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

Authors: A. M. Elaiw; A. S. Alsheri
Journal: Discrete Dynamics in Nature and Society
 (2013)

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## 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.


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## Body

## 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]
(1)x˙=λ-dx-βxv,(2)y˙=βxv-ay,(3)v˙=ky-cv,
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
(4)x˙i=ri(xi)-ψi(xi,v),i=1,2,(5)y˙i=βi′∫0hifi(τ)ψi(xi(t-τ),v(t-τ))dτ-aiyi,i=1,2,(6)v˙=∑i=12‍pi∫0ligi(ω)yi(t-ω)dω-cv,
where xi and yi represent the populations of the uninfected and infected cells, respectively, where i=1 and 2 correspond to CD4+ T cells and macrophages. Parameters βi′ and pi satisfy βi′<1 and pi<ki, 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 fi(τ) over the interval [0,hi] and hi 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 gi(ω) over the interval [0,li] and li is limit superior of this delay [18]. The probability distribution functions fi(τ):[0,hi]→ℝ+ and gi(ω):[0,li]→ℝ+ are integral functions with
(7)∫0hifi(τ)dτ=∫0ligi(ω)dω=1,i=1,2.
Function ψi(xi,v) represents the rate for the uninfected target cell of class i to be infected by the mature viruses. Special forms of function ψi(xi,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 functionri(xi). The following particular forms of function ri(xi) have widely been used in the literature of HIV dynamics:
(8)r¯i(xi)=λi-dxi,r~i(xi)=λi-dxi+bixi(1-xixi,max),
where bi is the maximum proliferation rate of the target cells of class i and xi,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
(9)x1(θ)=φ1(θ),x2(θ)=φ2(θ),y1(θ)=φ3(θ),y2(θ)=φ4(θ),v(θ)=φ5(θ),φj(θ)≥0,θ∈-[ℓ,0),φj(0)>0,j=1,…,5,ℓ=max{h1,h2,l1,l2},
where (φ1(θ),φ2(θ),…,φ5(θ))∈C and C=C([-ℓ,0],ℝ+5) is the Banach space of continuous functions mapping the interval [-ℓ,0] into ℝ+5. 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 functionsri and ψi satisfy the following assumptions.Assumption 1.
Functionri:[0,∞)→ℝ satisfies the following:(i)
ri(xi) is continuous and differentiable and ri(0)>0,(ii)
there exits anxi0>0 such that
(10)ri(xi0)=0,ri′(xi0)<0,(xi-xi0)ri(xi)≤0,xi≠xi0.Assumption 2.
(i)ψi(xi,v) is positive, continuous, and differentiable for all xi>0, and v>0,
(ii)ψi(0,v)=ψi(xi,0)=0, ∂ψi(xi,0)/∂v>0 for any x>0, v>0, i=1,2,
(iii)∂ψi(xi,v)/∂xi>0, ∂ψi(xi,v)/∂v>0 for all xi>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 Assumptions1 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 thatxi(t)>0, i=1,2, for all t≥0. Assume that xi(t) loses its nonnegativity on some local existence interval [0,ρ] for some constant ρ and let ti*∈[0,ρ] be such that xi(ti*)=0, i=1,2. From (4), we have
(11)x˙i(ti*)=ri(xi(ti*))-ψi(xi(ti*),v(ti*))=ri(0)-ψi(0,v(ti*))=ri(0)>0,i=1,2.
Hence, xi(t)>0 for some t∈(ti*,ti*+ε), where ε>0 is sufficiently small. This leads to a contradiction and hence xi(t)>0 for all t≥0. Further, from (5) and (6), we have
(12)yi(t)=yi(0)e-ait+βi′∫0te-ai(t-η)×∫0hifi(τ)ψi(xi(η-τ),v(η-τ))dτdη,i=1,2,v(t)=v(0)e-ct+∑i=12‍pi∫0te-c(t-η)∫0ligi(ω)yi(η-ω)dωdη,
confirming that yi(t)≥0, i=1,2, and v(t)≥0 for all t∈[0,ℓ]. By a recursive argument, we obtain yi(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 limsupt→∞xi(t)≤xi0. It follows that
(13)∫0hifi(τ)xi(t-τ)dτ≤xi0.
