Global Stability of HIV Infection of CD4⁺ T Cells and Macrophages with CTL Immune Response and Distributed Delays

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 𝑅₀ and the immune response reproduction number 𝑅^∗₀. We have proven that, if 𝑅₀≤1, then the uninfected steady state is globally asymptotically stable (GAS), if 𝑅^∗₀≤1<𝑅₀, then the infected steady state without CTL immune response is GAS, and, if 𝑅^∗₀>1, 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 𝑥̇(𝑡)(𝑡)(𝑡)𝑣(𝑡),𝑦(̇𝑡)(𝑡)𝑣(𝑡)(𝑡)(𝑡)𝑧(𝑡),𝑣(̇𝑡)(𝑡)(𝑡),𝑧(̇𝑡)(𝑡)𝑧(𝑡)(𝑡).=𝜆−𝑑𝑥−𝛽𝑥=𝛽𝑥−𝑎𝑦−𝑝𝑦=𝑘𝑦−𝑟𝑣=𝑐𝑦−𝑏𝑧 (1)(2)(3)(4) The state variables describe the plasma concentrations of𝑥(𝑡), the uninfected CD4⁺ T cells;𝑦(𝑡), the infected CD4⁺ T cells;𝑣(𝑡), the free virus particles; and𝑧(𝑡), the CTL cells at time𝑡. Here, (1) describes the population dynamics of the uninfected CD4⁺ T cells, where𝜆represents the rate of new uninfected cells that are generated from sources within the body,𝑑is the death rate constant, and𝛽is 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 constant𝑎and are killed by the CTL immune response with rate constant𝑝. Equation (3) describes the population dynamics of the free virus particles and shows that they are produced by the infected cells with rate constant𝑘and removed from the body with rate constant𝑟. Equation (4) describes the population dynamics of the CTL cells which are produced with rate constant𝑐and die with rate constant𝑏. 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 number𝑅₀and the immune response reproduction number𝑅^∗₀. We have shown that, if𝑅₀≤1, then the uninfected steady state is globally asymptotically stable (GAS), if𝑅^∗₀≤1<𝑅₀, then the infected steady state without CTL immune response is GAS, and, if𝑅^∗₀>1, 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 𝑥̇_𝑖(𝑡)=𝜆_𝑖−𝑑_𝑖𝑥_𝑖(𝑡)−𝛽_𝑖𝑥_𝑖(𝑡)𝑣(𝑡)𝑦̇,𝑖=1,2,_𝑖(𝑡)=𝛽_𝑖^ℎ𝑖₀𝑓_𝑖(𝜏)𝑒^{−𝑚𝑖𝜏}𝑥_𝑖(𝑡−𝜏)𝑣(𝑡−𝜏)𝑑𝜏−𝑎𝑦_𝑖(𝑡)−𝑝𝑦_𝑖(𝑡)𝑧(𝑡)𝑣(̇𝑡),𝑖=1,2,=𝑘^ℎ3₀𝑔(𝜏)𝑒^2−𝑛𝜏_𝑖=1𝑦_𝑖()(𝑡),𝑧(̇𝑡)𝑡−𝜏𝑑𝜏−𝑟𝑣=𝑐²_𝑖=1𝑦_𝑖(𝑡)𝑧(𝑡)−𝑏𝑧(𝑡). (5)(6)(7)(8) Here𝑖=1,2corresponds 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 let𝜏be the random variable that describes the time between viral entry and the transcription of viral RNA (stages (i) and (ii)) with a probability distribution𝑓_𝑖(𝜏)over the interval[0,ℎ_𝑖], andℎ_𝑖is limit superior to this delay. The factor𝑒^{−𝑚𝑖𝜏}accounts for the loss of target cells during this delay period, where𝑚_𝑖is constant. On the other hand, to consider the delay between viral RNA transcription and viral release and maturation, we let𝜏be the random variable; that is, the time between these two events with a probability distribution𝑔(𝜏)over the interval[0,ℎ₃], andℎ₃is limit superior to this delay [17]. The factor𝑒^−𝑛𝜏accounts for the loss of infected cells during this delay period, where𝑛is constant.

The probability distribution functions𝑓_𝑖(𝜏),  𝑖=1,2, and𝑔(𝜏)are assumed to satisfy𝑓_𝑖(𝜏)>0, and𝑔(𝜏)>0, and ^ℎ𝑖₀𝑓_𝑖(𝜏)𝑑𝜏=1,^ℎ𝑖₀𝑓_𝑖(𝑟)𝑒^𝑠𝑟𝑑𝑟<∞,𝑖=1,2,_ℎ30𝑔(𝜏)𝑑𝜏=1,^ℎ3₀𝑔(𝑟)𝑒^𝑠𝑟𝑑𝑟<∞, (9) where𝑠is a positive constant. Let 𝐹_𝑖=^ℎ𝑖₀𝑓_𝑖(𝜏)𝑒^{−𝑚𝑖𝜏}𝑑𝜏,𝑖=1,2,𝐺=^ℎ3₀𝑔(𝜏)𝑒^−𝑛𝜏𝑑𝜏. (10) Then 0<𝐹_𝑖≤1,0<𝐺≤1,𝑖=1,2. (11) The initial conditions for system (5)–(8) take the form 𝑥₁(𝜃)=𝜑₁(𝜃),𝑦₁(𝜃)=𝜑₂(𝜃),𝑥₂(𝜃)=𝜑₃(𝜃),𝑦₂(𝜃)=𝜑₄(𝜃),𝑣(𝜃)=𝜑₅(𝜃),𝑧(𝜃)=𝜑₆(𝜃),𝜑_𝑗(𝜃)[)𝜑≥0,𝜃∈−ℎ,0,𝑗=1,…,6,_𝑗(0)>0,𝑗=1,…,6, (12) whereℎ=max{ℎ₁,ℎ₂,ℎ₃}and(𝜑₁(𝜃),𝜑₂(𝜃),…,𝜑₆(𝜃))∈𝐶([−ℎ,0],𝑅⁶₊),where𝐶is the Banach space of continuous functions mapping the interval[−ℎ,0]into𝑅⁶₊. 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. Let(𝑥₁(𝑡),𝑦₁(𝑡),𝑥₂(𝑡),𝑦₂(𝑡),𝑣(𝑡),𝑧(𝑡))be any solution of (5)–(8) satisfying the initial conditions (12); then𝑥₁(𝑡),𝑦₁(𝑡),𝑥₂(𝑡),𝑦₂(𝑡),𝑣(𝑡), and𝑧(𝑡)are all non-negative for𝑡≥0and ultimately bounded.

