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

Authors: A. M. Elaiw
Journal: Discrete Dynamics in Nature and Society
 (2012)

---

## 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.


---

## Body

## 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
(1.1)x˙(t)=λ-dx(t)-(1-urt)β¯x(t)v(t),(1.2)y˙(t)=(1-urt)β¯∫0∞f(τ)e-mτx(t-τ)v(t-τ)dτ-ay(t),(1.3)v˙(t)=(1-up)p¯∫0∞g(τ)y(t-τ)dτ-cv(t),
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 urt∈[0,1] and up∈[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:(2.1)x˙1(t)=λ1-d1x1(t)-β1x1(t)v(t)1+α1v(t),(2.2)y˙1(t)=β1∫0∞f1(τ)e-m1τx1(t-τ)v(t-τ)1+α1v(t-τ)dτ-a1y1(t),(2.3)x˙2(t)=λ2-d2x2(t)-β2x2(t)v(t)1+α2v(t),(2.4)y˙2(t)=β2∫0∞f2(τ)e-m2τx2(t-τ)v(t-τ)1+α2v(t-τ)dτ-a2y2(t),(2.5)v˙(t)=p1∫0∞g1(τ)e-n1τy1(t-τ)dτ+p2∫0∞g2(τ)e-n2τy2(t-τ)dτ-cv(t).
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, αi,  i=1,2 are positive constants, βi=(1-urt)β¯i, and pi=(1-up)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 functionsfi(τ) and gi(τ) are assumed to satisfy fi(τ)>0 and gi(τ)>0,  i=1,2 and
(2.6)∫0∞fi(τ)dτ=∫0∞gi(τ)dτ=1,i=1,2,∫0∞fi(r)esrdr<∞,∫0∞gi(r)esrdr<∞,i=1,2,
where s is a positive number. Then
(2.7)0<∫0∞fi(τ)e-miτdτ≤1,formi≥0,i=1,2,0<∫0∞gi(τ)e-niτdτ≤1,forni≥0,i=1,2.
The initial conditions for system (2.1)–(2.5) take the form
(2.8)x1(θ)=φ1(θ),y1(θ)=φ2(θ),x2(θ)=φ3(θ),y2(θ)=φ4(θ),v(θ)=φ5(θ),φj(θ)≥0,θ∈(-∞,0),j=1,…,5,φj(0)>0,j=1,…,5,
where (φ1(θ),φ2(θ),…,φ5(θ))∈UC((-∞,0],ℝ+5), and UC is the Banach space of fading memory type defined as [30]
(2.9)UC((-∞,0],R+5)={φ∈C((-∞,0],R+5):φ(r)esrisuniformlycontinuouson⁡(-∞,0],‖φ‖=sup⁡r≤0φ(r)esr<∞},
where C((-∞,0],ℝ+5) is the Banach space of continuous functions mapping the interval (-∞,0]into ℝ+5. 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
(2.10)xi(t)=xi(0)e-∫0t‍[di+βiv(ξ)/(1+αiv(ξ))]dξ+λi∫0te-∫ηt‍[di+βiv(ξ)/(1+αiv(ξ))]dξdη,i=1,2,
which indicates that xi(t)≥0, for all t≥0. Now from (2.2), (2.4), and (2.5) we have
(2.11)yi(t)=yi(0)e-ait+βi∫0te-ai(t-η)∫0∞fi(τ)e-miτxi(η-τ)v(η-τ)1+αiv(η-τ)dτdη,i=1,2,v(t)=v(0)e-ct+p1∫0te-c(t-η)∫0∞g1(τ)e-n1τy1(η-τ)dτdη+p2∫0te-c(t-η)∫0∞g2(τ)e-n2τy2(η-τ)dτdη,
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 x˙i(t)≤λi-dixi(t), i=1,2. This implies limsupt→∞xi(t)≤λi/di,  i=1,2.