Let Ti(t)=βi′∫0hifi(τ)xi(t-τ)dτ+yi(t), Si=supxi∈[0,xi0]ri(xi) and a¯i≤min{ai,Si/xi0}, i=1,2; then
(14)T˙i(t)=∫0hiβi′fi(τ)(ri(xi(t-τ))-ψi(xi(t-τ),v(t-τ)))dτ+β′∫0hifi(τ)ψi(xi(t-τ),v(t-τ))dτ-aiyi(t)≤βi′Si-aiyi(t)≤βi′Si-aiyi(t)+βi′Si-a¯iβi′∫0hifi(τ)xi(t-τ)dτ≤2βi′Si-a¯iTi(t).
Hence, limsupt→∞Ti(t)≤Li, where Li=2βi′Si/a¯i. Since ∫0hifi(τ)xi(t-τ)dτ>0, we get limsupt→∞yi(t)≤Li. On the other hand,
(15)v˙(t)≤∑i=12‍piLi∫0ligi(τ)dτ-cv=∑i=12‍piLi-cv.
Then limsupt→∞v(t)≤L*, where L*=∑i=12(piLi/c). Therefore, X(t) is ultimately bounded.


### 2.2. Steady States

Let Assumptions1 and 2 be satisfied; then system (4)–(6) has an uninfected steady state E0(x10,x20,y10,y20,v0), where xi0 is defined in Assumption 1, yi0=0, i=1,2, and v0=0. The system can also have another steady state E*(x1*,x2*,y1*,y2*,v*) which is the infected steady state; the coordinates of the infected steady state, if they exist, satisfy the equalities
(16)ri(xi*)=ψi(xi*,v*),i=1,2,aiyi*=βi′ψi(xi*,v*),i=1,2,cv*=∑i=12‍piyi*.
We define the basic reproduction number for system (4)–(6) as
(17)R0=∑i=12‍Ri=∑i=12‍piβi′aic∂ψi(xi0,0)∂v.
The term ∂ψi(xi0,0)/∂v denotes the maximal average number of target cells i that each virus infects, and Ri 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∈{xi,yi,v,i=1,2}. We also define a function H:(0,∞)→[0,∞) as
(18)H(z)=z-1-lnz.
It is clear that H(z)≥0 for any z>0 and H has the global minimum H(1)=0.Assumption 4.
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∂
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0, xi>0.The global stability of the uninfected steady state will be established in the next theorem.Theorem 6.
If Assumptions1–5 hold true and R0≤1, then E0 is GAS.Proof.
Define a Lyapunov functionalW0 as follows:
(19)W0=∑i=12‍piai[βi′(xi-xi0-∫xi0xilimv→0+ψi(xi0,v)ψi(θ,v)dθ)+yi+βi′∫0hiδfi,hi(τ)ψi(xi(t-τ),v(t-τ))dτ+ai∫0liδgi,li(ω)yi(t-ω)dωψi(xi0,v)ψi(θ,v)]+v,
where
(20)δfi,hi(τ)=∫τhifi(σ)dσ,δgi,li(ω)=∫ωligi(σ)dσ,i=1,2.
Functions δfi,hi and δgi,li satisfy
(21)δfi,hi(0)=1,δfi,hi(hi)=0,dδfi,hi(τ)dτ=-fi(τ),δgi,li(0)=1,δgi,li(li)=0,dδgi,li(ω)dω=-gi(ω).