Proof . First, we prove that𝑥_𝑖(𝑡)>0,  𝑖=1,2,for all𝑡≥0.Assume that𝑥_𝑖(𝑡)loses its nonnegativity on some local existence interval[0,𝜌]for some constant𝜌and let𝑡^∗_𝑖∈[0,𝜌]be such that𝑥_𝑖(𝑡^∗_𝑖)=0.From (5) we have𝑥̇_𝑖(𝑡^∗_𝑖)=𝜆_𝑖>0.Hence𝑥_𝑖(𝑡)>0for some𝑡∈(𝑡^∗_𝑖,𝑡^∗_𝑖+𝜖),where𝜖>0is sufficiently small. This leads to a contradiction and hence𝑥_𝑖(𝑡)>0,for all𝑡≥0.Further, from (6) and (7) we have 𝑦_𝑖(𝑡)=𝑦_𝑖(0)𝑒^{−∫𝑡0[𝑎+𝑝𝑧(𝜉)]𝑑𝜉}+𝛽_𝑖^𝑡₀𝑒^{−∫𝑡𝜂[𝑎+𝑝𝑧(𝜉)]𝑑𝜉}×^ℎ𝑖₀𝑓_𝑖(𝜏)𝑒^{−𝑚𝑖𝜏}𝑥_𝑖𝑣𝑣(𝑡)(0)𝑒𝜂−𝜏𝜂−𝜏𝑑𝜏𝑑𝜂,𝑖=1,2,=𝑣^−𝑟𝑡+𝑘^𝑡₀𝑒^{−𝑟(𝑡−𝜂)ℎ3}₀𝑔(𝜏)𝑒^2−𝑛𝜏_𝑖=1𝑦_𝑖𝜂−𝜏𝑑𝜏𝑑𝜂, (13) confirming that𝑦_𝑖(𝑡)≥0,  𝑖=1,2,and𝑣(𝑡)≥0for all𝑡∈[0,ℎ]. By a recursive argument, we obtain𝑦_𝑖(𝑡)≥0,  𝑖=1,2, and𝑣(𝑡)≥0for all𝑡≥0.Now from (8) we have 𝑧(𝑡)=𝑧(0)𝑒^{−𝑏𝑡+𝑐∑2𝑖=1∫𝑡0𝑦𝑖(𝜂)𝑑𝜂}. (14) Then𝑧(𝑡)≥0,for all𝑡≥0.
Next we show the boundedness of the solutions of system (5)–(8). From (5) we have𝑥̇_𝑖(𝑡)≤𝜆_𝑖−𝑑_𝑖𝑥_𝑖(𝑡),  𝑖=1,2. This implieslimsup_𝑡→∞𝑥_𝑖(𝑡)≤𝜆_𝑖/𝑑_𝑖,  𝑖=1,2.
Let𝑋(𝑡)=∑²_𝑖=1[𝐹_𝑖𝑥_𝑖(𝑡)+𝑦_𝑖(𝑡+𝜏)]+(𝑝/𝑐)𝑧(𝑡+𝜏). Then 𝑋̇(𝑡)=²_𝑖=1𝐹_𝑖𝜆_𝑖−𝐹_𝑖𝑑_𝑖𝑥_𝑖(𝑡)−𝐹_𝑖𝛽_𝑖𝑥_𝑖(𝑡)𝑣(𝑡)+𝐹_𝑖𝛽_𝑖𝑥_𝑖(𝑡)𝑣(𝑡)−𝑎𝑦_𝑖(𝑡+𝜏)−𝑝𝑦_𝑖(𝑡+𝜏)𝑧(𝑡+𝜏)+𝑝²_𝑖=1𝑦_𝑖()𝑧()−𝑝𝑡+𝜏𝑡+𝜏𝑐()=𝑏𝑧𝑡+𝜏_2𝑖=1𝐹_𝑖𝜆_𝑖−𝐹_𝑖𝑑_𝑖𝑥_𝑖(𝑡)−𝑎𝑦_𝑖(−𝑝𝑡+𝜏)𝑐()≤𝑏𝑧𝑡+𝜏_2𝑖=1𝐹_𝑖𝜆_𝑖−𝜎²_𝑖=1𝐹_𝑖𝑥_𝑖(𝑡)+𝑦_𝑖(+𝑝𝑡+𝜏)𝑐𝑧()=𝑡+𝜏_2𝑖=1𝐹_𝑖𝜆_𝑖−𝜎𝑋(𝑡), (15) where𝜎=min{𝑑₁,𝑑₂,𝑎,𝑏}. Hencelimsup_𝑡→∞𝑋(𝑡)≤𝐿, where𝐿=∑²_𝑖=1𝐿_𝑖=∑²_𝑖=1(𝐹_𝑖𝜆_𝑖/𝜎). Since𝑥_𝑖(𝑡)>0,𝑦_𝑖(𝑡)≥0and𝑧(𝑡)≥0 thenlimsup_𝑡→∞∑²_𝑖=1𝑦_𝑖(𝑡)≤𝐿,limsup_𝑡→∞𝑦_𝑖(𝑡)≤𝐿,  𝑖=1,2andlimsup_𝑡→∞𝑧(𝑡)≤𝑐𝐿/𝑝. On the other hand, 𝑣(̇𝑡)≤𝑘^ℎ3₀𝑔(𝜏)𝑒^−𝑛𝜏(𝐿)𝑑𝜏−𝑟𝑣=𝑘𝐺𝐿−𝑟𝑣. (16) Thenlimsup_𝑡→∞𝑣(𝑡)≤𝑘𝐺𝐿/𝑟. Therefore,𝑥₁(𝑡),𝑦₁(𝑡),𝑥₂(𝑡),𝑦₂(𝑡),𝑣(𝑡), and𝑧(𝑡)are ultimately bounded.

1.3. Steady States

First we define the basic reproduction number𝑅₀and immune response reproduction number𝑅^∗₀of system (5)–(8) as 𝑅₀=²_𝑖=1𝑅_0𝑖=²_𝑖=1𝑘𝐺𝐹_𝑖𝛽_𝑖𝜆_𝑖𝑎𝑟𝑑_𝑖,𝑅^∗₀=²_𝑖=1𝐹_𝑖𝛽_𝑖𝜆_𝑖𝑘𝐺𝑐𝑎𝑑_𝑖𝑟𝑐+𝛽_𝑖.𝑘𝐺𝑏 (17) We can rewrite𝑅^∗₀as 𝑅^∗₀=²_𝑖=1𝑅^∗_0𝑖=²_𝑖=1𝐼_𝑖𝑅_0𝑖𝐼_𝑖+𝑅_0𝑖/𝐹_𝑖,𝐼_𝑖=𝑐𝜆_𝑖,𝑎𝑏 (18) where𝐼₁and𝐼₂are the immune strengths of CD4⁺ T cells and macrophages, respectively. Clearly𝑅₀>𝑅^∗₀.

Lemma 2. (i) If𝑅₀≤1, then there exists only one uninfected steady state𝐸₀=(𝑥⁰₁,0,𝑥⁰₂,0,0,0).
(ii) If𝑅₀>1, then there exist𝐸₀and an infected steady state without CTL immune response𝐸₁=(𝑥₁,𝑦₁,𝑥₂,𝑦₂,𝑣,0).
(iii) If𝑅^∗₀>1, then there exist𝐸₀,𝐸₁, and an infected steady state with CTL immune response𝐸₂=(𝑥^∗₁,𝑦^∗₁,𝑥^∗₂,𝑦^∗₂,𝑣^∗,𝑧^∗).

