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

Authors: A. M. Elaiw; R. M. Abukwaik; E. O. Alzahrani
Journal: Computational and Mathematical Methods in Medicine
 (2013)

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## 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 R0∗. We have proven that, if R0≤1, then the uninfected steady state is globally asymptotically stable (GAS), if R0*≤1<R0, then the infected steady state without CTL immune response is GAS, and, if R0*>1, then the infected steady state with CTL immune response is GAS.


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## Body

## 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
(1)x˙(t)=λ-dx(t)-βx(t)v(t),(2)y˙(t)=βx(t)v(t)-ay(t)-py(t)z(t),(3)v˙(t)=ky(t)-rv(t),(4)z˙(t)=cy(t)z(t)-bz(t).
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 numberR0*. We have shown that, ifR0≤1, then the uninfected steady state is globally asymptotically stable (GAS), ifR0*≤1<R0, then the infected steady state without CTL immune response is GAS, and, ifR0*>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
(5)x˙i(t)=λi-dixi(t)-βixi(t)v(t),i=1,2,(6)y˙i(t)=βi∫0hifi(τ)e-miτxi(t-τ)v(t-τ)dτ-ayi(t)-pyi(t)z(t),i=1,2,(7)v˙(t)=k∫0h3g(τ)e-nτ∑i=12‍yi(t-τ)dτ-rv(t),(8)z˙(t)=c∑i=12‍yi(t)z(t)-bz(t).
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 distributionfi(τ)over the interval[0,hi], andhiis limit superior to this delay. The factore-miτaccounts for the loss of target cells during this delay period, wheremiis 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 functionsfi(τ),  i=1,2, andg(τ)are assumed to satisfyfi(τ)>0, andg(τ)>0, and
(9)∫0hifi(τ)dτ=1,∫0hifi(r)esrdr<∞,i=1,2,∫0h3g(τ)dτ=1,∫0h3g(r)esrdr<∞,
wheresis a positive constant. Let
(10)Fi=∫0hifi(τ)e-miτdτ,i=1,2,G=∫0h3g(τ)e-nτdτ.
Then
(11)0<Fi≤1,0<G≤1,i=1,2.
The initial conditions for system (5)–(8) take the form
(12)x1(θ)=φ1(θ),y1(θ)=φ2(θ),x2(θ)=φ3(θ),y2(θ)=φ4(θ),v(θ)=φ5(θ),z(θ)=φ6(θ),φj(θ)≥0,θ∈[-h,0),j=1,…,6,φj(0)>0,j=1,…,6,
whereh=max{h1,h2,h3}and(φ1(θ),φ2(θ),…,φ6(θ))∈C([-h,0],R+6),whereCis the Banach space of continuous functions mapping the interval[-h,0]intoR+6. 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 thatxi(t)>0,  i=1,2,for allt≥0.Assume thatxi(t)loses its nonnegativity on some local existence interval[0,ρ]for some constantρand letti*∈[0,ρ]be such thatxi(ti*)=0.From (5) we havex˙i(ti*)=λi>0.Hencexi(t)>0for somet∈(ti*,ti*+ϵ),whereϵ>0is sufficiently small. This leads to a contradiction and hencexi(t)>0,for allt≥0.Further, from (6) and (7) we have
(13)yi(t)=yi(0)e-∫0t[a+pz(ξ)]dξ+βi∫0te-∫ηt[a+pz(ξ)]dξ×∫0hifi(τ)e-miτxi(η-τ)v(η-τ)dτdη,i=1,2,v(t)=v(0)e-rt+k∫0te-r(t-η)∫0h3g(τ)e-nτ∑i=12‍yi(η-τ)dτdη,
confirming thatyi(t)≥0,  i=1,2,andv(t)≥0for allt∈[0,h]. By a recursive argument, we obtainyi(t)≥0,  i=1,2, andv(t)≥0for allt≥0.Now from (8) we have
(14)z(t)=z(0)e-bt+c∑i=12∫0tyi(η)dη.
Thenz(t)≥0,for allt≥0.
Next we show the boundedness of the solutions of system (5)–(8). From (5) we havex˙i(t)≤λi-dixi(t),  i=1,2. This implieslimsupt→∞xi(t)≤λi/di,  i=1,2.
LetX(t)=∑i=12[Fixi(t)+yi(t+τ)]+(p/c)z(t+τ). Then
(15)X˙(t)=∑i=12‍[Fiλi-Fidixi(t)-Fiβixi(t)v(t)+Fiβixi(t)v(t)-ayi(t+τ)-pyi(t+τ)z(t+τ)]+p∑i=12‍yi(t+τ)z(t+τ)-pcbz(t+τ)=∑i=12‍[Fiλi-Fidixi(t)-ayi(t+τ)]-pcbz(t+τ)≤∑i=12‍Fiλi-σ(∑i=12‍[Fixi(t)+yi(t+τ)]+pcz(t+τ))=∑i=12‍Fiλi-σX(t),
whereσ=min{d1,d2,a,b}. Hencelimsupt→∞X(t)≤L, whereL=∑i=12Li=∑i=12(Fiλi/σ). Sincexi(t)>0,yi(t)≥0andz(t)≥0 thenlimsupt→∞∑i=12yi(t)≤L,limsupt→∞yi(t)≤L,  i=1,2andlimsupt→∞z(t)≤cL/p. On the other hand,
(16)v˙(t)≤k∫0h3g(τ)e-nτ(L)dτ-rv=kGL-rv.
Thenlimsupt→∞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 numberR0*of system (5)–(8) as
(17)R0=∑i=12‍R0i=∑i=12‍kGFiβiλiardi,R0*=∑i=12‍FiβiλikGca(dirc+βikGb).
We can rewriteR0*as
(18)R0*=∑i=12‍R0i*=∑i=12‍IiR0iIi+R0i/Fi,Ii=cλiab,
whereI1andI2are the immune strengths of CD4+ T cells and macrophages, respectively. ClearlyR0>R0*.Lemma 2.
(i) IfR0≤1, then there exists only one uninfected steady stateE0=(x10,0,x20,0,0,0).
(ii) IfR0>1, then there existE0and an infected steady state without CTL immune responseE1=(x~1,y~1,x~2,y~2,v~,0).
(iii) IfR0*>1, then there existE0,E1, and an infected steady state with CTL immune responseE2=(x1*,y1*,x2*,y2*,v*,z*).Proof .