LetXi(t)=∫0∞fi(τ)e-miτxi(t-τ)dτ+yi(t),  i=1,2, then
(2.12)X˙i(t)=∫0∞fi(τ)e-miτ(λi-dixi(t-τ)-βixi(t-τ)v(t-τ)1+αiv(t-τ))dτ+∫0∞fi(τ)e-miτβixi(t-τ)v(t-τ)1+αiv(t-τ)dτ-aiyi(t)=λi∫0∞fi(τ)e-miτdτ-di∫0∞fi(τ)e-miτxi(t-τ)dτ-aiyi(t)≤λi∫0∞fi(τ)e-miτdτ-σi[∫0∞fi(τ)e-miτxi(t-τ)dτ+yi(t)]=λi∫0∞fi(τ)e-miτdτ-σiXi(t)≤λi-σiXi(t),
where σi=min{di,ai}. Hence limsupt→∞Xi(t)≤Li, where Li=λi/σi, i=1,2. On the other hand,
(2.13)v˙(t)≤p1L1∫0∞g1(τ)e-n1τdτ+p2L2∫0∞g2(τ)e-n2τdτ-cv≤p1L1+p2L2-cv,
then limsupt→∞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=(x10,0,x20,0,0), where xi0=λi/di, i=1,2. In addition to E0, the system can also have a positive infected steady state E1(x1*,y1*,x2*,y2*,v*). The coordinates of the infected steady state, if they exist, satisfy the following equalities:
(2.14)λi=dixi*+βixi*v*1+αiv*,i=1,2,(2.15)aiyi*=Fiβixi*v*1+αiv*,i=1,2,(2.16)cv*=G1p1y1*+G2p2y2*,
where
(2.17)Fi=∫0∞fi(τ)e-miτdτ,Gi=∫0∞gi(τ)e-niτdτ,i=1,2.
Following van den Driessche and Watmough [32], we define the basic reproduction number for system (2.1)–(2.5) as
(2.18)R0=∑i=12‍Ri=∑i=12‍FiGiβipiλiaidic,
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.
IfR0>1, then there exists a positive steady state E1.Proof.
From (2.14) and (2.15) we have
(2.19)xi*=xi0(1+αiv*)(1+δiv*),i=1,2,(2.20)yi*=Fiβixi0v*ai(1+δiv*),i=1,2,
where δi=αi+βi/di. From (2.20) into (2.16) we get
(2.21)1=F1G1p1β1x10a1c(1+δ1v*)+F2G2p2β2x20a2c(1+δ2v*)=R11+δ1v*+R21+δ2v*.
Equation (2.21) can be written as
(2.22)δ1δ2v*2+(δ1R1+δ2R2+(1-R0)(δ1+δ2))v*+1-R0=0.
If R0>1, then the positive solution of (2.21) is given by:
(2.23)v*=-(δ1R1+δ2R2+(1-R0)(δ1+δ2))+(δ1R1+δ2R2+(1-R0)(δ1+δ2))2-4δ1δ2(1-R0)2δ1δ2.
It follows that, if R0>1 then x1*,  y1*,  x2*,  y2* 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
(2.24)H(z)=z-1-ln⁡z.
It is clear that H(z)≥0 for any z>0 and H has the global minimum H(1)=0.Theorem 2.3.
IfR0≤1, then E0 is GAS.Proof.
Define a Lyapunov functionalW1 as follows:
(2.25)W1=∑i=12‍γi[xi0H(xixi0)+1Fiyi+βiFi∫0∞fi(τ)e-miτ∫0τxi(t-θ)v(t-θ)1+αiv(t-θ)dθdτ+aiFiGi∫0∞gi(τ)e-niτ∫0τyi(t-θ)dθdτ(xixi0)]+v,
where γi=piFiGi/ai, i=1,2.
The time derivative ofW1 along the trajectories of (2.1)–(2.5) satisfies
(2.26)dW1dt=∑i=12‍γi[(1-xi0xi)(λi-dixi-βixiv1+αiv)+βiFi∫0∞fi(τ)e-miτxi(t-τ)v(t-τ)1+αiv(t-τ)dτ-aiFiyi+βiFi∫0∞fi(τ)e-miτ(xiv1+αiv-xi(t-τ)v(t-τ)1+αiv(t-τ))dτ+aiFiGi∫0∞gi(τ)e-niτ(yi-yi(t-τ))dτ(1-xi0xi)]+∑i=12‍pi∫0∞gi(τ)e-niτyi(t-τ)dτ-cv.
Collecting terms of (2.26) we get
(2.27)dW1dt=∑i=12‍γi(λi-dixi-λixi0xi+dixi0+βixi0v1+αiv)-cv=∑i=12‍γiλi(2-xixi0-xi0xi)-cv+cv∑i=12‍FiGipiβixi0aic(1+αiv)=-∑i=12‍γidi(xi-xi0)2xi-cv+cv∑i=12‍Ri1+αiv=-∑i=12‍γidi(xi-xi0)2xi-∑i=12‍Riαicv21+αiv+(R0-1)cv.