Proof . The steady states of (5)–(8) satisfy the following equations: 𝜆_𝑖−𝑑_𝑖𝑥_𝑖−𝛽_𝑖𝑥_𝑖𝐹𝑣=0,_𝑖𝛽_𝑖𝑥_𝑖𝑣−𝑎𝑦_𝑖−𝑝𝑦_𝑖𝑧=0,_2𝑖=1𝑘𝐺𝑦_𝑖−𝑟𝑣=0,_2𝑖=1𝑐𝑦_𝑖𝑧−𝑏𝑧=0. (19)(20)(21)(22) From (22) we have ²_𝑖=1𝑐𝑦_𝑖−𝑏𝑧=0. (23) Equation (23) has two possible solutions,𝑧=0or ²_𝑖=1𝑐𝑦_𝑖−𝑏=0. (24) If𝑧=0,then from (19) and (20) we obtain𝑥_𝑖and𝑦_𝑖as 𝑥_𝑖=𝜆_𝑖𝑑_𝑖+𝛽_𝑖𝑣,𝑦_𝑖=𝐹_𝑖𝛽_𝑖𝜆_𝑖𝑎𝑑_𝑖+𝛽_𝑖𝑣𝑣, (25) and inserting them into (21) we obtain ²_𝑖=1𝑘𝐺𝐹_𝑖𝛽_𝑖𝜆_𝑖𝑎𝑑_𝑖+𝛽_𝑖𝑣−𝑟𝑣=0. (26) Equation (26) has two possible solutions𝑣=0or∑²_𝑖=1(𝑘𝐺𝐹_𝑖𝛽_𝑖𝜆_𝑖/𝑎(𝑑_𝑖+𝛽_𝑖𝑣))−𝑟=0.
If𝑣=0, then substituting it in (25) leads to an uninfected steady state𝐸₀=(𝑥⁰₁,0,𝑥⁰₂,0,0,0), where𝑥⁰_𝑖=𝜆_𝑖/𝑑_𝑖,𝑖=1,2. If𝑣≠0,then we have ²_𝑖=1𝑘𝐺𝐹_𝑖𝛽_𝑖𝜆_𝑖𝑎𝑟𝑑_𝑖+𝛽_𝑖𝑣−1=0. (27) Now we can write (27) as 𝑅₀₁1+𝛿₁𝑣+𝑅₀₂1+𝛿₂𝑣−1=0,⟹𝛿₁𝛿₂𝑣²+𝛿₁𝑅₀₁+𝛿₂𝑅₀₂+1−𝑅₀𝛿₁+𝛿₂𝑣+1−𝑅₀=0, (28) where𝛿_𝑖=𝛽_𝑖/𝑑_𝑖, 𝑖=1,2.The solution of (28) is given by 𝑣^±=−𝛿₁𝑅₀₁+𝛿₂𝑅₀₂+1−𝑅₀𝛿₁+𝛿₂±𝛿₁𝑅₀₁+𝛿₂𝑅₀₂+1−𝑅₀𝛿₁+𝛿₂²−4𝛿₁𝛿₂1−𝑅₀^1/2×2𝛿₁𝛿₂⁻¹. (29) Clearly if𝑅₀>1, then𝑣⁺>0and𝑣⁻<0: then we choose𝑣=𝑣⁺.Therefore, if𝑅₀>1, then system (5)–(8) has an infected steady state without CTL immune response𝐸₁=(𝑥₁,𝑦₁,𝑥₂,𝑦₂,𝑣,0),where𝑣=𝑣⁺and 𝑥_𝑖=𝜆_𝑖𝑑_𝑖+𝛽_𝑖𝑣,𝑦_𝑖=𝐹_𝑖𝛽_𝑖𝜆_𝑖𝑎𝑑_𝑖+𝛽_𝑖𝑣𝑣,𝑖=1,2. (30) If𝑧≠0, then from (24) we have ²_𝑖=1𝑦^∗_𝑖=𝑏𝑐, (31) and inserting it into (21) we obtain 𝑣^∗=𝑘𝐺𝑏.𝑟𝑐 (32) From (32) to (19) and (20) we get 𝑥^∗_𝑖=𝜆_𝑖𝑟𝑐𝑑_𝑖𝑟𝑐+𝛽_𝑖,𝑦𝐺𝑘𝑏_∗𝑖=𝑘𝐺𝑏𝐹_𝑖𝛽_𝑖𝑥^∗_𝑖𝑟𝑐𝑎+𝑝𝑧^∗=𝑘𝐺𝑏𝐹_𝑖𝛽_𝑖𝜆_𝑖𝑎+𝑝𝑧^∗𝑑_𝑖𝑟𝑐+𝛽_𝑖,𝐺𝑘𝑏 (33) and inserting (31) into (20) we get ⟹𝑧^∗=𝑎𝑝²_𝑖=1𝐹_𝑖𝛽_𝑖𝜆_𝑖𝑘𝐺𝑐𝑎𝑑_𝑖𝑟𝑐+𝛽_𝑖=𝑎𝑘𝐺𝑏−1𝑝𝑅^∗₀−1. (34) We have𝑥^∗_𝑖>0, 𝑣^∗>0and if𝑅^∗₀>1, then𝑧^∗>0and𝑦^∗_𝑖>0. It follows that, if𝑅^∗₀>1, then there exists an infected steady state with CTL immune response𝐸₂=(𝑥^∗₁,𝑦^∗₁,𝑥^∗₂,𝑦^∗₂,𝑣^∗,𝑧^∗).
Hence, if𝑅₀≤1, then there exists only one steady state𝐸₀, if𝑅₀>1, then there exist two steady states𝐸₀and𝐸₁, and, if𝑅^∗₀>1, then there exist three steady states𝐸₀,𝐸₁, and𝐸₂.

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:𝑢=𝑢(𝑡)for any𝑢∈{𝑥_𝑖,𝑦_𝑖,𝑣,𝑧,𝑖=1,2}. We also define a function𝐻∶(0,∞)→[0,∞)as𝐻(𝑢)=𝑢−1−ln𝑢. It is clear that𝐻(𝑢)≥0for any𝑢>0and𝐻has the global minimum𝐻(1)=0.

Theorem 3. If𝑅₀≤1, then𝐸₀is GAS.

Proof . Define a Lyapunov functional𝑊₀as follows: 𝑊₀=²_𝑖=1𝛾_𝑖𝑥⁰_𝑖𝐻𝑥_𝑖𝑥⁰_𝑖+1𝐹_𝑖𝑦_𝑖+𝛽_𝑖𝐹_𝑖^ℎ𝑖₀𝑓_𝑖(𝜏)𝑒^{−𝑚𝑖𝜏0}_−𝜏𝑥_𝑖()()+𝑎𝑡+𝜃×𝑣𝑡+𝜃𝑑𝜃𝑑𝜏𝐹_𝑖𝐺^ℎ3₀𝑔(𝜏)𝑒^−𝑛𝜏0_−𝜏𝑦_𝑖(𝑡+𝜃)𝑑𝜃𝑑𝜏+𝑣+𝑘𝐺𝑝𝑎𝑐𝑧, (35) where𝛾_𝑖=𝑘𝐺𝐹_𝑖/𝑎,𝑖=1,2.
The time derivative of𝑊₀along the trajectories of (5)–(8) satisfies 𝑑𝑊₀=𝑑𝑡²_𝑖=1𝛾_𝑖𝑥1−^0𝑖𝑥_𝑖𝜆_𝑖−𝑑_𝑖𝑥_𝑖−𝛽_𝑖𝑥_𝑖𝑣+𝛽_𝑖𝐹_𝑖^ℎ𝑖₀𝑓_𝑖(𝜏)𝑒^{−𝑚𝑖𝜏}𝑥_𝑖()𝑣()𝑎𝑡−𝜏𝑡−𝜏𝑑𝜏−𝐹_𝑖𝑦_𝑖−𝑝𝐹_𝑖𝑦_𝑖𝛽𝑧+^𝑖𝐹_𝑖^ℎ𝑖₀𝑓_𝑖(𝜏)𝑒^{−𝑚𝑖𝜏}×𝑥_𝑖𝑣−𝑥_𝑖()𝑣()+𝑎𝑡−𝜏𝑡−𝜏𝑑𝜏𝐹_𝑖𝐺^ℎ3₀𝑔(𝜏)𝑒^−𝑛𝜏𝑦_𝑖−𝑦_𝑖()+𝑡−𝜏𝑑𝜏_2𝑖=1𝑘^ℎ3₀𝑔(𝜏)𝑒^−𝑛𝜏𝑦_𝑖()+𝑡−𝜏𝑑𝜏−𝑟𝑣_2𝑖=1𝑘𝐺𝑝𝑎𝑦_𝑖𝑧−𝑘𝐺𝑝𝑏𝑎𝑐𝑧. (36) Collecting terms of (36) we get 𝑑𝑊₀=𝑑𝑡_2𝑖=1𝛾_𝑖𝜆_𝑖−𝑑_𝑖𝑥_𝑖−𝜆_𝑖𝑥⁰_𝑖𝑥_𝑖+𝑑_𝑖𝑥⁰_𝑖+𝛽_𝑖𝑥⁰_𝑖𝑣−𝑟𝑣−𝑘𝐺𝑝𝑏𝑧=𝑎𝑐_2𝑖=1𝛾_𝑖𝜆_𝑖𝑥2−^𝑖𝑥⁰_𝑖−𝑥⁰_𝑖𝑥_𝑖−𝑘𝐺𝑝𝑏𝑎𝑐𝑧−𝑟𝑣+𝑟𝑣²_𝑖=1𝑘𝐺𝐹_𝑖𝛽_𝑖𝑥⁰_𝑖𝑎𝑟=−²_𝑖=1𝛾_𝑖𝑑_𝑖𝑥_𝑖−𝑥⁰_𝑖²𝑥_𝑖−𝑘𝐺𝑝𝑏𝑎𝑐𝑧−𝑟𝑣+𝑟𝑣²_𝑖=1𝑅_0𝑖=−²_𝑖=1𝛾_𝑖𝑑_𝑖𝑥_𝑖−𝑥⁰_𝑖²𝑥_𝑖−𝑘𝐺𝑝𝑏𝑅𝑎𝑐𝑧+₀−1𝑟𝑣. (37) If𝑅₀≤1, then𝑑𝑊₀/𝑑𝑡≤0for all𝑥₁,𝑥₂,𝑣,𝑧>0. By Theorem5.3.1in [26], the solutions of system (5)–(8) are limited to𝑀, the largest invariant subset of{𝑑𝑊₀/𝑑𝑡=0}. Clearly, it follows from (37) that𝑑𝑊₀/𝑑𝑡=0if and only if𝑥_𝑖=𝑥⁰_𝑖,𝑖=1,2,𝑣=0, and𝑧=0. Noting that𝑀is invariant, for each element of𝑀, we have𝑣=0; then𝑣̇=0. From (7) we drive that 𝑘^ℎ3₀𝑔(𝜏)𝑒^2−𝑛𝜏_𝑖=1𝑦_𝑖(𝑡−𝜏)𝑑𝜏=0. (38) Hence, this yields∑²_𝑖=1𝑦_𝑖(𝑡−𝜏)=0. Since𝑦_𝑖≥0for𝑖=1,2, then𝑦₁=𝑦₂=0. Hence𝑑𝑊₀/𝑑𝑡=0if and only if𝑥_𝑖=𝑥⁰_𝑖,𝑦_𝑖=0,𝑖=1,2,  𝑣=0, and𝑧=0. From LaSalle's Invariance Principle,𝐸₀is GAS.