The steady states of (5)–(8) satisfy the following equations:
(19)λi-dixi-βixiv=0,(20)Fiβixiv-ayi-pyiz=0,(21)∑i=12‍kGyi-rv=0,(22)∑i=12‍cyiz-bz=0.
From (22) we have
(23)(∑i=12‍cyi-b)z=0.
Equation (23) has two possible solutions,z=0or
(24)∑i=12‍cyi-b=0.
Ifz=0,then from (19) and (20) we obtainxiandyias
(25)xi=λidi+βiv,yi=Fiβiλia(di+βiv)v,
and inserting them into (21) we obtain
(26)(∑i=12‍kGFiβiλia(di+βiv)-r)v=0.
Equation (26) has two possible solutionsv=0or∑i=12(kGFiβiλi/a(di+βiv))-r=0.
Ifv=0, then substituting it in (25) leads to an uninfected steady stateE0=(x10,0,x20,0,0,0), wherexi0=λi/di,i=1,2. Ifv≠0,then we have
(27)∑i=12‍kGFiβiλiar(di+βiv)-1=0.
Now we can write (27) as
(28)R011+δ1v+R021+δ2v-1=0,⟹δ1δ2v2+[δ1R01+δ2R02+(1-R0)(δ1+δ2)]v+(1-R0)=0,
whereδi=βi/di, i=1,2.The solution of (28) is given by
(29)v±=([δ1R01+δ2R02+(1-R0)(δ1+δ2)]2-4δ1δ2(1-R0)-[δ1R01+δ2R02+(1-R0)(δ1+δ2)]±([δ1R01+δ2R02+(1-R0)(δ1+δ2)]2-4δ1δ2(1-R0))1/2)×(2δ1δ2)-1.
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=(x~1,y~1,x~2,y~2,v~,0),wherev~=v+and
(30)x~i=λidi+βiv~,y~i=Fiβiλia(di+βiv~)v~,i=1,2.
Ifz≠0, then from (24) we have
(31)∑i=12‍yi*=bc,
and inserting it into (21) we obtain
(32)v*=kGbrc.
From (32) to (19) and (20) we get
(33)xi*=λircdirc+βiGkb,yi*=kGbFiβixi*rc(a+pz*)=kGbFiβiλi(a+pz*)(dirc+βiGkb),
and inserting (31) into (20) we get
(34)⟹z*=ap(∑i=12‍FiβiλikGca(dirc+βikGb)-1)=ap(R0*-1).
We havexi*>0, v*>0and ifR0*>1, thenz*>0andyi*>0. It follows that, ifR0*>1, then there exists an infected steady state with CTL immune responseE2=(x1*,y1*,x2*,y2*,v*,z*).
Hence, ifR0≤1, then there exists only one steady stateE0, ifR0>1, then there exist two steady statesE0andE1, and, ifR0*>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∈{xi,yi,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:
(35)W0=∑i=12‍γi[xi0H(xixi0)+1Fiyi+βiFi∫0hifi(τ)e-miτ∫-τ0xi(t+θ)×v(t+θ)dθdτ+aFiG∫0h3g(τ)e-nτ∫-τ0yi(t+θ)dθdτ]+v+kGpacz,
whereγi=kGFi/a,i=1,2.
The time derivative ofW0along the trajectories of (5)–(8) satisfies
(36)dW0dt=∑i=12‍γi[(1-xi0xi)(λi-dixi-βixiv)+βiFi∫0hifi(τ)e-miτxi(t-τ)v(t-τ)dτ-aFiyi-pFiyiz+βiFi∫0hifi(τ)e-miτ×(xiv-xi(t-τ)v(t-τ))dτ+aFiG∫0h3g(τ)e-nτ(yi-yi(t-τ))dτ]+∑i=12‍k∫0h3g(τ)e-nτyi(t-τ)dτ-rv+∑i=12‍kGpayiz-kGpbacz.
Collecting terms of (36) we get
(37)dW0dt=∑i=12‍γi(λi-dixi-λixi0xi+dixi0+βixi0v)-rv-kGpbacz=∑i=12‍γiλi(2-xixi0-xi0xi)-kGpbacz-rv+rv∑i=12‍kGFiβixi0ar=-∑i=12‍γidi(xi-xi0)2xi-kGpbacz-rv+rv∑i=12‍R0i=-∑i=12‍γidi(xi-xi0)2xi-kGpbacz+(R0-1)rv.
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 ifxi=xi0,i=1,2,v=0, andz=0. Noting thatMis invariant, for each element ofM, we havev=0; thenv˙=0. From (7) we drive that
(38)k∫0h3g(τ)e-nτ∑i=12‍yi(t-τ)dτ=0.
Hence, this yields∑i=12yi(t-τ)=0. Sinceyi≥0fori=1,2, theny1=y2=0. HencedW0/dt=0if and only ifxi=xi0,yi=0,i=1,2,  v=0, andz=0. From LaSalle's Invariance Principle,E0is GAS.Theorem 4.
IfR0>1≥R0*, thenE1is GAS.Proof.
We construct the following Lyapunov functional:(39)W1=∑i=12‍γi[x~iH(xix~i)+1Fiy~iH(yiy~i)+1Fiβix~iv~×∫0hifi(τ)e-miτ∫-τ0H(1Fi((xi(t+θ))×(v(t+θ)))×(x~iv~)-11Fi)dθdτ+ay~iFiG∫0h3g(τ)e-nτ∫-τ0H(yi(t+θ)y~i)dθdτ]+v~H(vv~)+kGpacz.
Differentiating with respect to time yields
(40)dW1dt=∑i=12‍γi[(1-x~ixi)(λi-dixi-βixiv)+1Fi(1-y~iyi)×(βi∫0hifi(τ)e-miτxi(t-τ)v(t-τ)dτ-ayi-pyiz∫hi)+βiFi∫0hifi(τ)e-miτ×((xi(t-τ)v(t-τ)xiv)xiv-xi(t-τ)v(t-τ)+x~iv~ln(xi(t-τ)v(t-τ)xiv))dτ+aFiG∫0h3g(τ)e-nτ((yi(t-τ)yi)yi-yi(t-τ)+y~iln(yi(t-τ)yi))dτ]+(1-v~v)(∑i=12‍k∫0h3g(τ)e-nτyi(t-τ)dτ-rv)+kGpac(∑i=12‍cyiz-bz).
