Global properties of a delayed HIV infection model with CTL immune response

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Abstract

In this paper, we study a delayed six-dimensional human immunodeficiency virus (HIV) model with Cytotoxic T Lymphocytes (CTLs) immune response. Our model describes the interaction of HIV with two target cells: CD4+ T cells and macrophages. We derive that the global asymptotic attractivity of the model is completely determined by the basic reproduction number R0 and the immune reproduction number R0 for the viral infection. By constructing Lyapunov functionals, we have shown that the infection-free equilibrium E0, the immune-free equilibrium E1 and the chronic-infection equilibrium E2 are globally asymptotically attractive when R01,R01<R0 and R0>R0>1, respectively.

Introduction

It is well known that the initial infection with HIV generally occurs after transfer of the body fluid from an infected individual to an uninfected one, such as sexual intercourse, contaminated needles used for intravenous drug delivery and use of blood or blood products. And the immune response plays an important role in eliminating or controlling the disease after human body is infected by virus [1], [2], [3], [4], [5], [6], [7], [21], [28]. Recently, many mathematical models have been developed to study the dynamics of human immunodeficiency virus (HIV) infection (for example, [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [22] and so on). These models have been used to explain different phenomena. Moreover, some of these models are mainly modelled on the interaction of the HIV with the CD4+ T cells. For more references and some detailed mathematical analysis on such models, we refer to the survey papers by Kirschner [19] and Perelson and Nelson [20]. Other HIV models consider the interaction process of the HIV not only with the CD4+ T cells but also with the macrophages (see, e.g. [20], [30], [31], [32], [33]). The importance of considering these HIV models is due to the observation of Perleson el al. [34].

The basic properties of the two target cells HIV models are not well studied in the literature, Elaiw [35] studied that the HIV infection model describes two co-circulation populations of target cells, potentially representing CD4+ T cells and macrophages as follows:x˙(t)=λ1-d1x(t)-β1x(t)v(t),x˙1(t)=β1x(t)v(t)-ax1(t),y˙(t)=λ2-d2y(t)-β2y(t)v(t),y˙1(t)=β2y(t)v(t)-ay1(t),v˙(t)=k(x1(t)+y1(t))-rv(t),where the state variables and parameters are described in Table 1.

To account for the time between viral entry into a target cell and the production of new virus particles, typically lasts for around 1 day for the HIV-1 infection [23], Nelson and Perlson [26] assumed that there are two types of delays that occur between the administration of drug and the observed decline in viral load: a pharmacological delay that occurs between the ingestion of drug and its appearance within cells and an intracellular delay that is between initial infection of a cell by HIV and the release of new virions. Herz et al. [24] used a discrete delay to model the intracellular delay in a HIV model and showed that the incorporation of a delay would substantially shorten the estimate for the half-life of free virus. Mittler et al. [25] argued that a γ-distribution delay would be more realistic to model the intracellular delay phenomenon and introduced such a delay into the model of Perelson et al. [29].

In order to incorporate the intracellular phase of the virus life-cycle, we assume that virus production occurs after the virus entry by the constant delays τ1,τ2. The recruitment of virus-producing cells at time t is given by the number of the uninfected CD4+ T cells and the macrophages that were newly infected at time t-τ1,t-τ2 and are still alive at time t [38], [39]. If we assume constant death rates s1,s2 for infected CD4+ T cells and infected macrophages but not yet produce virus particles, the probability of surviving the time period from t-τi to t is e-siτi,i=1,2. Therefore, based on the discussion above, we obtain the following DDE model with CTL response:x˙(t)=λ1-d1x(t)-β1x(t)v(t),x˙1(t)=β1e-s1τ1x(t-τ1)v(t-τ1)-ax1(t)-px1(t)z(t),y˙(t)=λ2-d2y(t)-β2y(t)v(t),y˙1(t)=β2e-s2τ2y(t-τ2)v(t-τ2)-ay1(t)-py1(t)z(t),v˙(t)=k(x1(t)+y1(t))-rv(t),z˙(t)=c(x1(t)+y1(t))z(t)-bz(t),where the state variables and parameters are described in Table 1. All the parameters of the model (1.2) are supposed to be positive.

Section snippets

Well-posedness and equilibria of system (1.2)

Let C=C([-τ,0],R6) be the Banach space of continuous functions from [-τ,0] to R6 equipped with the sup-norm, here τ=max{τ1,τ2}. It is biologically reasonable to consider the following initial conditions for (1.2):(x(θ),x1(θ),y(θ),y1(θ),v(θ),z(θ))C,θ[-τ,0],x(θ)0,x1(θ)0,y(θ)0,y1(θ)0,v(θ)0,z(θ)0.By the results in Hale and Verdugn [27], we know that there is a unique solution (x(t),x1(t),y(t), y1(t),v(t),z(t)) of (1.2) with initial conditions (2.1). The following theorem establishes the

Global attractivity of E0

The global attractivity of the infected-free equilibrium by using a Lyapunov approach can be established as follows:

Theorem 3.1

If R01, then E0 is globally asymptotically attractive.

Proof

We consider a Lyapunov functionalL0=L0xt,x1(t),yt,y1(t),vt,zt=es2τ2xt(0)-x0lnxt(0)x0+es1τ1+s2τ2x1(t)+es1τ1(yt(0)-y0lnyt(0)y0)+es1τ1+s2τ2y1(t)+akes1τ1+s2τ2vt(0)+pces1τ1+s2τ2zt(0)+β1es2τ2-τ10xt(θ)vt(θ)dθ+β2es1τ1-τ20yt(θ)vt(θ)dθ,where xt(θ)=x(t+θ),yt(θ)=y(t+θ),vt(θ)=v(t+θ),zt(θ)=z(t+θ) for θ[-τ,0],τ=max{τ1,τ2}. Therefore, x(t)=

Conclusions

The global dynamics of the six-dimensional that describes the interactions of the HIV infection model with two target cells, CD4+ T-cells and macrophages and takes into account the Cytotoxic T Lymphocytes (CTL) immune response were studied in this paper. We obtain the following results by constructing Lyapunov functionals:

  • (1)

    if R01, then all solutions converge to the viral-free equilibrium E0;

  • (2)

    if R01<R0, then the immune-free equilibrium E1 is globally asymptotically stable;

  • (3)

    if R0>R0>1, then all

Acknowledgements

This work is supported by the National Natural Science Foundation of China (11171284) and program for Innovative Research Team (in Science and Technology) in University of Henan Province (2010IRTSTHN006) and Innovation Scientists and Technicians Troop Construction Projects of Henan Province (104200510011).

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