Theorem 4. If𝑅₀>1≥𝑅^∗₀, then𝐸₁is GAS.

Proof. We construct the following Lyapunov functional: 𝑊₁=²_𝑖=1𝛾_𝑖𝑥_𝑖𝐻𝑥_𝑖𝑥_𝑖+1𝐹_𝑖𝑦_𝑖𝐻𝑦_𝑖𝑦_𝑖+1𝐹_𝑖𝛽_𝑖𝑥_𝑖𝑣×^ℎ𝑖₀𝑓_𝑖(𝜏)𝑒^{−𝑚𝑖𝜏0}_−𝜏𝐻𝑥_𝑖()×(𝑣(×𝑥𝑡+𝜃𝑡+𝜃)))_𝑖𝑣⁻¹+𝑎𝑦𝑑𝜃𝑑𝜏_𝑖𝐹_𝑖𝐺^ℎ3₀𝑔(𝜏)𝑒^−𝑛𝜏0_−𝜏𝐻𝑦_𝑖(𝑡+𝜃)𝑦_𝑖+𝑣𝑑𝜃𝑑𝜏𝑣𝐻𝑣+𝑘𝐺𝑝𝑎𝑐𝑧. (39) Differentiating with respect to time yields 𝑑𝑊₁=𝑑𝑡_2𝑖=1𝛾_𝑖𝑥1−^𝑖𝑥_𝑖𝜆_𝑖−𝑑_𝑖𝑥_𝑖−𝛽_𝑖𝑥_𝑖𝑣+1𝐹_𝑖𝑦1−^𝑖𝑦_𝑖×𝛽_𝑖^ℎ𝑖₀𝑓_𝑖(𝜏)𝑒^{−𝑚𝑖𝜏}𝑥_𝑖(𝑡−𝜏)𝑣(𝑡−𝜏)𝑑𝜏−𝑎𝑦_𝑖−𝑝𝑦_𝑖𝑧+𝛽_𝑖𝐹_𝑖^ℎ𝑖₀𝑓_𝑖(𝜏)𝑒^{−𝑚𝑖𝜏}×𝑥_𝑖𝑣−𝑥_𝑖()𝑣()+𝑥𝑡−𝜏𝑡−𝜏_𝑖𝑥𝑣ln^𝑖(𝑡−𝜏)𝑣(𝑡−𝜏)𝑥_𝑖𝑣+𝑎𝑑𝜏𝐹_𝑖𝐺^ℎ3₀𝑔(𝜏)𝑒^−𝑛𝜏𝑦_𝑖−𝑦_𝑖()+𝑦𝑡−𝜏_𝑖𝑦ln^𝑖(𝑡−𝜏)𝑦_𝑖+𝑣𝑑𝜏1−𝑣²_𝑖=1𝑘^ℎ3₀𝑔(𝜏)𝑒^−𝑛𝜏𝑦_𝑖()+𝑡−𝜏𝑑𝜏−𝑟𝑣𝑘𝐺𝑝𝑎𝑐²_𝑖=1𝑐𝑦_𝑖𝑧−𝑏𝑧. (40) Collecting terms we obtain 𝑑𝑊₁=𝑑𝑡²_𝑖=1𝛾_𝑖𝜆_𝑖−𝑑_𝑖𝑥_𝑖−𝜆_𝑖𝑥_𝑖𝑥_𝑖+𝑑_𝑖𝑥_𝑖+𝛽_𝑖𝑥_𝑖𝑣−𝛽_𝑖𝑦_𝑖𝐹_𝑖𝑦_𝑖^ℎ𝑖₀𝑓_𝑖(𝜏)𝑒^{−𝑚𝑖𝜏}𝑥_𝑖()𝑣()+𝑎𝑡−𝜏𝑡−𝜏𝑑𝜏𝐹_𝑖𝑦_𝑖+𝑝𝐹_𝑖𝑦_𝑖𝑧+1𝐹_𝑖𝛽_𝑖𝑥_𝑖𝑣^ℎ𝑖₀𝑓_𝑖(𝜏)𝑒^{−𝑚𝑖𝜏}𝑥×ln^𝑖(𝑡−𝜏)𝑣(𝑡−𝜏)𝑥_𝑖𝑣+𝑎𝑦𝑑𝜏_𝑖𝐹_𝑖𝐺^ℎ3₀𝑔(𝜏)𝑒^−𝑛𝜏𝑦ln^𝑖(𝑡−𝜏)𝑦_𝑖𝑣𝑑𝜏−𝑟𝑣−𝑣²_𝑖=1𝑘^ℎ3₀𝑔(𝜏)𝑒^−𝑛𝜏𝑦_𝑖(𝑡−𝜏)𝑑𝜏+𝑟𝑣−𝑘𝐺𝑝𝑏𝑎𝑐𝑧. (41) From (19)–(21) we have 𝜆_𝑖=𝑑_𝑖𝑥_𝑖+𝛽_𝑖𝑥_𝑖𝑣,𝐹_𝑖𝛽_𝑖𝑥_𝑖𝑣=𝑎𝑦_𝑖,𝑟𝑣=²_𝑖=1𝐺𝑘𝑦_𝑖. (42) Using (42) and the following equality: 𝑣𝑣𝑟𝑣=𝑟𝑣=𝑣𝑣²_𝑖=1𝐺𝑘𝑦_𝑖=𝑣𝑣²_𝑖=1𝛾_𝑖𝑎𝐹_𝑖𝑦_𝑖=𝑣𝑣²_𝑖=1𝛾_𝑖𝛽_𝑖𝑥_𝑖𝑣, (43) we obtain 𝑑𝑊₁=𝑑𝑡²_𝑖=1𝛾_𝑖𝑑_𝑖𝑥_𝑖+𝛽_𝑖𝑥_𝑖𝑣−𝑑_𝑖𝑥_𝑖−𝑥_𝑖𝑥_𝑖𝑑_𝑖𝑥_𝑖+𝛽_𝑖𝑥_𝑖𝑣+𝑑_𝑖𝑥_𝑖+𝛽_𝑖𝑥_𝑖𝑣𝑣𝑣−1𝐹_𝑖𝛽_𝑖𝑥_𝑖𝑣^ℎ𝑖₀𝑓_𝑖(𝜏)𝑒^{−𝑚𝑖𝜏}𝑦_𝑖𝑥_𝑖(𝑡−𝜏)𝑣(𝑡−𝜏)𝑦_𝑖𝑥_𝑖𝑣𝑑𝜏+𝛽_𝑖𝑥_𝑖𝑝𝑣+𝐹_𝑖𝑦_𝑖𝑧+1𝐹_𝑖𝛽_𝑖𝑥_𝑖𝑣^ℎ𝑖₀𝑓_𝑖(𝜏)𝑒^{−𝑚𝑖𝜏}𝑥×ln^𝑖(𝑡−𝜏)𝑣(𝑡−𝜏)𝑥_𝑖𝑣+1𝑑𝜏𝐺𝛽_𝑖𝑥_𝑖𝑣_𝑖^ℎ3₀𝑔(𝜏)𝑒^−𝑛𝜏𝑦ln^𝑖(𝑡−𝜏)𝑦_𝑖𝑑𝜏−𝛽_𝑖𝑥_𝑖𝑣𝑣𝑣−1𝐺𝛽_𝑖𝑥_𝑖𝑣^ℎ3₀𝑔(𝜏)𝑒^−𝑛𝜏𝑣𝑦_𝑖(𝑡−𝜏)𝑣𝑦_𝑖𝑑𝜏+𝛽_𝑖𝑥_𝑖𝑣−𝑘𝐺𝑝𝑏𝑎𝑐𝑧. (44) Then collecting terms of (44) and using the following equalities: 𝑥ln^𝑖(𝑡−𝜏)𝑣(𝑡−𝜏)𝑥_𝑖𝑣𝑦=ln^𝑖𝑥_𝑖(𝑡−𝜏)𝑣(𝑡−𝜏)𝑦_𝑖𝑥_𝑖𝑣𝑥+ln^𝑖𝑥_𝑖+ln𝑣𝑦_𝑖𝑣𝑦_𝑖𝑦,𝑖=1,2,ln^𝑖(𝑡−𝜏)𝑦_𝑖𝑣𝑦=ln^𝑖𝑣𝑦_𝑖+ln𝑣𝑦_𝑖(𝑡−𝜏)𝑣𝑦_𝑖,𝑖=1,2, (45) we obtain 