Collecting terms we obtain
(41)dW1dt=∑i=12‍γi[∫(yi(t-τ)yi)λi-dixi-λix~ixi+dix~i+βix~iv-βiy~iFiyi∫0hifi(τ)e-miτxi(t-τ)v(t-τ)dτ+aFiy~i+pFiy~iz+1Fiβix~iv~∫0hifi(τ)e-miτ×ln(xi(t-τ)v(t-τ)xiv)dτ+ay~iFiG∫0h3g(τ)e-nτln(yi(t-τ)yi)dτ]-rv-v~v∑i=12‍k∫0h3g(τ)e-nτyi(t-τ)dτ+rv~-kGpbacz.
From (19)–(21) we have
(42)λi=dix~i+βix~iv~,Fiβix~iv~=ay~i,rv~=∑i=12‍Gky~i.
Using (42) and the following equality:
(43)rv=rv~vv~=vv~∑i=12‍Gky~i=vv~∑i=12‍γiaFiy~i=vv~∑i=12‍γiβix~iv~,
we obtain
(44)dW1dt=∑i=12‍γi[∫0hifi(τ)e-miτln(xi(t-τ)v(t-τ)xiv)dix~i+βix~iv~-dixi-x~ixi(dix~i+βix~iv~)+dix~i+βix~iv~vv~-1Fiβix~iv~∫0hifi(τ)e-miτy~ixi(t-τ)v(t-τ)yix~iv~dτ+βix~iv~+pFiy~iz+1Fiβix~iv~∫0hifi(τ)e-miτ×ln(xi(t-τ)v(t-τ)xiv)dτ+1Gβix~iv~i∫0h3g(τ)e-nτln(yi(t-τ)yi)dτ-βix~iv~vv~-1Gβix~iv~∫0h3g(τ)e-nτv~yi(t-τ)vy~idτ+βix~iv~∫0hifi(τ)e-miτln(xi(t-τ)v(t-τ)xiv)]-kGpbacz.
Then collecting terms of (44) and using the following equalities:
(45)ln(xi(t-τ)v(t-τ)xiv)=ln(y~ixi(t-τ)v(t-τ)yix~iv~)+ln(x~ixi)+ln(v~yivy~i),i=1,2,ln(yi(t-τ)yi)=ln(vy~iv~yi)+ln(v~yi(t-τ)vy~i),i=1,2,
we obtain
(46)dW1dt=∑i=12‍γi[∫0h3fi(τ)e-miτ(ln(y~ixi(t-τ)v(t-τ)yix~iv~)dix~i(2-x~ixi-xix~i)+βix~iv~(1-x~ixi)+2βix~iv~-1Fiβix~iv~∫0hifi(τ)e-miτ×y~ixi(t-τ)v(t-τ)yix~iv~dτ+1Fiβix~iv~∫0hifi(τ)e-miτ×(ln(y~ixi(t-τ)v(t-τ)yix~iv~)+ln(x~ixi)+ln(v~yivy~i))dτ+1Gβix~iv~∫0h3g(τ)e-nτ×(ln(vy~iv~yi)+ln(v~yi(t-τ)vy~i))dτ-1Gβix~iv~∫0h3g(τ)e-nτv~yi(t-τ)vy~idτ∫0hifi(τ)e-miτ(ln(y~ixi(t-τ)v(t-τ)yix~iv~)]+∑i=12‍kGpay~iz-kGpbacz.
Equation (46) can be rewritten as
(47)dW1dt=∑i=12‍γi[∫((v~yi(t-τ)vy~i)dix~i(2-x~ixi-xix~i)-βix~iv~(x~ixi-1-ln(x~ixi))-1Fiβix~iv~∫0hifi(τ)e-miτ×((v~yi(t-τ)vy~i)y~ixi(t-τ)v(t-τ)yix~iv~-1-ln(y~ixi(t-τ)v(t-τ)yix~iv~))dτ-1Gβix~iv~∫0h3g(τ)e-nτ((v~yi(t-τ)vy~i)v~yi(t-τ)vy~i-1-ln(v~yi(t-τ)vy~i))dτ]+kGpa(∑i=12‍y~i-bc)z.
We can rewritedW1/dtas:
(48)dW1dt=-∑i=12γi[∫di(xi-x~i)2xi+βix~iv~H(x~ixi)+1Fiβix~iv~×∫0hifi(τ)e-miτH(1Fi(1G(y~ixi(t-τ))×(v(t-τ))1G)×(yix~iv~)-11Fi)dτ+1Gβix~iv~∫0h3g(τ)e-nτH((v~yi(t-τ)vy~i)v~yi(t-τ)vy~i)dτ]+kGpa(∑i=12‍y~i-bc)z.
Now we show that, ifR0*≤1, then∑i=12y~i<b/c. Assume thatR0*≤1; then
(49)∑i=12‍kGcFiβiλiarcdi+aβiGkb≤1.
From (27) we have
(50)∑i=12‍kGFiβiλiardi+aβirv~=1,⟹∑i=12‍kGcFiβiλiarcdi+aβicrv~=1.
Comparing (49) and (50) we get
(51)∑i=12‍kGcFiβiλiarcdi+aβiGkb≤∑i=12‍kGcFiβiλiarcdi+aβicrv~.
Then
(52)v~≤Gkbrc.
Using (42) we have
(53)Gkr∑i=12‍yi~=v~≤Gkbrc.
Then
(54)∑i=12‍yi~≤bc.
Now ifR0>1, thenx~i,y~i,v~>0.It follows that, ifR0>1≥R0*, 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 ifxi=x~i,z=0, andH=0: that is,
(55)y~ixi(t-τ)v(t-τ)yix~iv~=v~yi(t-τ)vy~i=1foralmostallτ∈[0,h].
Ifxi=x~i, then from (19) we havev=v~and from (55) we haveyi=y~i. It follows thatdW1/dt=0atE1. LaSalle's Invariance Principle implies the global stability ofE1.Theorem 5.
IfR0>R0*>1, thenE2is GAS.Proof .
We construct the following Lyapunov functional:(56)W2=∑i=12‍γi[xi*H(xixi*)+1Fiyi*H(yiyi*)+1Fiβixi*v*×∫0hifi(τ)e-miτ∫-τ0H(1G(1G(xi(t+θ))×(v(t+θ))1G)×(xi*v*)-11G1G1Fi)dθdτ+(ayi*+pyi*z*)FiG∫0h3g(τ)e-nτ×∫-τ0H(yi(t+θ)yi*)dθdτ]+(1+paz*)v*H(vv*)+kGpacz*H(zz*).