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
(22)dW0dt=∑i=12‍piai[βi′(1-limv→0+ψi(xi0,v)ψi(xi,v))(ri(xi)-ψi(xi,v))+βi′∫0hifi(τ)ψi(xi(t-τ),v(t-τ))dτ-aiyi+βi′∫0hifi(τ)ψi(xi,v)-ψi(xi(t-τ),v(t-τ))dτ+ai∫0ligi(ω)(yi-yi(t-ω))dωβi′(1-limv→0+ψi(xi0,v)ψi(xi,v))]+∑i=12‍pi∫0ligi(ω)yi(t-ω)dω-cv=∑i=12‍piai[βi′(1-limv→0+ψi(xi0,v)ψi(xi,v))ri(xi)+βi′ψi(xi,v)limv→0+ψi(xi0,v)ψi(xi,v)]-cv=∑i=12‍piai[βi′(1-∂ψi(xi0,0)/∂v∂ψi(xi,0)/∂v)ri(xi)+βi′ψi(xi,v)∂ψi(xi0,0)/∂v∂ψi(xi,0)/∂v]-cv≤∑i=12‍piai[βi′(1-∂ψi(xi0,0)/∂v∂ψi(xi,0)/∂v)ri(xi)+βi′v∂ψi(xi,0)∂v(∂ψi(xi0,0)/∂v∂ψi(xi,0)/∂v)]-cv=∑i=12‍piaiβi′(1-∂ψi(xi0,0)/∂v∂ψi(xi,0)/∂v)ri(xi)+(∑i=12‍piβi′aic∂ψi(xi0,0)∂v-1)cv=∑i=12‍piaiβi′(1-∂ψi(xi0,0)/∂v∂ψi(xi,0)/∂v)ri(xi)+(R0-1)cv.
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 xi,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 xi=xi0, i=1,2, and v=0. Noting that M is invariant, for each element of M, we have v=0, and then v˙=0. From (6), we derive that
(23)0=v˙=∑i=12‍pi∫0lig(ω)yi(t-ω)dω.
Since yi(t-θ)≥0 for all θ∈[0,ℓ], then ∑i=12pi∫0lig(ω)yi(t-ω)dω=0 if and only if yi(t-ω)=0, i=1,2. Hence, dW0/dt=0 if and only if xi=xi0, yi=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.
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0.Theorem 9.
Assume that Assumptions1, 2, 7, and 8 hold true and E* exists; then E* is GAS.Proof.
Define Lyapunov functionalW1 as follows:
(24)W1=∑i=12‍piai[βi′(xi-xi*-∫xi*xiψi(xi*,v*)ψi(θ,v*)dθ)+yi*(yiyi*-1-lnyiyi*)+βi′ψi(xi*,v*)×∫0hiδfi,hi(τ)H(ψi(xi(t-τ),v(t-τ))ψi(xi*,v*))dτ+aiyi*∫0liδgi,li(ω)H(yi(t-ω)yi*)dω]+v*(vv*-1-lnvv*).
By calculating the time derivative along (4)–(6), we get
(25)dW1dt=∑i=12‍piai[βi′(1-ψi(xi*,v*)ψi(xi,v*))(ri(xi)-ψi(xi,v))+(1-yi*yi)×(βi′∫0hi‍fi(τ)ψi(xi(t-τ),v(t-τ))dτ-aiyi)+βi′ψi(xi*,v*)×∫0hi‍fi(τ)(ψi(xi,v)ψi(xi*,v*)-ψi(xi(t-τ),v(t-τ))ψi(xi*,v*)+lnψi(xi(t-τ),v(t-τ))ψi(xi,v))dτ+aiyi*∫0li‍gi(ω)(yiyi*-yi(t-ω)yi*+lnyi(t-ω)yi)dω]+(1-v*v)(∑i=12‍pi∫0ligi(ω)yi(t-ω)dω-cv).
Collecting terms of (25), we obtain
(26)dW1dt=∑i=12‍piai×[βi′(1-ψi(xi*,v*)ψi(xi,v*))(ri(xi)-ri(xi*))+βi′ri(xi*)(1-ψi(xi*,v*)ψi(xi,v*))-βi′ψi(xi,v)+βi′ψi(xi*,v*)ψi(xi,v)ψi(xi,v*)+βi′∫0hifi(τ)ψi(xi(t-τ),v(t-τ))dτ-aiyi-yi*yiβi′∫0hifi(τ)ψi(xi(t-τ),v(t-τ))dτ+aiyi*+βi′ψi(xi,v)-βi′∫0hifi(τ)ψi(xi(t-τ),v(t-τ))dτ+βi′ψi(xi*,v*)×∫0hifi(τ)ln(ψi(xi(t-τ),v(t-τ))ψi(xi,v))dτ+aiyi-ai∫0ligi(ω)(yi(t-ω))dω+aiyi*∫0ligi(ω)ln(yi(t-ω)yi)dω]+∑i=12‍pi∫0ligi(ω)yi(t-ω)dω-cv-∑i=12‍pi∫0ligi(ω)v*yi(t-ω)vdω+cv*.