𝑑𝑊₁=𝑑𝑡²_𝑖=1𝛾_𝑖𝑑_𝑖𝑥_𝑖𝑥2−^𝑖𝑥_𝑖−𝑥_𝑖𝑥_𝑖+𝛽_𝑖𝑥_𝑖𝑣𝑥1−^𝑖𝑥_𝑖+2𝛽_𝑖𝑥_𝑖𝑣−1𝐹_𝑖𝛽_𝑖𝑥_𝑖𝑣^ℎ𝑖₀𝑓_𝑖(𝜏)𝑒^{−𝑚𝑖𝜏}×𝑦_𝑖𝑥_𝑖(𝑡−𝜏)𝑣(𝑡−𝜏)𝑦_𝑖𝑥_𝑖𝑣+1𝑑𝜏𝐹_𝑖𝛽_𝑖𝑥_𝑖𝑣^ℎ𝑖₀𝑓_𝑖(𝜏)𝑒^{−𝑚𝑖𝜏}×𝑦ln^𝑖𝑥_𝑖(𝑡−𝜏)𝑣(𝑡−𝜏)𝑦_𝑖𝑥_𝑖𝑣𝑥+ln^𝑖𝑥_𝑖+ln𝑣𝑦_𝑖𝑣𝑦_𝑖+1𝑑𝜏𝐺𝛽_𝑖𝑥_𝑖𝑣^ℎ3₀𝑔(𝜏)𝑒^−𝑛𝜏×𝑣𝑦ln^𝑖𝑣𝑦_𝑖+ln𝑣𝑦_𝑖(𝑡−𝜏)𝑣𝑦_𝑖−1𝑑𝜏𝐺𝛽_𝑖𝑥_𝑖𝑣₀^ℎ3𝑔(𝜏)𝑒^−𝑛𝜏𝑣𝑦_𝑖(𝑡−𝜏)𝑣𝑦_𝑖+𝑑𝜏_2𝑖=1𝑘𝐺𝑝𝑎𝑦_𝑖𝑧−𝑘𝐺𝑝𝑏𝑎𝑐𝑧. (46) Equation (46) can be rewritten as 𝑑𝑊₁=𝑑𝑡_2𝑖=1𝛾_𝑖𝑑_𝑖𝑥_𝑖𝑥2−^𝑖𝑥_𝑖−𝑥_𝑖𝑥_𝑖−𝛽_𝑖𝑥_𝑖𝑣𝑥_𝑖𝑥_𝑖𝑥−1−ln^𝑖𝑥_𝑖−1𝐹_𝑖𝛽_𝑖𝑥_𝑖𝑣^ℎ𝑖₀𝑓_𝑖(𝜏)𝑒^{−𝑚𝑖𝜏}×𝑦_𝑖𝑥_𝑖(𝑡−𝜏)𝑣(𝑡−𝜏)𝑦_𝑖𝑥_𝑖𝑣𝑦−1−ln^𝑖𝑥_𝑖(𝑡−𝜏)𝑣(𝑡−𝜏)𝑦_𝑖𝑥_𝑖𝑣−1𝑑𝜏𝐺𝛽_𝑖𝑥_𝑖𝑣₀^ℎ3𝑔(𝜏)𝑒^−𝑛𝜏𝑣𝑦_𝑖(𝑡−𝜏)𝑣𝑦_𝑖−1−ln𝑣𝑦_𝑖(𝑡−𝜏)𝑣𝑦_𝑖+𝑑𝜏𝑘𝐺𝑝𝑎²_𝑖=1𝑦_𝑖−𝑏𝑐𝑧. (47) We can rewrite𝑑𝑊₁/𝑑𝑡as: 𝑑𝑊₁𝑑𝑡=−²_𝑖=1𝛾_𝑖𝑑_𝑖𝑥_𝑖−𝑥_𝑖²𝑥_𝑖+𝛽_𝑖𝑥_𝑖𝑥𝑣𝐻^𝑖𝑥_𝑖+1𝐹_𝑖𝛽_𝑖𝑥_𝑖𝑣×^ℎ𝑖₀𝑓_𝑖(𝜏)𝑒^{−𝑚𝑖𝜏}𝐻𝑦_𝑖𝑥_𝑖()×(𝑣(×𝑦𝑡−𝜏𝑡−𝜏))_𝑖𝑥_𝑖𝑣⁻¹+1𝑑𝜏𝐺𝛽_𝑖𝑥_𝑖𝑣^ℎ3₀𝑔(𝜏)𝑒^−𝑛𝜏𝐻𝑣𝑦_𝑖(𝑡−𝜏)𝑣𝑦_𝑖+𝑑𝜏𝑘𝐺𝑝𝑎²_𝑖=1𝑦_𝑖−𝑏𝑐𝑧. (48) Now we show that, if𝑅^∗₀≤1, then∑²_𝑖=1𝑦_𝑖<𝑏/𝑐. Assume that𝑅^∗₀≤1; then ²_𝑖=1𝑘𝐺𝑐𝐹_𝑖𝛽_𝑖𝜆_𝑖𝑎𝑟𝑐𝑑_𝑖+𝑎𝛽_𝑖𝐺𝑘𝑏≤1. (49) From (27) we have ²_𝑖=1𝑘𝐺𝐹_𝑖𝛽_𝑖𝜆_𝑖𝑎𝑟𝑑_𝑖+𝑎𝛽_𝑖𝑟𝑣⟹=1,_2𝑖=1𝑘𝐺𝑐𝐹_𝑖𝛽_𝑖𝜆_𝑖𝑎𝑟𝑐𝑑_𝑖+𝑎𝛽_𝑖𝑐𝑟𝑣=1. (50) Comparing (49) and (50) we get ²_𝑖=1𝑘𝐺𝑐𝐹_𝑖𝛽_𝑖𝜆_𝑖𝑎𝑟𝑐𝑑_𝑖+𝑎𝛽_𝑖≤𝐺𝑘𝑏²_𝑖=1𝑘𝐺𝑐𝐹_𝑖𝛽_𝑖𝜆_𝑖𝑎𝑟𝑐𝑑_𝑖+𝑎𝛽_𝑖𝑣.𝑐𝑟 (51) Then 𝑣≤𝐺𝑘𝑏.𝑟𝑐 (52) Using (42) we have 𝐺𝑘𝑟²_𝑖=1𝑦_𝑖=𝑣≤𝐺𝑘𝑏.𝑟𝑐 (53) Then ²_𝑖=1𝑦_𝑖≤𝑏𝑐. (54) Now if𝑅₀>1, then𝑥_𝑖,𝑦_𝑖,𝑣>0.It follows that, if𝑅₀>1≥𝑅^∗₀, then𝑑𝑊₁/𝑑𝑡≤0. By Theorem5.3.1in [26], the solutions of system (5)–(8) are limited to𝑀, the largest invariant subset of{𝑑𝑊₁/𝑑𝑡=0}. It can be seen that𝑑𝑊₁/𝑑𝑡=0if and only if𝑥_𝑖=𝑥_𝑖,𝑧=0, and𝐻=0: that is, 𝑦_𝑖𝑥_𝑖(𝑡−𝜏)𝑣(𝑡−𝜏)𝑦_𝑖𝑥_𝑖𝑣=𝑣𝑦_𝑖(𝑡−𝜏)𝑣𝑦_𝑖[].=1foralmostall𝜏∈0,ℎ (55) If𝑥_𝑖=𝑥_𝑖, then from (19) we have𝑣=𝑣and from (55) we have𝑦_𝑖=𝑦_𝑖. It follows that𝑑𝑊₁/𝑑𝑡=0at𝐸₁. LaSalle's Invariance Principle implies the global stability of𝐸₁.