Differentiating with respect to time yields
(57)dW2dt=∑i=12‍γi[(1-xi*xi)(λi-dixi-βixiv)+1Fi(1-yi*yi)(βi∫0hifi(τ)e-miτxi(t-τ)×v(t-τ)dτ-ayi-pyiz∫0hi)+βiFi∫0hifi(τ)e-miτ×((xi(t-τ)v(t-τ)xiv)xiv-xi(t-τ)v(t-τ)+xi*v*ln(xi(t-τ)v(t-τ)xiv))dτ+aFiG∫0h3g(τ)e-nτ×(yi-yi(t-τ)+yi*ln(yi(t-τ)yi))dτ+pFiG∫0h3g(τ)e-nτ×((yi(t-τ)yi)yiz*-yi(t-τ)z*+yi*z*ln(yi(t-τ)yi))dτ]+(1+paz*)(1-v*v)×(∑i=12‍k∫0h3g(τ)e-nτyi(t-τ)dτ-rv)+kGpac(1-z*z)(∑i=12‍cyiz-bz).
Collecting terms we obtain
(58)dW2dt=∑i=12‍γi[λi-dixi-λixi*xi+dixi*+βixi*v-βiyi*Fiyi∫0hifi(τ)e-miτxi(t-τ)v(t-τ)dτ+aFiyi*+pFiyi*z+1Fiβixi*v*×∫0hifi(τ)e-miτln(xi(t-τ)v(t-τ)xiv)dτ+ayi*FiG∫0h3g(τ)e-nτln(yi(t-τ)yi)dτ+pyi*z*FiG∫0h3g(τ)e-nτln(yi(t-τ)yi)dτ]-rv-v*v∑i=12‍k∫0h3g(τ)e-nτyi(t-τ)dτ+rv*-pz*arv-v*v∑i=12‍ka∫0h3g(τ)e-nτpyi(t-τ)z*dτ+pz*arv*-kGpbacz+kGpbacz*.
Using (19)–(21), we have equalities
(59)rv=rv*(vv*)=vv*∑i=12‍Gkyi*=vv*∑i=12‍γiFiayi*,pz*arv*=∑i=12‍kGapyi*z*=∑i=12‍γiFipyi*z*,pz*arv=pz*arv*(vv*)=vv*∑i=12‍γiFipyi*z*,kGpbacz*=∑i=12‍kGapyi*z*=∑i=12‍γiFipyi*z*,since∑i=12‍yi*=bc,kGpbacz=kGpbacz*(zz*)=∑i=12‍γiFipyi*z.
We obtain
(60)dW2dt=∑i=12‍γi[dixi*+βixi*v*-dixi-xi*xi(dixi*+βixi*v*)+dixi*+βixi*v*vv*-1Fiβixi*v*×∫0hifi(τ)e-miτyi*xi(t-τ)v(t-τ)yixi*v*dτ+ayi*Fi+pyi*zFi+1Fiβixi*v*∫0hifi(τ)e-miτ×ln(xi(t-τ)v(t-τ)xiv)dτ+ayi*FiG∫0h3g(τ)e-nτln(yi(t-τ)yi)dτ+pyi*z*FiG∫0h3g(τ)e-nτln(yi(t-τ)yi)dτ-ayi*Fivv*-ayi*FiG∫0h3g(τ)e-nτ×yi(t-τ)v*yi*vdτ+ayi*Fi-pyi*z*Fivv*-pyi*z*FiG∫0h3g(τ)e-nτyi(t-τ)v*yi*vdτ+pyi*z*Fi-pyi*zFi+pyi*z*Fi].
Then collecting terms of (60)
(61)dW2dt=∑i=12‍γi[dixi*(2-xi*xi-xixi*)+βixi*v*(1-xi*xi)+βixi*v*vv*-1Fiβixi*v*∫0hifi(τ)e-miτ×yi*xi(t-τ)v(t-τ)yixi*v*dτ+2Fi(ayi*+pyi*z*)+1Fiβixi*v*∫0hifi(τ)e-miτ×ln(xi(t-τ)v(t-τ)xiv)dτ+1FiG(ayi*+pyi*z*)∫0h3g(τ)e-nτ×ln(yi(t-τ)yi)dτ-1FiG(ayi*+pyi*z*)∫0h3g(τ)e-nτ×yi(t-τ)v*yi*vdτ-1Fi(ayi*+pyi*z*)vv*].
Using the following equalities:
(62)βixi*v*=1Fi(ayi*+pyi*z*),ln(xi(t-τ)v(t-τ)xiv)=ln(yi*xi(t-τ)v(t-τ)yixi*v*)+ln(xi*xi)+ln(v*yivyi*),i=1,2,ln(yi(t-τ)yi)=ln(vyi*v*yi)+ln(v*yi(t-τ)vyi*),i=1,2,
we obtain
(63)dW2dt=∑i=12‍γi[dixi*(2-xi*xi-xixi*)+βixi*v*×(1-xi*xi)+2βixi*v*-1Fiβixi*v*∫0hifi(τ)e-miτ×yi*xi(t-τ)v(t-τ)yixi*v*dτ+1Fiβixi*v*∫0hifi(τ)e-miτ×(ln(yi*xi(t-τ)v(t-τ)yixi*v*)+ln(xi*xi)+ln(v*yivyi*))dτ+1Gβixi*v*∫0h3g(τ)e-nτ×(ln(vyi*v*yi)+ln(v*yi(t-τ)vyi*))dτ-1Gβixi*v*∫0h3g(τ)e-nτv*yi(t-τ)vyi*dτ].
Equation (63) can be rewritten as
(64)dW2dt=∑i=12‍γi[dixi*(2-xi*xi-xixi*)-βixi*v*(xi*xi-1-ln(xi*xi))-1Fiβixi*v*∫0hifi(τ)e-miτ×(yi*xi(t-τ)v(t-τ)yixi*v*-1-ln(yi*xi(t-τ)v(t-τ)yixi*v*))dτ-1Gβixi*v*∫0h3g(τ)e-nτ×(v*yi(t-τ)vyi*-1-ln(v*yi(t-τ)vyi*))dτ]=-∑i=12‍γi[di(xi-xi*)2xi+βixi*v*H(xi*xi)+1Fiβixi*v*∫0hifi(τ)e-miτ×H(yi*xi(t-τ)v(t-τ)yixi*v*)dτ+1Gβixi*v*×∫0h3g(τ)e-nτH(v*yi(t-τ)vyi*)dτ].
It can be easily seen that, ifxi*,yi*,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 ifxi=xi*andH=0; that is,
(65)yi*xi(t-τ)v(t-τ)yixi*v*=v*yi(t-τ)vyi*=1foralmostallτ∈[0,h].