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 xi=xi0,  i=1,2, and  v=0. Noting that M is invariant, for each element of M we have v=0, then v˙=0. From (2.5) we drive that
(2.28)0=v˙=p1∫0∞g1(τ)e-n1τy1(t-τ)dτ+p2∫0∞g2(τ)e-n2τy2(t-τ)dτ.
This yields y1=y2=0. Hence dW1/dt=0 if and only if xi=xi0,  yi=0,  i=1,2, and v=0. From La Salle's Invariance Principle, E0 is GAS.Theorem 2.4.
IfR0>1, then E1 is GAS.Proof.
We construct the following Lyapunov functional:(2.29)W2=∑i=12‍γi[xi*H(xixi*)+1Fiyi*H(yiyi*)+1Fiβixi*v*1+αiv*∫0∞fi(τ)e-miτ∫0τH(xi(t-θ)v(t-θ)(1+αiv*)xi*v*(1+αiv(t-θ)))dθdτ+aiyi*FiGi∫0∞gi(τ)e-niτ∫0τH(yi(t-θ)yi*)dθdτ]+v*H(vv*).
Differentiating with respect to time yields
(2.30)dW2dt=∑i=12‍γi[(1-xi*xi)(λi-dixi-βixiv1+αiv)+1Fi(1-yi*yi)(βi∫0∞fi(τ)e-miτxi(t-τ)v(t-τ)1+αiv(t-τ)dτ-aiyi)+βiFi∫0∞fi(τ)e-miτ×(xiv1+αiv-xi(t-τ)v(t-τ)1+αiv(t-τ)+xi*v*1+αiv*ln⁡(xi(t-τ)v(t-τ)(1+αiv)xiv(1+αiv(t-τ))))dτ+aiFiGi∫0∞gi(τ)e-niτ(yi-yi(t-τ)+yi*ln⁡(yi(t-τ)yi))dτ]+(1-v*v)(∑i=12‍pi∫0∞gi(τ)e-niτyi(t-τ)dτ-cv).
Collecting terms we obtain
(2.31)dW2dt=∑i=12‍γi[λi-dixi-λixi*xi+dixi*+βixi*v1+αiv-βiyi*Fiyi∫0∞fi(τ)e-miτxi(t-τ)v(t-τ)1+αiv(t-τ)dτ+aiFiyi*+1Fiβixi*v*1+αiv*∫0∞fi(τ)e-miτln⁡(xi(t-τ)v(t-τ)(1+αiv)xiv(1+αiv(t-τ)))dτ+aiyi*FiGi∫0∞gi(τ)e-niτln⁡(yi(t-τ)yi)dτ]-cv-v*v∑i=12‍pi∫0∞gi(τ)e-niτyi(t-τ)dτ+cv*.
Using the infected steady state conditions (2.14)–(2.16), and the following equality:
(2.32)cv=cv*vv*=vv*∑i=12‍Gipiyi*=vv*∑i=12‍γiaiFiyi*,
we obtain
(2.33)dW2dt=∑i=12‍γi[dixi*+aiFiyi*-dixi-xi*xi(dixi*+aiFiyi*)+dixi*+aiFiyi*v(1+αiv*)v*(1+αiv)-aiFi2yi*∫0∞fi(τ)e-miτyi*xi(t-τ)v(t-τ)(1+αiv*)yixi*v*(1+αiv(t-τ))dτ+aiFiyi*+aiFi2yi*∫0∞fi(τ)e-miτln⁡(xi(t-τ)v(t-τ)(1+αiv)xiv(1+αiv(t-τ)))dτ+aiFiGiyi*∫0∞gi(τ)e-niτln⁡(yi(t-τ)yi)dτ-aiFiyi*vv*-aiFiGiyi*∫0∞gi(τ)e-niτv*yi(t-τ)vyi*dτ+aiFiyi*].