Using the infected steady state conditions (16), we get
(27)dW1dt=∑i=12‍piai[βi′(1-ψi(xi*,v*)ψi(xi,v*))(ri(xi)-ri(xi*))+aiyi*(1-ψi(xi*,v*)ψi(xi,v*))+aiyi*ψi(xi,v)ψi(xi,v*)-aiyi*∫0hifi(τ)yi*ψi(xi(t-τ),v(t-τ))yiψi(xi*,v*)dτ+2aiyi*+aiyi*×∫0hifi(τ)ln(ψi(xi(t-τ),v(t-τ))ψi(xi,v))dτ-aiyi*∫0ligi(ω)v*yi(t-ω)vyi*dτ+aiyi*∫0ligi(ω)ln(yi(t-ω)yi)dω-aiyi*vv*]=∑i=12‍piai[βi′(1-ψi(xi*,v*)ψi(xi,v*))(ri(xi)-ri(xi*))+aiyi*×(-1+ψi(xi,v)ψi(xi,v*)-vv*+vv*ψi(xi,v*)ψi(xi,v))-aiyi*ψi(xi*,v*)ψi(xi,v*)-aiyi*vv*ψi(xi,v*)ψi(xi,v)-aiyi*∫0hifi(τ)yi*ψi(xi(t-τ),v(t-τ))yiψi(xi*,v*)dτ+4aiyi*+aiyi*×∫0hifi(τ)ln(ψi(xi(t-τ),v(t-τ))ψi(xi,v))dτ-aiyi*∫0ligi(ω)v*yi(t-ω)vyi*dτ+aiyi*∫0ligi(ω)ln(yi(t-ω)yi)dω].
Using the following equalities:
(28)ln(ψi(xi(t-τ),v(t-τ))ψi(xi,v))=ln(yi*ψi(xi(t-τ),v(t-τ))yiψi(xi*,v*))+ln(ψi(xi*,v*)ψi(xi,v*))+ln(vv*ψi(xi,v*)ψi(xi,v))+ln(v*yivyi*),ln(yi(t-ω)yi)=ln(vyi*v*yi)+ln(v*yi(t-ω)vyi*),
we obtain
(29)dW1dt=∑i=12‍piai[aiyi*(1-ψi(xi*,v*)ψi(xi,v*))(ri(xi)ri(xi*)-1)+aiyi*(ψi(xi,v)ψi(xi,v*)-vv*)(1-ψi(xi,v*)ψi(xi,v))-aiyi*H(ψi(xi*,v*)ψi(xi,v*))-aiyi*H(vv*ψi(xi,v*)ψi(xi,v))-aiyi*∫0hifi(τ)×H(yi*ψi(xi(t-τ),v(t-τ))yiψi(xi*,v*))dτ-aiyi*∫0ligi(ω)H(v*yi(t-ω)vyi*)dω].
Clearly, if E* exists, then dW1/dt≤0 for all xi,yi,v>0, i=1,2, and dW1/dt=0 if and only if xi=xi*, yi=yi*, and v=v*, which is the infected steady state E*. It follows that E* is GAS.


## 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 Assumptions1 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 thatxi(t)>0, i=1,2, for all t≥0. Assume that xi(t) loses its nonnegativity on some local existence interval [0,ρ] for some constant ρ and let ti*∈[0,ρ] be such that xi(ti*)=0, i=1,2. From (4), we have
(11)x˙i(ti*)=ri(xi(ti*))-ψi(xi(ti*),v(ti*))=ri(0)-ψi(0,v(ti*))=ri(0)>0,i=1,2.