Ifxi=xi*, then from (19) we havev=v*and from (65) we havey=y*.Moreover, from (20) we have
(66)0=βiFixi*v*-ayi*-pyi*z⟹z=βiFixi*v*pyi*-ap=z*.
HencedW2/dt=0atE2. LaSalle's Invariance Principle implies the global stability ofE2.


## 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
(5)x˙i(t)=λi-dixi(t)-βixi(t)v(t),i=1,2,(6)y˙i(t)=βi∫0hifi(τ)e-miτxi(t-τ)v(t-τ)dτ-ayi(t)-pyi(t)z(t),i=1,2,(7)v˙(t)=k∫0h3g(τ)e-nτ∑i=12‍yi(t-τ)dτ-rv(t),(8)z˙(t)=c∑i=12‍yi(t)z(t)-bz(t).
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 distributionfi(τ)over the interval[0,hi], andhiis limit superior to this delay. The factore-miτaccounts for the loss of target cells during this delay period, wheremiis 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 functionsfi(τ),  i=1,2, andg(τ)are assumed to satisfyfi(τ)>0, andg(τ)>0, and
(9)∫0hifi(τ)dτ=1,∫0hifi(r)esrdr<∞,i=1,2,∫0h3g(τ)dτ=1,∫0h3g(r)esrdr<∞,
wheresis a positive constant. Let
(10)Fi=∫0hifi(τ)e-miτdτ,i=1,2,G=∫0h3g(τ)e-nτdτ.
Then
(11)0<Fi≤1,0<G≤1,i=1,2.
The initial conditions for system (5)–(8) take the form
(12)x1(θ)=φ1(θ),y1(θ)=φ2(θ),x2(θ)=φ3(θ),y2(θ)=φ4(θ),v(θ)=φ5(θ),z(θ)=φ6(θ),φj(θ)≥0,θ∈[-h,0),j=1,…,6,φj(0)>0,j=1,…,6,
whereh=max{h1,h2,h3}and(φ1(θ),φ2(θ),…,φ6(θ))∈C([-h,0],R+6),whereCis the Banach space of continuous functions mapping the interval[-h,0]intoR+6. 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 thatxi(t)>0,  i=1,2,for allt≥0.Assume thatxi(t)loses its nonnegativity on some local existence interval[0,ρ]for some constantρand letti*∈[0,ρ]be such thatxi(ti*)=0.From (5) we havex˙i(ti*)=λi>0.Hencexi(t)>0for somet∈(ti*,ti*+ϵ),whereϵ>0is sufficiently small. This leads to a contradiction and hencexi(t)>0,for allt≥0.Further, from (6) and (7) we have
(13)yi(t)=yi(0)e-∫0t[a+pz(ξ)]dξ+βi∫0te-∫ηt[a+pz(ξ)]dξ×∫0hifi(τ)e-miτxi(η-τ)v(η-τ)dτdη,i=1,2,v(t)=v(0)e-rt+k∫0te-r(t-η)∫0h3g(τ)e-nτ∑i=12‍yi(η-τ)dτdη,
confirming thatyi(t)≥0,  i=1,2,andv(t)≥0for allt∈[0,h]. By a recursive argument, we obtainyi(t)≥0,  i=1,2, andv(t)≥0for allt≥0.Now from (8) we have
(14)z(t)=z(0)e-bt+c∑i=12∫0tyi(η)dη.
Thenz(t)≥0,for allt≥0.
Next we show the boundedness of the solutions of system (5)–(8). From (5) we havex˙i(t)≤λi-dixi(t),  i=1,2. This implieslimsupt→∞xi(t)≤λi/di,  i=1,2.
LetX(t)=∑i=12[Fixi(t)+yi(t+τ)]+(p/c)z(t+τ). Then
(15)X˙(t)=∑i=12‍[Fiλi-Fidixi(t)-Fiβixi(t)v(t)+Fiβixi(t)v(t)-ayi(t+τ)-pyi(t+τ)z(t+τ)]+p∑i=12‍yi(t+τ)z(t+τ)-pcbz(t+τ)=∑i=12‍[Fiλi-Fidixi(t)-ayi(t+τ)]-pcbz(t+τ)≤∑i=12‍Fiλi-σ(∑i=12‍[Fixi(t)+yi(t+τ)]+pcz(t+τ))=∑i=12‍Fiλi-σX(t),
whereσ=min{d1,d2,a,b}. Hencelimsupt→∞X(t)≤L, whereL=∑i=12Li=∑i=12(Fiλi/σ). Sincexi(t)>0,yi(t)≥0andz(t)≥0 thenlimsupt→∞∑i=12yi(t)≤L,limsupt→∞yi(t)≤L,  i=1,2andlimsupt→∞z(t)≤cL/p. On the other hand,
(16)v˙(t)≤k∫0h3g(τ)e-nτ(L)dτ-rv=kGL-rv.
Thenlimsupt→∞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 numberR0*of system (5)–(8) as
(17)R0=∑i=12‍R0i=∑i=12‍kGFiβiλiardi,R0*=∑i=12‍FiβiλikGca(dirc+βikGb).
We can rewriteR0*as
(18)R0*=∑i=12‍R0i*=∑i=12‍IiR0iIi+R0i/Fi,Ii=cλiab,
whereI1andI2are the immune strengths of CD4+ T cells and macrophages, respectively. ClearlyR0>R0*.Lemma 2.
(i) IfR0≤1, then there exists only one uninfected steady stateE0=(x10,0,x20,0,0,0).
(ii) IfR0>1, then there existE0and an infected steady state without CTL immune responseE1=(x~1,y~1,x~2,y~2,v~,0).
(iii) IfR0*>1, then there existE0,E1, and an infected steady state with CTL immune responseE2=(x1*,y1*,x2*,y2*,v*,z*).Proof .
The steady states of (5)–(8) satisfy the following equations:
(19)λi-dixi-βixiv=0,(20)Fiβixiv-ayi-pyiz=0,(21)∑i=12‍kGyi-rv=0,(22)∑i=12‍cyiz-bz=0.
From (22) we have
(23)(∑i=12‍cyi-b)z=0.
Equation (23) has two possible solutions,z=0or
(24)∑i=12‍cyi-b=0.
Ifz=0,then from (19) and (20) we obtainxiandyias
(25)xi=λidi+βiv,yi=Fiβiλia(di+βiv)v,
and inserting them into (21) we obtain
(26)(∑i=12‍kGFiβiλia(di+βiv)-r)v=0.