Theorem 5. If𝑅₀>𝑅^∗₀>1, then𝐸₂is 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^∗×^ℎ𝑖₀𝑓_𝑖(𝜏)𝑒^{−𝑚𝑖𝜏0}_−𝜏𝐻𝑥_𝑖()×(𝑣(×𝑥𝑡+𝜃𝑡+𝜃))_∗𝑖𝑣^∗−1+𝑑𝜃𝑑𝜏𝑎𝑦^∗_𝑖+𝑝𝑦^∗_𝑖𝑧^∗𝐹_𝑖𝐺^ℎ3₀𝑔(𝜏)𝑒^−𝑛𝜏×⁰_−𝜏𝐻𝑦_𝑖(𝑡+𝜃)𝑦^∗_𝑖+𝑝𝑑𝜃𝑑𝜏1+𝑎𝑧^∗𝑣^∗𝐻𝑣𝑣^∗+𝑘𝐺𝑝𝑧𝑎𝑐^∗𝐻𝑧𝑧^∗. (56) Differentiating with respect to time yields 𝑑𝑊₂=𝑑𝑡²_𝑖=1𝛾_𝑖𝑥1−^∗𝑖𝑥_𝑖𝜆_𝑖−𝑑_𝑖𝑥_𝑖−𝛽_𝑖𝑥_𝑖𝑣+1𝐹_𝑖𝑦1−^∗𝑖𝑦_𝑖𝛽_𝑖^ℎ𝑖₀𝑓_𝑖(𝜏)𝑒^{−𝑚𝑖𝜏}𝑥_𝑖()()𝑡−𝜏×𝑣𝑡−𝜏𝑑𝜏−𝑎𝑦_𝑖−𝑝𝑦_𝑖𝑧+𝛽_𝑖𝐹_𝑖^ℎ𝑖₀𝑓_𝑖(𝜏)𝑒^{−𝑚𝑖𝜏}×𝑥_𝑖𝑣−𝑥_𝑖(𝑡−𝜏)𝑣(𝑡−𝜏)+𝑥^∗_𝑖𝑣^∗𝑥ln^𝑖(𝑡−𝜏)𝑣(𝑡−𝜏)𝑥_𝑖𝑣+𝑎𝑑𝜏𝐹_𝑖𝐺^ℎ3₀𝑔(𝜏)𝑒^−𝑛𝜏×𝑦_𝑖−𝑦_𝑖(𝑡−𝜏)+𝑦^∗_𝑖𝑦ln^𝑖(𝑡−𝜏)𝑦_𝑖+𝑝𝑑𝜏𝐹_𝑖𝐺^ℎ3₀𝑔(𝜏)𝑒^−𝑛𝜏×𝑦_𝑖𝑧^∗−𝑦_𝑖(𝑡−𝜏)𝑧^∗+𝑦^∗_𝑖𝑧^∗𝑦ln^𝑖(𝑡−𝜏)𝑦_𝑖+𝑝𝑑𝜏1+𝑎𝑧^∗𝑣1−^∗𝑣ײ_𝑖=1𝑘^ℎ3₀𝑔(𝜏)𝑒^−𝑛𝜏𝑦_𝑖()+𝑡−𝜏𝑑𝜏−𝑟𝑣𝑘𝐺𝑝𝑧𝑎𝑐1−^∗𝑧²_𝑖=1𝑐𝑦_𝑖𝑧−𝑏𝑧. (57) Collecting terms we obtain 𝑑𝑊₂=𝑑𝑡²_𝑖=1𝛾_𝑖𝜆_𝑖−𝑑_𝑖𝑥_𝑖−𝜆_𝑖𝑥^∗_𝑖𝑥_𝑖+𝑑_𝑖𝑥^∗_𝑖+𝛽_𝑖𝑥^∗_𝑖𝑣−𝛽_𝑖𝑦^∗_𝑖𝐹_𝑖𝑦_𝑖^ℎ𝑖₀𝑓_𝑖(𝜏)𝑒^{−𝑚𝑖𝜏}𝑥_𝑖()𝑣()+𝑎𝑡−𝜏𝑡−𝜏𝑑𝜏𝐹_𝑖𝑦^∗_𝑖+𝑝𝐹_𝑖𝑦^∗_𝑖1𝑧+𝐹_𝑖𝛽_𝑖𝑥^∗_𝑖𝑣^∗×^ℎ𝑖₀𝑓_𝑖(𝜏)𝑒^{−𝑚𝑖𝜏}𝑥ln^𝑖(𝑡−𝜏)𝑣(𝑡−𝜏)𝑥_𝑖𝑣+𝑑𝜏𝑎𝑦^∗_𝑖𝐹_𝑖𝐺^ℎ3₀𝑔(𝜏)𝑒^−𝑛𝜏𝑦ln^𝑖(𝑡−𝜏)𝑦_𝑖+𝑑𝜏𝑝𝑦^∗_𝑖𝑧^∗𝐹_𝑖𝐺^ℎ3₀𝑔(𝜏)𝑒^−𝑛𝜏𝑦ln^𝑖(𝑡−𝜏)𝑦_𝑖−𝑣𝑑𝜏−𝑟𝑣_∗𝑣²_𝑖=1𝑘^ℎ3₀𝑔(𝜏)𝑒^−𝑛𝜏𝑦_𝑖(𝑡−𝜏)𝑑𝜏+𝑟𝑣^∗−𝑝𝑧^∗𝑎−𝑣𝑟𝑣_∗𝑣²_𝑖=1𝑘𝑎^ℎ3₀𝑔(𝜏)𝑒^−𝑛𝜏𝑝𝑦_𝑖(𝑡−𝜏)𝑧^∗𝑑𝜏+𝑝𝑧^∗𝑎𝑟𝑣^∗−𝑘𝐺𝑝𝑏𝑎𝑐𝑧+𝑘𝐺𝑝𝑏𝑧𝑎𝑐^∗. (58) Using (19)–(21), we have equalities 𝑟𝑣=𝑟𝑣^∗𝑣𝑣^∗=𝑣𝑣^∗2_𝑖=1𝐺𝑘𝑦^∗_𝑖=𝑣𝑣^∗2_𝑖=1𝛾_𝑖𝐹_𝑖𝑎𝑦^∗_𝑖,𝑝𝑧^∗𝑎𝑟𝑣^∗=²_𝑖=1𝑘𝐺𝑎𝑝𝑦^∗_𝑖𝑧^∗=²_𝑖=1𝛾_𝑖𝐹_𝑖𝑝𝑦^∗_𝑖𝑧^∗,𝑝𝑧^∗𝑎𝑟𝑣=𝑝𝑧^∗𝑎𝑟𝑣^∗𝑣𝑣^∗=𝑣𝑣^∗2_𝑖=1𝛾_𝑖𝐹_𝑖𝑝𝑦^∗_𝑖𝑧^∗,𝑘𝐺𝑝𝑏𝑧𝑎𝑐^∗=²_𝑖=1𝑘𝐺𝑎𝑝𝑦^∗_𝑖𝑧^∗=²_𝑖=1𝛾_𝑖𝐹_𝑖𝑝𝑦^∗_𝑖𝑧^∗,since²_𝑖=1𝑦^∗_𝑖=𝑏𝑐,𝑘𝐺𝑝𝑏𝑎𝑐𝑧=𝑘𝐺𝑝𝑏𝑧𝑎𝑐^∗𝑧𝑧^∗=²_𝑖=1𝛾_𝑖𝐹_𝑖𝑝𝑦^∗_𝑖𝑧. (59) We obtain 