Then collecting terms of (2.33) and using the following equalities:
(2.34)ln⁡(xi(t-τ)v(t-τ)(1+αiv)xiv(1+αiv(t-τ)))=ln⁡(yi*xi(t-τ)v(t-τ)(1+αiv*)yixi*v*(1+αiv(t-τ)))+ln⁡(xi*xi)+ln⁡(v*yivyi*)+ln⁡(1+αiv1+αiv*),i=1,2,ln⁡(yi(t-τ)yi)=ln⁡(vyi*v*yi)+ln⁡(v*yi(t-τ)vyi*),i=1,2ln⁡(v*yivyi*)+ln⁡(vyi*v*yi)=ln⁡(1)=0,i=1,2
we obtain
(2.35)dW2dt=∑i=12‍γi[dixi*(2-xi*xi-xixi*)+aiFiyi*(1-xi*xi)+2aiFiyi*+aiFiyi*(v(1+αiv*)v*(1+αiv)-vv*)-aiFi2yi*∫0∞fi(τ)e-miτyi*xi(t-τ)v(t-τ)(1+αiv*)yixi*v*(1+αiv(t-τ))dτ+aiFi2yi*∫0∞fi(τ)e-miτ×(ln⁡(yi*xi(t-τ)v(t-τ)(1+αiv*)yixi*v*(1+αiv(t-τ)))+ln⁡(xi*xi)+ln⁡(v*yivyi*)+ln⁡(1+αiv1+αiv*))dτ+aiFiGiyi*∫0∞gi(τ)e-niτ(ln⁡(vyi*v*yi)+ln⁡(v*yi(t-τ)vyi*))dτ-aiFiGiyi*∫0∞gi(τ)e-niτv*yi(t-τ)vyi*dτ].
Equation (2.35) can be rewritten as
(2.36)dW2dt=∑i=12‍γi[dixi*(2-xi*xi-xixi*)-aiFiyi*(xi*xi-1-ln⁡(xi*xi))+aiFiyi*(-1+v(1+αiv*)v*(1+αiv)-vv*+1+αiv1+αiv*)-aiFiyi*(1+αiv1+αiv*-1-ln⁡(1+αiv1+αiv*))-aiFi2yi*∫0∞fi(τ)e-miτ×(yi*xi(t-τ)v(t-τ)(1+αiv*)yixi*v*(1+αiv(t-τ))-1-ln⁡(yi*xi(t-τ)v(t-τ)(1+αiv*)yixi*v*(1+αiv(t-τ))))dτ-aiFiGiyi*∫0∞gi(τ)e-niτ(v*yi(t-τ)vyi*-1-ln⁡(v*yi(t-τ)vyi*))dτ].
Using the following equality:
(2.37)-1+v(1+αiv*)v*(1+αiv)-vv*+1+αiv1+αiv*=-αi(v-v*)2v*(1+αiv*)(1+αiv),i=1,2,
we can rewrite dW2/dt as
(2.38)dW2dt=-∑i=12‍γi[di(xi-xi*)2xi+aiFiyi*αi(v-v*)2v*(1+αiv*)(1+αiv)+aiFiyi*H(xi*xi)+aiFiyi*H(1+αiv1+αiv*)+aiyi*Fi2∫0∞fi(τ)e-miτH(yi*xi(t-τ)v(t-τ)(1+αiv*)yixi*v*(1+αiv(t-τ)))dτ+aiyi*FiGi∫0∞gi(τ)e-niτH(v*yi(t-τ)vyi*)dτ].
It is easy to see that if xi*,yi*,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 xi=xi*,  v=v*, and H=0, that is,
(2.39)yi*xi(t-τ)v(t-τ)(1+αiv*)yixi*v*(1+αiv(t-τ))=v*yi(t-τ)vyi*=1foralmostallτ∈(0,∞).
If v=v* then from (2.39) we have yi=yi*, and hence dW2/dt equal to zero at E1. LaSalle's Invariance Principle implies global stability of E1.


## 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
(2.10)xi(t)=xi(0)e-∫0t‍[di+βiv(ξ)/(1+αiv(ξ))]dξ+λi∫0te-∫ηt‍[di+βiv(ξ)/(1+αiv(ξ))]dξdη,i=1,2,
which indicates that xi(t)≥0, for all t≥0. Now from (2.2), (2.4), and (2.5) we have
(2.11)yi(t)=yi(0)e-ait+βi∫0te-ai(t-η)∫0∞fi(τ)e-miτxi(η-τ)v(η-τ)1+αiv(η-τ)dτdη,i=1,2,v(t)=v(0)e-ct+p1∫0te-c(t-η)∫0∞g1(τ)e-n1τy1(η-τ)dτdη+p2∫0te-c(t-η)∫0∞g2(τ)e-n2τy2(η-τ)dτdη,
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 x˙i(t)≤λi-dixi(t), i=1,2. This implies limsupt→∞xi(t)≤λi/di,  i=1,2.