Equation (26) has two possible solutionsv=0or∑i=12(kGFiβiλi/a(di+βiv))-r=0.
Ifv=0, then substituting it in (25) leads to an uninfected steady stateE0=(x10,0,x20,0,0,0), wherexi0=λi/di,i=1,2. Ifv≠0,then we have
(27)∑i=12‍kGFiβiλiar(di+βiv)-1=0.
Now we can write (27) as
(28)R011+δ1v+R021+δ2v-1=0,⟹δ1δ2v2+[δ1R01+δ2R02+(1-R0)(δ1+δ2)]v+(1-R0)=0,
whereδi=βi/di, i=1,2.The solution of (28) is given by
(29)v±=([δ1R01+δ2R02+(1-R0)(δ1+δ2)]2-4δ1δ2(1-R0)-[δ1R01+δ2R02+(1-R0)(δ1+δ2)]±([δ1R01+δ2R02+(1-R0)(δ1+δ2)]2-4δ1δ2(1-R0))1/2)×(2δ1δ2)-1.
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=(x~1,y~1,x~2,y~2,v~,0),wherev~=v+and
(30)x~i=λidi+βiv~,y~i=Fiβiλia(di+βiv~)v~,i=1,2.
Ifz≠0, then from (24) we have
(31)∑i=12‍yi*=bc,
and inserting it into (21) we obtain
(32)v*=kGbrc.
From (32) to (19) and (20) we get
(33)xi*=λircdirc+βiGkb,yi*=kGbFiβixi*rc(a+pz*)=kGbFiβiλi(a+pz*)(dirc+βiGkb),
and inserting (31) into (20) we get
(34)⟹z*=ap(∑i=12‍FiβiλikGca(dirc+βikGb)-1)=ap(R0*-1).
We havexi*>0, v*>0and ifR0*>1, thenz*>0andyi*>0. It follows that, ifR0*>1, then there exists an infected steady state with CTL immune responseE2=(x1*,y1*,x2*,y2*,v*,z*).
Hence, ifR0≤1, then there exists only one steady stateE0, ifR0>1, then there exist two steady statesE0andE1, and, ifR0*>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∈{xi,yi,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:
(35)W0=∑i=12‍γi[xi0H(xixi0)+1Fiyi+βiFi∫0hifi(τ)e-miτ∫-τ0xi(t+θ)×v(t+θ)dθdτ+aFiG∫0h3g(τ)e-nτ∫-τ0yi(t+θ)dθdτ]+v+kGpacz,
whereγi=kGFi/a,i=1,2.
The time derivative ofW0along the trajectories of (5)–(8) satisfies
(36)dW0dt=∑i=12‍γi[(1-xi0xi)(λi-dixi-βixiv)+βiFi∫0hifi(τ)e-miτxi(t-τ)v(t-τ)dτ-aFiyi-pFiyiz+βiFi∫0hifi(τ)e-miτ×(xiv-xi(t-τ)v(t-τ))dτ+aFiG∫0h3g(τ)e-nτ(yi-yi(t-τ))dτ]+∑i=12‍k∫0h3g(τ)e-nτyi(t-τ)dτ-rv+∑i=12‍kGpayiz-kGpbacz.
Collecting terms of (36) we get
(37)dW0dt=∑i=12‍γi(λi-dixi-λixi0xi+dixi0+βixi0v)-rv-kGpbacz=∑i=12‍γiλi(2-xixi0-xi0xi)-kGpbacz-rv+rv∑i=12‍kGFiβixi0ar=-∑i=12‍γidi(xi-xi0)2xi-kGpbacz-rv+rv∑i=12‍R0i=-∑i=12‍γidi(xi-xi0)2xi-kGpbacz+(R0-1)rv.
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 ifxi=xi0,i=1,2,v=0, andz=0. Noting thatMis invariant, for each element ofM, we havev=0; thenv˙=0. From (7) we drive that
(38)k∫0h3g(τ)e-nτ∑i=12‍yi(t-τ)dτ=0.
Hence, this yields∑i=12yi(t-τ)=0. Sinceyi≥0fori=1,2, theny1=y2=0. HencedW0/dt=0if and only ifxi=xi0,yi=0,i=1,2,  v=0, andz=0. From LaSalle's Invariance Principle,E0is GAS.Theorem 4.
IfR0>1≥R0*, thenE1is GAS.Proof.
We construct the following Lyapunov functional:(39)W1=∑i=12‍γi[x~iH(xix~i)+1Fiy~iH(yiy~i)+1Fiβix~iv~×∫0hifi(τ)e-miτ∫-τ0H(1Fi((xi(t+θ))×(v(t+θ)))×(x~iv~)-11Fi)dθdτ+ay~iFiG∫0h3g(τ)e-nτ∫-τ0H(yi(t+θ)y~i)dθdτ]+v~H(vv~)+kGpacz.
Differentiating with respect to time yields
(40)dW1dt=∑i=12‍γi[(1-x~ixi)(λi-dixi-βixiv)+1Fi(1-y~iyi)×(βi∫0hifi(τ)e-miτxi(t-τ)v(t-τ)dτ-ayi-pyiz∫hi)+βiFi∫0hifi(τ)e-miτ×((xi(t-τ)v(t-τ)xiv)xiv-xi(t-τ)v(t-τ)+x~iv~ln(xi(t-τ)v(t-τ)xiv))dτ+aFiG∫0h3g(τ)e-nτ((yi(t-τ)yi)yi-yi(t-τ)+y~iln(yi(t-τ)yi))dτ]+(1-v~v)(∑i=12‍k∫0h3g(τ)e-nτyi(t-τ)dτ-rv)+kGpac(∑i=12‍cyiz-bz).
Collecting terms we obtain
(41)dW1dt=∑i=12‍γi[∫(yi(t-τ)yi)λi-dixi-λix~ixi+dix~i+βix~iv-βiy~iFiyi∫0hifi(τ)e-miτxi(t-τ)v(t-τ)dτ+aFiy~i+pFiy~iz+1Fiβix~iv~∫0hifi(τ)e-miτ×ln(xi(t-τ)v(t-τ)xiv)dτ+ay~iFiG∫0h3g(τ)e-nτln(yi(t-τ)yi)dτ]-rv-v~v∑i=12‍k∫0h3g(τ)e-nτyi(t-τ)dτ+rv~-kGpbacz.
From (19)–(21) we have
(42)λi=dix~i+βix~iv~,Fiβix~iv~=ay~i,rv~=∑i=12‍Gky~i.