𝑑𝑊₂=𝑑𝑡²_𝑖=1𝛾_𝑖𝑑_𝑖𝑥^∗_𝑖+𝛽_𝑖𝑥^∗_𝑖𝑣^∗−𝑑_𝑖𝑥_𝑖−𝑥^∗_𝑖𝑥_𝑖𝑑_𝑖𝑥^∗_𝑖+𝛽_𝑖𝑥^∗_𝑖𝑣^∗+𝑑_𝑖𝑥^∗_𝑖+𝛽_𝑖𝑥^∗_𝑖𝑣^∗𝑣𝑣^∗−1𝐹_𝑖𝛽_𝑖𝑥^∗_𝑖𝑣^∗×^ℎ𝑖₀𝑓_𝑖(𝜏)𝑒^{−𝑚𝑖𝜏}𝑦^∗_𝑖𝑥_𝑖(𝑡−𝜏)𝑣(𝑡−𝜏)𝑦_𝑖𝑥^∗_𝑖𝑣^∗+𝑑𝜏𝑎𝑦^∗_𝑖𝐹_𝑖+𝑝𝑦^∗_𝑖𝑧𝐹_𝑖+1𝐹_𝑖𝛽_𝑖𝑥^∗_𝑖𝑣^∗ℎ𝑖₀𝑓_𝑖(𝜏)𝑒^−𝑚i𝜏𝑥×ln^𝑖(𝑡−𝜏)𝑣(𝑡−𝜏)𝑥_𝑖𝑣+𝑑𝜏𝑎𝑦^∗_𝑖𝐹_𝑖𝐺^ℎ3₀𝑔(𝜏)𝑒^−𝑛𝜏𝑦ln^𝑖(𝑡−𝜏)𝑦_𝑖+𝑑𝜏𝑝𝑦^∗_𝑖𝑧^∗𝐹_𝑖𝐺^ℎ3₀𝑔(𝜏)𝑒^−𝑛𝜏𝑦ln^𝑖(𝑡−𝜏)𝑦_𝑖−𝑑𝜏𝑎𝑦^∗_𝑖𝐹_𝑖𝑣𝑣^∗−𝑎𝑦^∗_𝑖𝐹_𝑖𝐺^ℎ3₀𝑔(𝜏)𝑒^−𝑛𝜏×𝑦_𝑖(𝑡−𝜏)𝑣^∗𝑦^∗_𝑖𝑣+𝑑𝜏𝑎𝑦^∗_𝑖𝐹_𝑖−𝑝𝑦^∗_𝑖𝑧^∗𝐹_𝑖𝑣𝑣^∗−𝑝𝑦^∗_𝑖𝑧^∗𝐹_𝑖𝐺^ℎ3₀𝑔(𝜏)𝑒^−𝑛𝜏𝑦_𝑖(𝑡−𝜏)𝑣^∗𝑦^∗_𝑖𝑣+𝑑𝜏𝑝𝑦^∗_𝑖𝑧^∗𝐹_𝑖−𝑝𝑦^∗_𝑖𝑧𝐹_𝑖+𝑝𝑦^∗_𝑖𝑧^∗𝐹_𝑖. (60) Then collecting terms of (60) 𝑑𝑊₂=𝑑𝑡²_𝑖=1𝛾_𝑖𝑑_𝑖𝑥^∗_𝑖𝑥2−^∗𝑖𝑥_𝑖−𝑥_𝑖𝑥^∗_𝑖+𝛽_𝑖𝑥^∗_𝑖𝑣^∗𝑥1−^∗i𝑥_𝑖+𝛽_𝑖𝑥^∗_𝑖𝑣^∗𝑣𝑣^∗−1𝐹_𝑖𝛽_𝑖𝑥^∗_𝑖𝑣^∗ℎ𝑖₀𝑓_𝑖(𝜏)𝑒^{−𝑚𝑖𝜏}×𝑦^∗_𝑖𝑥_𝑖(𝑡−𝜏)𝑣(𝑡−𝜏)𝑦_𝑖𝑥^∗_𝑖𝑣^∗+2𝑑𝜏𝐹_𝑖𝑎𝑦^∗_𝑖+𝑝𝑦^∗_𝑖𝑧^∗+1𝐹_𝑖𝛽_𝑖𝑥^∗_𝑖𝑣^∗ℎ𝑖₀𝑓_𝑖(𝜏)𝑒^{−𝑚𝑖𝜏}𝑥×ln^𝑖(𝑡−𝜏)𝑣(𝑡−𝜏)𝑥_𝑖𝑣+1𝑑𝜏𝐹_𝑖𝐺𝑎𝑦^∗_𝑖+𝑝𝑦^∗_𝑖𝑧^∗ℎ3₀𝑔(𝜏)𝑒^−𝑛𝜏𝑦×ln^𝑖(𝑡−𝜏)𝑦_𝑖−1𝑑𝜏𝐹_𝑖𝐺𝑎𝑦^∗_𝑖+𝑝𝑦^∗_𝑖𝑧^∗ℎ3₀𝑔(𝜏)𝑒^−𝑛𝜏×𝑦_𝑖(𝑡−𝜏)𝑣^∗𝑦^∗_𝑖𝑣−1𝑑𝜏𝐹_𝑖𝑎𝑦^∗_𝑖+𝑝𝑦^∗_𝑖𝑧^∗𝑣𝑣^∗. (61) Using the following equalities: 𝛽_𝑖𝑥^∗_𝑖𝑣^∗=1𝐹_𝑖𝑎𝑦^∗_𝑖+𝑝𝑦^∗_𝑖𝑧^∗,𝑥ln^𝑖(𝑡−𝜏)𝑣(𝑡−𝜏)𝑥_𝑖𝑣𝑦=ln^∗𝑖𝑥_𝑖(𝑡−𝜏)𝑣(𝑡−𝜏)𝑦_𝑖𝑥^∗_𝑖𝑣^∗𝑥+ln^∗𝑖𝑥_𝑖𝑣+ln^∗𝑦_𝑖𝑣𝑦^∗_𝑖𝑦,𝑖=1,2,ln^𝑖(𝑡−𝜏)𝑦_𝑖=ln𝑣𝑦^∗_𝑖𝑣^∗𝑦_𝑖𝑣+ln^∗𝑦_𝑖(𝑡−𝜏)𝑣𝑦^∗_𝑖,𝑖=1,2, (62) we obtain 𝑑𝑊₂=𝑑𝑡²_𝑖=1𝛾_𝑖𝑑_𝑖𝑥^∗_𝑖𝑥2−^∗𝑖𝑥_𝑖−𝑥_𝑖𝑥^∗_𝑖+𝛽_𝑖𝑥^∗_𝑖𝑣^∗×𝑥1−^∗𝑖𝑥_𝑖+2𝛽_𝑖𝑥^∗_𝑖𝑣^∗−1𝐹_𝑖𝛽_𝑖𝑥^∗_𝑖𝑣^∗ℎ𝑖₀𝑓_𝑖(𝜏)𝑒^{−𝑚𝑖𝜏}×𝑦^∗_𝑖𝑥_𝑖(𝑡−𝜏)𝑣(𝑡−𝜏)𝑦_𝑖𝑥^∗_𝑖𝑣^∗+1𝑑𝜏𝐹_𝑖𝛽_𝑖𝑥^∗_𝑖𝑣^∗ℎ𝑖₀𝑓_𝑖(𝜏)𝑒^{−𝑚𝑖𝜏}×𝑦ln^∗𝑖𝑥_𝑖(𝑡−𝜏)𝑣(𝑡−𝜏)𝑦_𝑖𝑥^∗_𝑖𝑣^∗𝑥+ln^∗𝑖𝑥_𝑖𝑣+ln^∗𝑦_𝑖𝑣𝑦^∗_𝑖+1𝑑𝜏𝐺𝛽_𝑖𝑥^∗_𝑖𝑣^∗ℎ3₀𝑔(𝜏)𝑒^−𝑛𝜏×ln𝑣𝑦^∗_𝑖𝑣^∗𝑦_𝑖𝑣+ln^∗𝑦_𝑖(𝑡−𝜏)𝑣𝑦^∗_𝑖−1𝑑𝜏𝐺𝛽_𝑖𝑥^∗_𝑖𝑣^∗ℎ3₀𝑔(𝜏)𝑒^−𝑛𝜏𝑣^∗𝑦_𝑖(𝑡−𝜏)𝑣𝑦^∗_𝑖𝑑𝜏. (63) Equation (63) can