LetXi(t)=∫0∞fi(τ)e-miτxi(t-τ)dτ+yi(t),  i=1,2, then
(2.12)X˙i(t)=∫0∞fi(τ)e-miτ(λi-dixi(t-τ)-βixi(t-τ)v(t-τ)1+αiv(t-τ))dτ+∫0∞fi(τ)e-miτβixi(t-τ)v(t-τ)1+αiv(t-τ)dτ-aiyi(t)=λi∫0∞fi(τ)e-miτdτ-di∫0∞fi(τ)e-miτxi(t-τ)dτ-aiyi(t)≤λi∫0∞fi(τ)e-miτdτ-σi[∫0∞fi(τ)e-miτxi(t-τ)dτ+yi(t)]=λi∫0∞fi(τ)e-miτdτ-σiXi(t)≤λi-σiXi(t),
where σi=min{di,ai}. Hence limsupt→∞Xi(t)≤Li, where Li=λi/σi, i=1,2. On the other hand,
(2.13)v˙(t)≤p1L1∫0∞g1(τ)e-n1τdτ+p2L2∫0∞g2(τ)e-n2τdτ-cv≤p1L1+p2L2-cv,
then limsupt→∞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=(x10,0,x20,0,0), where xi0=λi/di, i=1,2. In addition to E0, the system can also have a positive infected steady state E1(x1*,y1*,x2*,y2*,v*). The coordinates of the infected steady state, if they exist, satisfy the following equalities:
(2.14)λi=dixi*+βixi*v*1+αiv*,i=1,2,(2.15)aiyi*=Fiβixi*v*1+αiv*,i=1,2,(2.16)cv*=G1p1y1*+G2p2y2*,
where
(2.17)Fi=∫0∞fi(τ)e-miτdτ,Gi=∫0∞gi(τ)e-niτdτ,i=1,2.
Following van den Driessche and Watmough [32], we define the basic reproduction number for system (2.1)–(2.5) as
(2.18)R0=∑i=12‍Ri=∑i=12‍FiGiβipiλiaidic,
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.
IfR0>1, then there exists a positive steady state E1.Proof.
From (2.14) and (2.15) we have
(2.19)xi*=xi0(1+αiv*)(1+δiv*),i=1,2,(2.20)yi*=Fiβixi0v*ai(1+δiv*),i=1,2,
where δi=αi+βi/di. From (2.20) into (2.16) we get
(2.21)1=F1G1p1β1x10a1c(1+δ1v*)+F2G2p2β2x20a2c(1+δ2v*)=R11+δ1v*+R21+δ2v*.
Equation (2.21) can be written as
(2.22)δ1δ2v*2+(δ1R1+δ2R2+(1-R0)(δ1+δ2))v*+1-R0=0.
If R0>1, then the positive solution of (2.21) is given by:
(2.23)v*=-(δ1R1+δ2R2+(1-R0)(δ1+δ2))+(δ1R1+δ2R2+(1-R0)(δ1+δ2))2-4δ1δ2(1-R0)2δ1δ2.
It follows that, if R0>1 then x1*,  y1*,  x2*,  y2* 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
(2.24)H(z)=z-1-ln⁡z.
It is clear that H(z)≥0 for any z>0 and H has the global minimum H(1)=0.Theorem 2.3.
IfR0≤1, then E0 is GAS.Proof.
Define a Lyapunov functionalW1 as follows:
(2.25)W1=∑i=12‍γi[xi0H(xixi0)+1Fiyi+βiFi∫0∞fi(τ)e-miτ∫0τxi(t-θ)v(t-θ)1+αiv(t-θ)dθdτ+aiFiGi∫0∞gi(τ)e-niτ∫0τyi(t-θ)dθdτ(xixi0)]+v,
where γi=piFiGi/ai, i=1,2.
