𝑥_𝑖,𝑣𝑑𝜏+𝑎_𝑖𝑦^∗_𝑖^𝑙𝑖₀𝑔_𝑖(𝜔)𝑦_𝑖𝑦^∗_𝑖−𝑦_𝑖(𝑡−𝜔)𝑦^∗_𝑖𝑦+ln^𝑖(𝑡−𝜔)𝑦_𝑖+𝑣𝑑𝜔1−^∗𝑣²_𝑖=1𝑝_𝑖^𝑙𝑖₀𝑔_𝑖(𝜔)𝑦_𝑖(𝑡−𝜔)𝑑𝜔−𝑐𝑣. (25) Collecting terms of (25), we obtain 𝑑𝑊₁=𝑑𝑡²_𝑖=1𝑝_𝑖𝑎_𝑖×𝛽^′_𝑖𝜓1−^𝑖𝑥^∗_𝑖,𝑣^∗𝜓_𝑖𝑥_𝑖,𝑣^∗𝑟_𝑖𝑥_𝑖−𝑟_𝑖𝑥^∗_𝑖+𝛽^′_𝑖𝑟_𝑖𝑥^∗_𝑖𝜓1−^𝑖𝑥^∗_𝑖,𝑣^∗𝜓_𝑖𝑥_𝑖,𝑣^∗−𝛽^′_𝑖𝜓_𝑖𝑥_𝑖,𝑣+𝛽^′_𝑖𝜓_𝑖𝑥^∗_𝑖,𝑣^∗𝜓_𝑖𝑥_𝑖,𝑣𝜓_𝑖𝑥_𝑖,𝑣^∗+𝛽^′_𝑖^ℎ𝑖₀𝑓_𝑖(𝜏)𝜓_𝑖𝑥_𝑖(𝑡−𝜏),𝑣(𝑡−𝜏)𝑑𝜏−𝑎_𝑖𝑦_𝑖−𝑦^∗_𝑖𝑦_𝑖𝛽^′_𝑖^ℎ𝑖₀𝑓_𝑖(𝜏)𝜓_𝑖𝑥_𝑖(𝑡−𝜏),𝑣(𝑡−𝜏)𝑑𝜏+𝑎_𝑖𝑦^∗_𝑖+𝛽^′_𝑖𝜓_𝑖𝑥_𝑖,𝑣−𝛽^′_𝑖^ℎ𝑖₀𝑓_𝑖(𝜏)𝜓_𝑖𝑥_𝑖(𝑡−𝜏),𝑣(𝑡−𝜏)𝑑𝜏+𝛽^′_𝑖𝜓_𝑖𝑥^∗_𝑖,𝑣^∗×^ℎ𝑖₀𝑓_𝑖(𝜏)𝜓ln^𝑖𝑥_𝑖(𝑡−𝜏),𝑣(𝑡−𝜏)𝜓_𝑖𝑥_𝑖,𝑣𝑑𝜏+𝑎_𝑖𝑦_𝑖−𝑎_𝑖^𝑙𝑖₀𝑔_𝑖(𝜔)𝑦_𝑖(𝑡−𝜔)𝑑𝜔+𝑎_𝑖𝑦^∗_𝑖^𝑙𝑖₀𝑔_𝑖(𝜔)𝑦ln^𝑖(𝑡−𝜔)𝑦_𝑖+𝑑𝜔_2𝑖=1𝑝_𝑖^𝑙𝑖₀𝑔_𝑖(𝜔)𝑦_𝑖()−𝑡−𝜔𝑑𝜔−𝑐𝑣_2𝑖=1𝑝_𝑖^𝑙𝑖₀𝑔_𝑖(𝜔)𝑣^∗𝑦_𝑖(𝑡−𝜔)𝑣𝑑𝜔+𝑐𝑣^∗. (26) Using the infected steady state conditions (16), we get 𝑑𝑊₁=𝑑𝑡_2𝑖=1𝑝_𝑖𝑎_𝑖𝛽^′_𝑖𝜓1−^𝑖𝑥^∗_𝑖,𝑣^∗𝜓_𝑖𝑥_𝑖,𝑣^∗𝑟_𝑖𝑥_𝑖−𝑟_𝑖𝑥^∗_𝑖+𝑎_𝑖𝑦^∗_𝑖𝜓1−^𝑖𝑥^∗_𝑖,𝑣^∗𝜓_𝑖𝑥_𝑖,𝑣^∗+𝑎_𝑖𝑦^∗_𝑖𝜓_𝑖𝑥_𝑖,𝑣𝜓_𝑖𝑥_𝑖,𝑣^∗−𝑎_𝑖𝑦^∗_𝑖^ℎ𝑖₀𝑓_𝑖(𝜏)𝑦^∗_𝑖𝜓_𝑖𝑥_𝑖(𝑡−𝜏),𝑣(𝑡−𝜏)𝑦_𝑖𝜓_𝑖𝑥^∗_𝑖,𝑣^∗𝑑𝜏+2𝑎_𝑖𝑦^∗_𝑖+𝑎_𝑖𝑦^∗_𝑖×^ℎ𝑖₀𝑓_𝑖(𝜏)𝜓ln^𝑖𝑥_𝑖(𝑡−𝜏),𝑣(𝑡−𝜏)𝜓_𝑖𝑥_𝑖,𝑣𝑑𝜏−𝑎_𝑖𝑦^∗_𝑖^𝑙𝑖₀𝑔_𝑖(𝜔)𝑣^∗𝑦_𝑖(𝑡−𝜔)𝑣𝑦^∗_𝑖𝑑𝜏+𝑎_𝑖𝑦^∗_𝑖^𝑙𝑖₀𝑔_𝑖(𝜔)𝑦ln^𝑖(𝑡−𝜔)𝑦_𝑖𝑑𝜔−𝑎_𝑖𝑦^∗_𝑖𝑣𝑣^∗=²_𝑖=1𝑝_𝑖𝑎_𝑖𝛽^′_𝑖𝜓1−^𝑖𝑥^∗_𝑖,𝑣^∗𝜓_𝑖𝑥_𝑖,