be rewritten as 𝑑𝑊₂=𝑑𝑡²_𝑖=1𝛾_𝑖𝑑_𝑖𝑥^∗_𝑖𝑥2−^∗𝑖𝑥_𝑖−𝑥_𝑖𝑥^∗_𝑖−𝛽_𝑖𝑥^∗_𝑖𝑣^∗𝑥^∗_𝑖𝑥_𝑖𝑥−1−ln^∗𝑖𝑥_𝑖−1𝐹_𝑖𝛽_𝑖𝑥^∗_𝑖𝑣^∗ℎ𝑖₀𝑓_𝑖(𝜏)𝑒^{−𝑚𝑖𝜏}×𝑦^∗_𝑖𝑥_𝑖(𝑡−𝜏)𝑣(𝑡−𝜏)𝑦_𝑖𝑥^∗_𝑖𝑣^∗𝑦−1−ln^∗𝑖𝑥_𝑖(𝑡−𝜏)𝑣(𝑡−𝜏)𝑦_𝑖𝑥^∗_𝑖𝑣^∗−1𝑑𝜏𝐺𝛽_𝑖𝑥^∗_𝑖𝑣^∗ℎ3₀𝑔(𝜏)𝑒^−𝑛𝜏×𝑣^∗𝑦_𝑖(𝑡−𝜏)𝑣𝑦^∗_𝑖𝑣−1−ln^∗𝑦_𝑖(𝑡−𝜏)𝑣𝑦^∗_𝑖𝑑𝜏=−²_𝑖=1𝛾_𝑖𝑑_𝑖𝑥_𝑖−𝑥^∗_𝑖²𝑥_𝑖+𝛽_𝑖𝑥^∗_𝑖𝑣^∗𝐻𝑥^∗_𝑖𝑥_𝑖+1𝐹_𝑖𝛽_𝑖𝑥^∗_𝑖𝑣^∗ℎ𝑖₀𝑓_𝑖(𝜏)𝑒^{−𝑚𝑖𝜏}𝑦×𝐻^∗𝑖𝑥_𝑖(𝑡−𝜏)𝑣(𝑡−𝜏)𝑦_𝑖𝑥^∗_𝑖𝑣^∗+1𝑑𝜏𝐺𝛽_𝑖𝑥^∗_𝑖𝑣^∗×^ℎ3₀𝑔(𝜏)𝑒^−𝑛𝜏𝐻𝑣^∗𝑦_𝑖(𝑡−𝜏)𝑣𝑦^∗_𝑖𝑑𝜏. (64) It can be easily seen that, if𝑥^∗_𝑖,𝑦^∗_𝑖,𝑣^∗>0,  𝑖=1,2, then𝑑𝑊₂/𝑑𝑡≤0. By Theorem5.3.1in [26], the solutions of system (5)–(8) are limited to𝑀, the largest invariant subset of{𝑑𝑊₂/𝑑𝑡=0}. It can be seen that𝑑𝑊₂/𝑑𝑡=0if and only if𝑥_𝑖=𝑥^∗_𝑖and𝐻=0; that is, 𝑦^∗_𝑖𝑥_𝑖(𝑡−𝜏)𝑣(𝑡−𝜏)𝑦_𝑖𝑥^∗_𝑖𝑣^∗=𝑣^∗𝑦_𝑖(𝑡−𝜏)𝑣𝑦^∗_𝑖[].=1foralmostall𝜏∈0,ℎ (65) If𝑥_𝑖=𝑥^∗_𝑖, then from (19) we have𝑣=𝑣^∗and from (65) we have𝑦=𝑦^∗.Moreover, from (20) we have 0=𝛽_𝑖𝐹_𝑖𝑥^∗_𝑖𝑣^∗−𝑎𝑦^∗_𝑖−𝑝𝑦^∗_𝑖𝛽𝑧⟹𝑧=^𝑖𝐹_𝑖𝑥^∗_𝑖𝑣^∗𝑝𝑦^∗_𝑖−𝑎𝑝=𝑧^∗. (66) Hence𝑑𝑊₂/𝑑𝑡=0at𝐸₂. LaSalle's Invariance Principle implies the global stability of𝐸₂.

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)if𝑅₀≤1, then the uninfected steady state𝐸₀is GAS;(ii)if𝑅₀>1≥𝑅^∗₀, then the infected steady state without CTL immune response𝐸₁is GAS;(iii)if𝑅₀>𝑅^∗₀>1, then the infected steady state with CTL immune response𝐸₂is GAS.

Conflict of Interests

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

Acknowledgments

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