Hence, xi(t)>0 for some t∈(ti*,ti*+ε), where ε>0 is sufficiently small. This leads to a contradiction and hence xi(t)>0 for all t≥0. Further, from (5) and (6), we have
(12)yi(t)=yi(0)e-ait+βi′∫0te-ai(t-η)×∫0hifi(τ)ψi(xi(η-τ),v(η-τ))dτdη,i=1,2,v(t)=v(0)e-ct+∑i=12‍pi∫0te-c(t-η)∫0ligi(ω)yi(η-ω)dωdη,
confirming that yi(t)≥0, i=1,2, and v(t)≥0 for all t∈[0,ℓ]. By a recursive argument, we obtain yi(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 limsupt→∞xi(t)≤xi0. It follows that
(13)∫0hifi(τ)xi(t-τ)dτ≤xi0.
Let Ti(t)=βi′∫0hifi(τ)xi(t-τ)dτ+yi(t), Si=supxi∈[0,xi0]ri(xi) and a¯i≤min{ai,Si/xi0}, i=1,2; then
(14)T˙i(t)=∫0hiβi′fi(τ)(ri(xi(t-τ))-ψi(xi(t-τ),v(t-τ)))dτ+β′∫0hifi(τ)ψi(xi(t-τ),v(t-τ))dτ-aiyi(t)≤βi′Si-aiyi(t)≤βi′Si-aiyi(t)+βi′Si-a¯iβi′∫0hifi(τ)xi(t-τ)dτ≤2βi′Si-a¯iTi(t).
Hence, limsupt→∞Ti(t)≤Li, where Li=2βi′Si/a¯i. Since ∫0hifi(τ)xi(t-τ)dτ>0, we get limsupt→∞yi(t)≤Li. On the other hand,
(15)v˙(t)≤∑i=12‍piLi∫0ligi(τ)dτ-cv=∑i=12‍piLi-cv.
Then limsupt→∞v(t)≤L*, where L*=∑i=12(piLi/c). Therefore, X(t) is ultimately bounded.


## 2.2. Steady States

Let Assumptions1 and 2 be satisfied; then system (4)–(6) has an uninfected steady state E0(x10,x20,y10,y20,v0), where xi0 is defined in Assumption 1, yi0=0, i=1,2, and v0=0. The system can also have another steady state E*(x1*,x2*,y1*,y2*,v*) which is the infected steady state; the coordinates of the infected steady state, if they exist, satisfy the equalities
(16)ri(xi*)=ψi(xi*,v*),i=1,2,aiyi*=βi′ψi(xi*,v*),i=1,2,cv*=∑i=12‍piyi*.
We define the basic reproduction number for system (4)–(6) as
(17)R0=∑i=12‍Ri=∑i=12‍piβi′aic∂ψi(xi0,0)∂v.
The term ∂ψi(xi0,0)/∂v denotes the maximal average number of target cells i that each virus infects, and Ri 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∈{xi,yi,v,i=1,2}. We also define a function H:(0,∞)→[0,∞) as
(18)H(z)=z-1-lnz.
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
(
x
i
0
,
0
)
/
∂
v
)
/
(
∂
ψ
i
(
x
i
,
0
)
/
∂
v
)
)
r
i
(
x
i
)
≤
0, xi>0.The global stability of the uninfected steady state will be established in the next theorem.Theorem 6.
If Assumptions1–5 hold true and R0≤1, then E0 is GAS.Proof.
Define a Lyapunov functionalW0 as follows:
(19)W0=∑i=12‍piai[βi′(xi-xi0-∫xi0xilimv→0+ψi(xi0,v)ψi(θ,v)dθ)+yi+βi′∫0hiδfi,hi(τ)ψi(xi(t-τ),v(t-τ))dτ+ai∫0liδgi,li(ω)yi(t-ω)dωψi(xi0,v)ψi(θ,v)]+v,
where
(20)δfi,hi(τ)=∫τhifi(σ)dσ,δgi,li(ω)=∫ωligi(σ)dσ,i=1,2.
Functions δfi,hi and δgi,li satisfy
(21)δfi,hi(0)=1,δfi,hi(hi)=0,dδfi,hi(τ)dτ=-fi(τ),δgi,li(0)=1,δgi,li(li)=0,dδgi,li(ω)dω=-gi(ω).