The time derivative ofW1 along the trajectories of (2.1)–(2.5) satisfies
(2.26)dW1dt=∑i=12‍γi[(1-xi0xi)(λi-dixi-βixiv1+αiv)+βiFi∫0∞fi(τ)e-miτxi(t-τ)v(t-τ)1+αiv(t-τ)dτ-aiFiyi+βiFi∫0∞fi(τ)e-miτ(xiv1+αiv-xi(t-τ)v(t-τ)1+αiv(t-τ))dτ+aiFiGi∫0∞gi(τ)e-niτ(yi-yi(t-τ))dτ(1-xi0xi)]+∑i=12‍pi∫0∞gi(τ)e-niτyi(t-τ)dτ-cv.
Collecting terms of (2.26) we get
(2.27)dW1dt=∑i=12‍γi(λi-dixi-λixi0xi+dixi0+βixi0v1+αiv)-cv=∑i=12‍γiλi(2-xixi0-xi0xi)-cv+cv∑i=12‍FiGipiβixi0aic(1+αiv)=-∑i=12‍γidi(xi-xi0)2xi-cv+cv∑i=12‍Ri1+αiv=-∑i=12‍γidi(xi-xi0)2xi-∑i=12‍Riαicv21+αiv+(R0-1)cv.
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 xi=xi0,  i=1,2, and  v=0. Noting that M is invariant, for each element of M we have v=0, then v˙=0. From (2.5) we drive that
(2.28)0=v˙=p1∫0∞g1(τ)e-n1τy1(t-τ)dτ+p2∫0∞g2(τ)e-n2τy2(t-τ)dτ.
This yields y1=y2=0. Hence dW1/dt=0 if and only if xi=xi0,  yi=0,  i=1,2, and v=0. From La Salle's Invariance Principle, E0 is GAS.Theorem 2.4.
IfR0>1, then E1 is GAS.Proof.
We construct the following Lyapunov functional:(2.29)W2=∑i=12‍γi[xi*H(xixi*)+1Fiyi*H(yiyi*)+1Fiβixi*v*1+αiv*∫0∞fi(τ)e-miτ∫0τH(xi(t-θ)v(t-θ)(1+αiv*)xi*v*(1+αiv(t-θ)))dθdτ+aiyi*FiGi∫0∞gi(τ)e-niτ∫0τH(yi(t-θ)yi*)dθdτ]+v*H(vv*).
Differentiating with respect to time yields
(2.30)dW2dt=∑i=12‍γi[(1-xi*xi)(λi-dixi-βixiv1+αiv)+1Fi(1-yi*yi)(βi∫0∞fi(τ)e-miτxi(t-τ)v(t-τ)1+αiv(t-τ)dτ-aiyi)+βiFi∫0∞fi(τ)e-miτ×(xiv1+αiv-xi(t-τ)v(t-τ)1+αiv(t-τ)+xi*v*1+αiv*ln⁡(xi(t-τ)v(t-τ)(1+αiv)xiv(1+αiv(t-τ))))dτ+aiFiGi∫0∞gi(τ)e-niτ(yi-yi(t-τ)+yi*ln⁡(yi(t-τ)yi))dτ]+(1-v*v)(∑i=12‍pi∫0∞gi(τ)e-niτyi(t-τ)dτ-cv).
Collecting terms we obtain
(2.31)dW2dt=∑i=12‍γi[λi-dixi-λixi*xi+dixi*+βixi*v1+αiv-βiyi*Fiyi∫0∞fi(τ)e-miτxi(t-τ)v(t-τ)1+αiv(t-τ)dτ+aiFiyi*+1Fiβixi*v*1+αiv*∫0∞fi(τ)e-miτln⁡(xi(t-τ)v(t-τ)(1+αiv)xiv(1+αiv(t-τ)))dτ+aiyi*FiGi∫0∞gi(τ)e-niτln⁡(yi(t-τ)yi)dτ]-cv-v*v∑i=12‍pi∫0∞gi(τ)e-niτyi(t-τ)dτ+cv*.
Using the infected steady state conditions (2.14)–(2.16), and the following equality:
(2.32)cv=cv*vv*=vv*∑i=12‍Gipiyi*=vv*∑i=12‍γiaiFiyi*,
we obtain
(2.33)dW2dt=∑i=12‍γi[dixi*+aiFiyi*-dixi-xi*xi(dixi*+aiFiyi*)+dixi*+aiFiyi*v(1+αiv*)v*(1+αiv)-aiFi2yi*∫0∞fi(τ)e-miτyi*xi(t-τ)v(t-τ)(1+αiv*)yixi*v*(1+αiv(t-τ))dτ+aiFiyi*+aiFi2yi*∫0∞fi(τ)e-miτln⁡(xi(t-τ)v(t-τ)(1+αiv)xiv(1+αiv(t-τ)))dτ+aiFiGiyi*∫0∞gi(τ)e-niτln⁡(yi(t-τ)yi)dτ-aiFiyi*vv*-aiFiGiyi*∫0∞gi(τ)e-niτv*yi(t-τ)vyi*dτ+aiFiyi*].