𝑣^∗𝑟_𝑖𝑥_𝑖−𝑟_𝑖𝑥^∗_𝑖+𝑎_𝑖𝑦^∗_𝑖×𝜓−1+^𝑖𝑥_𝑖,𝑣𝜓_𝑖𝑥_𝑖,𝑣^∗−𝑣𝑣^∗+𝑣𝑣^∗𝜓_𝑖𝑥_𝑖,𝑣^∗𝜓_𝑖𝑥_𝑖,𝑣−𝑎_𝑖𝑦^∗_𝑖𝜓_𝑖𝑥^∗_𝑖,𝑣^∗𝜓_𝑖𝑥_𝑖,𝑣^∗−𝑎_𝑖𝑦^∗_𝑖𝑣𝑣^∗𝜓_𝑖𝑥_𝑖,𝑣^∗𝜓_𝑖𝑥_𝑖,𝑣−𝑎_𝑖𝑦^∗_𝑖^ℎ𝑖₀𝑓_𝑖(𝜏)𝑦^∗_𝑖𝜓_𝑖𝑥_𝑖(𝑡−𝜏),𝑣(𝑡−𝜏)𝑦_𝑖𝜓_𝑖𝑥^∗_𝑖,𝑣^∗𝑑𝜏+4𝑎_𝑖𝑦^∗_𝑖+𝑎_𝑖𝑦^∗_𝑖×^ℎ𝑖₀𝑓_𝑖(𝜏)𝜓ln^𝑖𝑥_𝑖(𝑡−𝜏),𝑣(𝑡−𝜏)𝜓_𝑖𝑥_𝑖,𝑣𝑑𝜏−𝑎_𝑖𝑦^∗_𝑖^𝑙𝑖₀𝑔_𝑖(𝜔)𝑣^∗𝑦_𝑖(𝑡−𝜔)𝑣𝑦^∗_𝑖𝑑𝜏+𝑎_𝑖𝑦^∗_𝑖^𝑙𝑖₀𝑔_𝑖(𝜔)𝑦ln^𝑖(𝑡−𝜔)𝑦_𝑖𝑑𝜔. (27) Using the following equalities: 𝜓ln^𝑖𝑥_𝑖(𝑡−𝜏),𝑣(𝑡−𝜏)𝜓_𝑖𝑥_𝑖𝑦,𝑣=ln^∗𝑖𝜓_𝑖𝑥_𝑖(𝑡−𝜏),𝑣(𝑡−𝜏)𝑦_𝑖𝜓_𝑖𝑥^∗_𝑖,𝑣^∗𝜓+ln^𝑖𝑥^∗_𝑖,𝑣^∗𝜓_𝑖𝑥_𝑖,𝑣^∗𝑣+ln𝑣^∗𝜓_𝑖𝑥_𝑖,𝑣^∗𝜓_𝑖𝑥_𝑖𝑣,𝑣+ln^∗𝑦_𝑖𝑣𝑦^∗_𝑖,𝑦ln^𝑖(𝑡−𝜔)𝑦_𝑖=ln𝑣𝑦^∗_𝑖𝑣^∗𝑦_𝑖𝑣+ln^∗𝑦_𝑖(𝑡−𝜔)𝑣𝑦^∗_𝑖, (28) we obtain 𝑑𝑊₁=𝑑𝑡_2𝑖=1𝑝_𝑖𝑎_𝑖𝑎_𝑖𝑦^∗_𝑖𝜓1−^𝑖𝑥^∗_𝑖,𝑣^∗𝜓_𝑖𝑥_𝑖,𝑣^∗𝑟_𝑖𝑥_𝑖𝑟_𝑖𝑥^∗_𝑖−1+𝑎_𝑖𝑦^∗_𝑖𝜓_𝑖𝑥_𝑖,𝑣𝜓_𝑖𝑥_𝑖,𝑣^∗−𝑣𝑣^∗𝜓1−^𝑖𝑥_𝑖,𝑣^∗𝜓_𝑖𝑥_𝑖,𝑣−𝑎_𝑖𝑦^∗_𝑖𝐻𝜓_𝑖𝑥^∗_𝑖,𝑣^∗𝜓_𝑖𝑥_𝑖,𝑣^∗−𝑎_𝑖𝑦^∗_𝑖𝐻𝑣𝑣^∗𝜓_𝑖𝑥_𝑖,𝑣^∗𝜓_𝑖𝑥_𝑖,𝑣−𝑎_𝑖𝑦^∗_𝑖^ℎ𝑖₀𝑓_𝑖(𝜏)𝑦×𝐻^∗𝑖𝜓_𝑖𝑥_𝑖(𝑡−𝜏),𝑣(𝑡−𝜏)𝑦_𝑖𝜓_𝑖𝑥^∗_𝑖,𝑣^∗𝑑𝜏−𝑎_𝑖𝑦^∗_𝑖^𝑙𝑖₀𝑔_𝑖(𝜔)𝐻𝑣^∗𝑦_𝑖(𝑡−𝜔)𝑣𝑦^∗_𝑖𝑑𝜔. (29) Clearly, if 𝐸_∗ exists, then 𝑑𝑊₁/𝑑𝑡≤0 for all 𝑥_𝑖,𝑦_𝑖,𝑣>0, 𝑖=1,2, and 𝑑𝑊₁/𝑑𝑡=0 if and only if 𝑥_𝑖=𝑥^∗_𝑖, 𝑦_𝑖=𝑦^∗_𝑖, and 𝑣=𝑣^∗, which is the infected steady state 𝐸_∗. It follows that 𝐸_∗ is GAS.

3. Conclusion

In this paper, we have proposed an HIV dynamics model describing the interaction of the HIV with CD4⁺ T cells and macrophages. The incidence rate of virus infection and the growth rate of the uninfected cells are given by general functions. Two types of distributed delays have been incorporated into the model to account for the time delay between the time the uninfected cells are contacted by the virus particle and the time for the emission of infectious (mature) virus particles. We have established a set of conditions which are sufficient for the global stability of the steady states of the model. Using Lyapunov functionals and LaSalle's invariant principle, we have proven that if 𝑅₀≤1, then the uninfected steady state is GAS, and if the infected steady state exists, then it is GAS.

Acknowledgments

This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks the DSR technical and financial support. The authors are grateful to the anonymous reviewers for constructive suggestions and valuable comments, which improve the quality of the paper.