Using (42) and the following equality:
(43)rv=rv~vv~=vv~∑i=12‍Gky~i=vv~∑i=12‍γiaFiy~i=vv~∑i=12‍γiβix~iv~,
we obtain
(44)dW1dt=∑i=12‍γi[∫0hifi(τ)e-miτln(xi(t-τ)v(t-τ)xiv)dix~i+βix~iv~-dixi-x~ixi(dix~i+βix~iv~)+dix~i+βix~iv~vv~-1Fiβix~iv~∫0hifi(τ)e-miτy~ixi(t-τ)v(t-τ)yix~iv~dτ+βix~iv~+pFiy~iz+1Fiβix~iv~∫0hifi(τ)e-miτ×ln(xi(t-τ)v(t-τ)xiv)dτ+1Gβix~iv~i∫0h3g(τ)e-nτln(yi(t-τ)yi)dτ-βix~iv~vv~-1Gβix~iv~∫0h3g(τ)e-nτv~yi(t-τ)vy~idτ+βix~iv~∫0hifi(τ)e-miτln(xi(t-τ)v(t-τ)xiv)]-kGpbacz.
Then collecting terms of (44) and using the following equalities:
(45)ln(xi(t-τ)v(t-τ)xiv)=ln(y~ixi(t-τ)v(t-τ)yix~iv~)+ln(x~ixi)+ln(v~yivy~i),i=1,2,ln(yi(t-τ)yi)=ln(vy~iv~yi)+ln(v~yi(t-τ)vy~i),i=1,2,
we obtain
(46)dW1dt=∑i=12‍γi[∫0h3fi(τ)e-miτ(ln(y~ixi(t-τ)v(t-τ)yix~iv~)dix~i(2-x~ixi-xix~i)+βix~iv~(1-x~ixi)+2βix~iv~-1Fiβix~iv~∫0hifi(τ)e-miτ×y~ixi(t-τ)v(t-τ)yix~iv~dτ+1Fiβix~iv~∫0hifi(τ)e-miτ×(ln(y~ixi(t-τ)v(t-τ)yix~iv~)+ln(x~ixi)+ln(v~yivy~i))dτ+1Gβix~iv~∫0h3g(τ)e-nτ×(ln(vy~iv~yi)+ln(v~yi(t-τ)vy~i))dτ-1Gβix~iv~∫0h3g(τ)e-nτv~yi(t-τ)vy~idτ∫0hifi(τ)e-miτ(ln(y~ixi(t-τ)v(t-τ)yix~iv~)]+∑i=12‍kGpay~iz-kGpbacz.
Equation (46) can be rewritten as
(47)dW1dt=∑i=12‍γi[∫((v~yi(t-τ)vy~i)dix~i(2-x~ixi-xix~i)-βix~iv~(x~ixi-1-ln(x~ixi))-1Fiβix~iv~∫0hifi(τ)e-miτ×((v~yi(t-τ)vy~i)y~ixi(t-τ)v(t-τ)yix~iv~-1-ln(y~ixi(t-τ)v(t-τ)yix~iv~))dτ-1Gβix~iv~∫0h3g(τ)e-nτ((v~yi(t-τ)vy~i)v~yi(t-τ)vy~i-1-ln(v~yi(t-τ)vy~i))dτ]+kGpa(∑i=12‍y~i-bc)z.
We can rewritedW1/dtas:
(48)dW1dt=-∑i=12γi[∫di(xi-x~i)2xi+βix~iv~H(x~ixi)+1Fiβix~iv~×∫0hifi(τ)e-miτH(1Fi(1G(y~ixi(t-τ))×(v(t-τ))1G)×(yix~iv~)-11Fi)dτ+1Gβix~iv~∫0h3g(τ)e-nτH((v~yi(t-τ)vy~i)v~yi(t-τ)vy~i)dτ]+kGpa(∑i=12‍y~i-bc)z.
Now we show that, ifR0*≤1, then∑i=12y~i<b/c. Assume thatR0*≤1; then
(49)∑i=12‍kGcFiβiλiarcdi+aβiGkb≤1.
From (27) we have
(50)∑i=12‍kGFiβiλiardi+aβirv~=1,⟹∑i=12‍kGcFiβiλiarcdi+aβicrv~=1.
Comparing (49) and (50) we get
(51)∑i=12‍kGcFiβiλiarcdi+aβiGkb≤∑i=12‍kGcFiβiλiarcdi+aβicrv~.
Then
(52)v~≤Gkbrc.
Using (42) we have
(53)Gkr∑i=12‍yi~=v~≤Gkbrc.
Then
(54)∑i=12‍yi~≤bc.
Now ifR0>1, thenx~i,y~i,v~>0.It follows that, ifR0>1≥R0*, 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 ifxi=x~i,z=0, andH=0: that is,
(55)y~ixi(t-τ)v(t-τ)yix~iv~=v~yi(t-τ)vy~i=1foralmostallτ∈[0,h].
Ifxi=x~i, then from (19) we havev=v~and from (55) we haveyi=y~i. It follows thatdW1/dt=0atE1. LaSalle's Invariance Principle implies the global stability ofE1.Theorem 5.
IfR0>R0*>1, thenE2is GAS.Proof .
We construct the following Lyapunov functional:(56)W2=∑i=12‍γi[xi*H(xixi*)+1Fiyi*H(yiyi*)+1Fiβixi*v*×∫0hifi(τ)e-miτ∫-τ0H(1G(1G(xi(t+θ))×(v(t+θ))1G)×(xi*v*)-11G1G1Fi)dθdτ+(ayi*+pyi*z*)FiG∫0h3g(τ)e-nτ×∫-τ0H(yi(t+θ)yi*)dθdτ]+(1+paz*)v*H(vv*)+kGpacz*H(zz*).
Differentiating with respect to time yields
(57)dW2dt=∑i=12‍γi[(1-xi*xi)(λi-dixi-βixiv)+1Fi(1-yi*yi)(βi∫0hifi(τ)e-miτxi(t-τ)×v(t-τ)dτ-ayi-pyiz∫0hi)+βiFi∫0hifi(τ)e-miτ×((xi(t-τ)v(t-τ)xiv)xiv-xi(t-τ)v(t-τ)+xi*v*ln(xi(t-τ)v(t-τ)xiv))dτ+aFiG∫0h3g(τ)e-nτ×(yi-yi(t-τ)+yi*ln(yi(t-τ)yi))dτ+pFiG∫0h3g(τ)e-nτ×((yi(t-τ)yi)yiz*-yi(t-τ)z*+yi*z*ln(yi(t-τ)yi))dτ]+(1+paz*)(1-v*v)×(∑i=12‍k∫0h3g(τ)e-nτyi(t-τ)dτ-rv)+kGpac(1-z*z)(∑i=12‍cyiz-bz).