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
(22)dW0dt=∑i=12‍piai[βi′(1-limv→0+ψi(xi0,v)ψi(xi,v))(ri(xi)-ψi(xi,v))+βi′∫0hifi(τ)ψi(xi(t-τ),v(t-τ))dτ-aiyi+βi′∫0hifi(τ)ψi(xi,v)-ψi(xi(t-τ),v(t-τ))dτ+ai∫0ligi(ω)(yi-yi(t-ω))dωβi′(1-limv→0+ψi(xi0,v)ψi(xi,v))]+∑i=12‍pi∫0ligi(ω)yi(t-ω)dω-cv=∑i=12‍piai[βi′(1-limv→0+ψi(xi0,v)ψi(xi,v))ri(xi)+βi′ψi(xi,v)limv→0+ψi(xi0,v)ψi(xi,v)]-cv=∑i=12‍piai[βi′(1-∂ψi(xi0,0)/∂v∂ψi(xi,0)/∂v)ri(xi)+βi′ψi(xi,v)∂ψi(xi0,0)/∂v∂ψi(xi,0)/∂v]-cv≤∑i=12‍piai[βi′(1-∂ψi(xi0,0)/∂v∂ψi(xi,0)/∂v)ri(xi)+βi′v∂ψi(xi,0)∂v(∂ψi(xi0,0)/∂v∂ψi(xi,0)/∂v)]-cv=∑i=12‍piaiβi′(1-∂ψi(xi0,0)/∂v∂ψi(xi,0)/∂v)ri(xi)+(∑i=12‍piβi′aic∂ψi(xi0,0)∂v-1)cv=∑i=12‍piaiβi′(1-∂ψi(xi0,0)/∂v∂ψi(xi,0)/∂v)ri(xi)+(R0-1)cv.
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 xi,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 xi=xi0, i=1,2, and v=0. Noting that M is invariant, for each element of M, we have v=0, and then v˙=0. From (6), we derive that
(23)0=v˙=∑i=12‍pi∫0lig(ω)yi(t-ω)dω.
Since yi(t-θ)≥0 for all θ∈[0,ℓ], then ∑i=12pi∫0lig(ω)yi(t-ω)dω=0 if and only if yi(t-ω)=0, i=1,2. Hence, dW0/dt=0 if and only if xi=xi0, yi=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 Assumptions1, 2, 7, and 8 hold true and E* exists; then E* is GAS.Proof.
Define Lyapunov functionalW1 as follows:
(24)W1=∑i=12‍piai[βi′(xi-xi*-∫xi*xiψi(xi*,v*)ψi(θ,v*)dθ)+yi*(yiyi*-1-lnyiyi*)+βi′ψi(xi*,v*)×∫0hiδfi,hi(τ)H(ψi(xi(t-τ),v(t-τ))ψi(xi*,v*))dτ+aiyi*∫0liδgi,li(ω)H(yi(t-ω)yi*)dω]+v*(vv*-1-lnvv*).
By calculating the time derivative along (4)–(6), we get
(25)dW1dt=∑i=12‍piai[βi′(1-ψi(xi*,v*)ψi(xi,v*))(ri(xi)-ψi(xi,v))+(1-yi*yi)×(βi′∫0hi‍fi(τ)ψi(xi(t-τ),v(t-τ))dτ-aiyi)+βi′ψi(xi*,v*)×∫0hi‍fi(τ)(ψi(xi,v)ψi(xi*,v*)-ψi(xi(t-τ),v(t-τ))ψi(xi*,v*)+lnψi(xi(t-τ),v(t-τ))ψi(xi,v))dτ+aiyi*∫0li‍gi(ω)(yiyi*-yi(t-ω)yi*+lnyi(t-ω)yi)dω]+(1-v*v)(∑i=12‍pi∫0ligi(ω)yi(t-ω)dω-cv).