Then collecting terms of (2.33) and using the following equalities:
(2.34)ln⁡(xi(t-τ)v(t-τ)(1+αiv)xiv(1+αiv(t-τ)))=ln⁡(yi*xi(t-τ)v(t-τ)(1+αiv*)yixi*v*(1+αiv(t-τ)))+ln⁡(xi*xi)+ln⁡(v*yivyi*)+ln⁡(1+αiv1+αiv*),i=1,2,ln⁡(yi(t-τ)yi)=ln⁡(vyi*v*yi)+ln⁡(v*yi(t-τ)vyi*),i=1,2ln⁡(v*yivyi*)+ln⁡(vyi*v*yi)=ln⁡(1)=0,i=1,2
we obtain
(2.35)dW2dt=∑i=12‍γi[dixi*(2-xi*xi-xixi*)+aiFiyi*(1-xi*xi)+2aiFiyi*+aiFiyi*(v(1+αiv*)v*(1+αiv)-vv*)-aiFi2yi*∫0∞fi(τ)e-miτyi*xi(t-τ)v(t-τ)(1+αiv*)yixi*v*(1+αiv(t-τ))dτ+aiFi2yi*∫0∞fi(τ)e-miτ×(ln⁡(yi*xi(t-τ)v(t-τ)(1+αiv*)yixi*v*(1+αiv(t-τ)))+ln⁡(xi*xi)+ln⁡(v*yivyi*)+ln⁡(1+αiv1+αiv*))dτ+aiFiGiyi*∫0∞gi(τ)e-niτ(ln⁡(vyi*v*yi)+ln⁡(v*yi(t-τ)vyi*))dτ-aiFiGiyi*∫0∞gi(τ)e-niτv*yi(t-τ)vyi*dτ].
Equation (2.35) can be rewritten as
(2.36)dW2dt=∑i=12‍γi[dixi*(2-xi*xi-xixi*)-aiFiyi*(xi*xi-1-ln⁡(xi*xi))+aiFiyi*(-1+v(1+αiv*)v*(1+αiv)-vv*+1+αiv1+αiv*)-aiFiyi*(1+αiv1+αiv*-1-ln⁡(1+αiv1+αiv*))-aiFi2yi*∫0∞fi(τ)e-miτ×(yi*xi(t-τ)v(t-τ)(1+αiv*)yixi*v*(1+αiv(t-τ))-1-ln⁡(yi*xi(t-τ)v(t-τ)(1+αiv*)yixi*v*(1+αiv(t-τ))))dτ-aiFiGiyi*∫0∞gi(τ)e-niτ(v*yi(t-τ)vyi*-1-ln⁡(v*yi(t-τ)vyi*))dτ].
Using the following equality:
(2.37)-1+v(1+αiv*)v*(1+αiv)-vv*+1+αiv1+αiv*=-αi(v-v*)2v*(1+αiv*)(1+αiv),i=1,2,
we can rewrite dW2/dt as
(2.38)dW2dt=-∑i=12‍γi[di(xi-xi*)2xi+aiFiyi*αi(v-v*)2v*(1+αiv*)(1+αiv)+aiFiyi*H(xi*xi)+aiFiyi*H(1+αiv1+αiv*)+aiyi*Fi2∫0∞fi(τ)e-miτH(yi*xi(t-τ)v(t-τ)(1+αiv*)yixi*v*(1+αiv(t-τ)))dτ+aiyi*FiGi∫0∞gi(τ)e-niτH(v*yi(t-τ)vyi*)dτ].
It is easy to see that if xi*,yi*,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 xi=xi*,  v=v*, and H=0, that is,
(2.39)yi*xi(t-τ)v(t-τ)(1+αiv*)yixi*v*(1+αiv(t-τ))=v*yi(t-τ)vyi*=1foralmostallτ∈(0,∞).
If v=v* then from (2.39) we have yi=yi*, 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.



---

*Source: 253703-2012-08-16.xml*