Collecting terms we obtain
(58)dW2dt=∑i=12‍γi[λi-dixi-λixi*xi+dixi*+βixi*v-βiyi*Fiyi∫0hifi(τ)e-miτxi(t-τ)v(t-τ)dτ+aFiyi*+pFiyi*z+1Fiβixi*v*×∫0hifi(τ)e-miτln(xi(t-τ)v(t-τ)xiv)dτ+ayi*FiG∫0h3g(τ)e-nτln(yi(t-τ)yi)dτ+pyi*z*FiG∫0h3g(τ)e-nτln(yi(t-τ)yi)dτ]-rv-v*v∑i=12‍k∫0h3g(τ)e-nτyi(t-τ)dτ+rv*-pz*arv-v*v∑i=12‍ka∫0h3g(τ)e-nτpyi(t-τ)z*dτ+pz*arv*-kGpbacz+kGpbacz*.
Using (19)–(21), we have equalities
(59)rv=rv*(vv*)=vv*∑i=12‍Gkyi*=vv*∑i=12‍γiFiayi*,pz*arv*=∑i=12‍kGapyi*z*=∑i=12‍γiFipyi*z*,pz*arv=pz*arv*(vv*)=vv*∑i=12‍γiFipyi*z*,kGpbacz*=∑i=12‍kGapyi*z*=∑i=12‍γiFipyi*z*,since∑i=12‍yi*=bc,kGpbacz=kGpbacz*(zz*)=∑i=12‍γiFipyi*z.
We obtain
(60)dW2dt=∑i=12‍γi[dixi*+βixi*v*-dixi-xi*xi(dixi*+βixi*v*)+dixi*+βixi*v*vv*-1Fiβixi*v*×∫0hifi(τ)e-miτyi*xi(t-τ)v(t-τ)yixi*v*dτ+ayi*Fi+pyi*zFi+1Fiβixi*v*∫0hifi(τ)e-miτ×ln(xi(t-τ)v(t-τ)xiv)dτ+ayi*FiG∫0h3g(τ)e-nτln(yi(t-τ)yi)dτ+pyi*z*FiG∫0h3g(τ)e-nτln(yi(t-τ)yi)dτ-ayi*Fivv*-ayi*FiG∫0h3g(τ)e-nτ×yi(t-τ)v*yi*vdτ+ayi*Fi-pyi*z*Fivv*-pyi*z*FiG∫0h3g(τ)e-nτyi(t-τ)v*yi*vdτ+pyi*z*Fi-pyi*zFi+pyi*z*Fi].
Then collecting terms of (60)
(61)dW2dt=∑i=12‍γi[dixi*(2-xi*xi-xixi*)+βixi*v*(1-xi*xi)+βixi*v*vv*-1Fiβixi*v*∫0hifi(τ)e-miτ×yi*xi(t-τ)v(t-τ)yixi*v*dτ+2Fi(ayi*+pyi*z*)+1Fiβixi*v*∫0hifi(τ)e-miτ×ln(xi(t-τ)v(t-τ)xiv)dτ+1FiG(ayi*+pyi*z*)∫0h3g(τ)e-nτ×ln(yi(t-τ)yi)dτ-1FiG(ayi*+pyi*z*)∫0h3g(τ)e-nτ×yi(t-τ)v*yi*vdτ-1Fi(ayi*+pyi*z*)vv*].
Using the following equalities:
(62)βixi*v*=1Fi(ayi*+pyi*z*),ln(xi(t-τ)v(t-τ)xiv)=ln(yi*xi(t-τ)v(t-τ)yixi*v*)+ln(xi*xi)+ln(v*yivyi*),i=1,2,ln(yi(t-τ)yi)=ln(vyi*v*yi)+ln(v*yi(t-τ)vyi*),i=1,2,
we obtain
(63)dW2dt=∑i=12‍γi[dixi*(2-xi*xi-xixi*)+βixi*v*×(1-xi*xi)+2βixi*v*-1Fiβixi*v*∫0hifi(τ)e-miτ×yi*xi(t-τ)v(t-τ)yixi*v*dτ+1Fiβixi*v*∫0hifi(τ)e-miτ×(ln(yi*xi(t-τ)v(t-τ)yixi*v*)+ln(xi*xi)+ln(v*yivyi*))dτ+1Gβixi*v*∫0h3g(τ)e-nτ×(ln(vyi*v*yi)+ln(v*yi(t-τ)vyi*))dτ-1Gβixi*v*∫0h3g(τ)e-nτv*yi(t-τ)vyi*dτ].
Equation (63) can be rewritten as
(64)dW2dt=∑i=12‍γi[dixi*(2-xi*xi-xixi*)-βixi*v*(xi*xi-1-ln(xi*xi))-1Fiβixi*v*∫0hifi(τ)e-miτ×(yi*xi(t-τ)v(t-τ)yixi*v*-1-ln(yi*xi(t-τ)v(t-τ)yixi*v*))dτ-1Gβixi*v*∫0h3g(τ)e-nτ×(v*yi(t-τ)vyi*-1-ln(v*yi(t-τ)vyi*))dτ]=-∑i=12‍γi[di(xi-xi*)2xi+βixi*v*H(xi*xi)+1Fiβixi*v*∫0hifi(τ)e-miτ×H(yi*xi(t-τ)v(t-τ)yixi*v*)dτ+1Gβixi*v*×∫0h3g(τ)e-nτH(v*yi(t-τ)vyi*)dτ].
It can be easily seen that, ifxi*,yi*,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 ifxi=xi*andH=0; that is,
(65)yi*xi(t-τ)v(t-τ)yixi*v*=v*yi(t-τ)vyi*=1foralmostallτ∈[0,h].
Ifxi=xi*, then from (19) we havev=v*and from (65) we havey=y*.Moreover, from (20) we have
(66)0=βiFixi*v*-ayi*-pyi*z⟹z=βiFixi*v*pyi*-ap=z*.
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≥R0*, then the infected steady state without CTL immune responseE1is GAS;(iii)
ifR0>R0*>1, then the infected steady state with CTL immune responseE2is GAS.



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*Source: 653204-2013-12-02.xml*