Collecting terms of (25), we obtain
(26)dW1dt=∑i=12‍piai×[βi′(1-ψi(xi*,v*)ψi(xi,v*))(ri(xi)-ri(xi*))+βi′ri(xi*)(1-ψi(xi*,v*)ψi(xi,v*))-βi′ψi(xi,v)+βi′ψi(xi*,v*)ψi(xi,v)ψi(xi,v*)+βi′∫0hifi(τ)ψi(xi(t-τ),v(t-τ))dτ-aiyi-yi*yiβi′∫0hifi(τ)ψi(xi(t-τ),v(t-τ))dτ+aiyi*+βi′ψi(xi,v)-βi′∫0hifi(τ)ψi(xi(t-τ),v(t-τ))dτ+βi′ψi(xi*,v*)×∫0hifi(τ)ln(ψi(xi(t-τ),v(t-τ))ψi(xi,v))dτ+aiyi-ai∫0ligi(ω)(yi(t-ω))dω+aiyi*∫0ligi(ω)ln(yi(t-ω)yi)dω]+∑i=12‍pi∫0ligi(ω)yi(t-ω)dω-cv-∑i=12‍pi∫0ligi(ω)v*yi(t-ω)vdω+cv*.
Using the infected steady state conditions (16), we get
(27)dW1dt=∑i=12‍piai[βi′(1-ψi(xi*,v*)ψi(xi,v*))(ri(xi)-ri(xi*))+aiyi*(1-ψi(xi*,v*)ψi(xi,v*))+aiyi*ψi(xi,v)ψi(xi,v*)-aiyi*∫0hifi(τ)yi*ψi(xi(t-τ),v(t-τ))yiψi(xi*,v*)dτ+2aiyi*+aiyi*×∫0hifi(τ)ln(ψi(xi(t-τ),v(t-τ))ψi(xi,v))dτ-aiyi*∫0ligi(ω)v*yi(t-ω)vyi*dτ+aiyi*∫0ligi(ω)ln(yi(t-ω)yi)dω-aiyi*vv*]=∑i=12‍piai[βi′(1-ψi(xi*,v*)ψi(xi,v*))(ri(xi)-ri(xi*))+aiyi*×(-1+ψi(xi,v)ψi(xi,v*)-vv*+vv*ψi(xi,v*)ψi(xi,v))-aiyi*ψi(xi*,v*)ψi(xi,v*)-aiyi*vv*ψi(xi,v*)ψi(xi,v)-aiyi*∫0hifi(τ)yi*ψi(xi(t-τ),v(t-τ))yiψi(xi*,v*)dτ+4aiyi*+aiyi*×∫0hifi(τ)ln(ψi(xi(t-τ),v(t-τ))ψi(xi,v))dτ-aiyi*∫0ligi(ω)v*yi(t-ω)vyi*dτ+aiyi*∫0ligi(ω)ln(yi(t-ω)yi)dω].
Using the following equalities:
(28)ln(ψi(xi(t-τ),v(t-τ))ψi(xi,v))=ln(yi*ψi(xi(t-τ),v(t-τ))yiψi(xi*,v*))+ln(ψi(xi*,v*)ψi(xi,v*))+ln(vv*ψi(xi,v*)ψi(xi,v))+ln(v*yivyi*),ln(yi(t-ω)yi)=ln(vyi*v*yi)+ln(v*yi(t-ω)vyi*),
we obtain
(29)dW1dt=∑i=12‍piai[aiyi*(1-ψi(xi*,v*)ψi(xi,v*))(ri(xi)ri(xi*)-1)+aiyi*(ψi(xi,v)ψi(xi,v*)-vv*)(1-ψi(xi,v*)ψi(xi,v))-aiyi*H(ψi(xi*,v*)ψi(xi,v*))-aiyi*H(vv*ψi(xi,v*)ψi(xi,v))-aiyi*∫0hifi(τ)×H(yi*ψi(xi(t-τ),v(t-τ))yiψi(xi*,v*))dτ-aiyi*∫0ligi(ω)H(v*yi(t-ω)vyi*)dω].
Clearly, if E* exists, then dW1/dt≤0 for all xi,yi,v>0, i=1,2, and dW1/dt=0 if and only if xi=xi*, yi=yi*, 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.



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*Source: 264759-2013-07-30.xml*