Discrete Dynamics in Nature and Society

Volume 2012 (2012), Article ID 253703, 13 pages

doi:10.1155/2012/253703

Research Article

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

A. M. Elaiw1,2

¹Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

²Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut 71511, Egypt

Received 14 March 2012; Revised 10 July 2012; Accepted 24 July 2012

Academic Editor: Cengiz Çinar

Copyright © 2012 A. M. Elaiw. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 is less than unity then the uninfected steady state is globally asymptotically stable, and if then the infected steady state exists and it is globally asymptotically stable.

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

where ,   and represent the populations of uninfected CD4⁺ T cells, infected cells, and free virus particles at time , respectively. Here, represents the rate of which new CD4⁺ T cells are generated from sources within the body, is the death rate constant, and 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 . The virus particles are produced by the infected cells with rate constant and are removed from the system with rate constant . 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 and 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 become infected cells at time , where is a random variable with a probability distribution . The factor accounts for the loss of target cells during time period . On the other hand, it is assumed that a cell infected at time starts to yield new infectious virus at time , where is distributed according to a probability distribution .

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 . If , then the uninfected steady state is globally asymptotically stable (GAS) and if , 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:

The state variables describes the plasma concentrations of: , the uninfected CD4⁺ T cells; , the infected CD4⁺ T cells; , the uninfected macrophages; , the infected macrophages; , the free virus particles. Here, ,   are positive constants, , and ,  . The factors ,   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 functions and are assumed to satisfy and ,   and

where is a positive number. Then

The initial conditions for system (2.1)–(2.5) take the form

where , and is the Banach space of fading memory type defined as [30]

where is the Banach space of continuous functions mapping the interval into . 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 be any solution of (2.1)–(2.5) satisfying the initial conditions (2.8), then ,  ,  ,   and are all nonnegative for and ultimately bounded.

Proof. From (2.1) and (2.3) we have

which indicates that , for all . Now from (2.2), (2.4), and (2.5) we have

confirming that ,  , and for all .

Next we show the boundedness of the solutions. From (2.1) and (2.3) we have , . This implies ,  .

Let ,  , then

where . Hence , where , . On the other hand,

then . Therefore, ,  ,  ,  , and are ultimately bounded.

2.2. Steady States

It is clear that system (2.1)–(2.5) has an uninfected steady state , where , . In addition to , the system can also have a positive infected steady state . The coordinates of the infected steady state, if they exist, satisfy the following equalities:

where

Following van den Driessche and Watmough [32], we define the basic reproduction number for system (2.1)–(2.5) as

where and 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. If , then there exists a positive steady state .

Proof. From (2.14) and (2.15) we have

where . From (2.20) into (2.16) we get

Equation (2.21) can be written as

If , then the positive solution of (2.21) is given by:

It follows that, if then ,  ,  ,   and 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: , for any . We also define a function as

It is clear that for any and has the global minimum .

Theorem 2.3. If , then is GAS.

Proof. Define a Lyapunov functional as follows:

where , .

The time derivative of along the trajectories of (2.1)–(2.5) satisfies

Collecting terms of (2.26) we get

If then for all . By Theorem in [31], the solutions of system (2.1)–(2.5) limit to , the largest invariant subset of . Clearly, it follows from (2.27) that if and only if ,  , and  . Noting that is invariant, for each element of we have , then . From (2.5) we drive that

This yields . Hence if and only if ,  ,  , and . From La Salle's Invariance Principle, is GAS.

Theorem 2.4. If , then is GAS.

Proof. We construct the following Lyapunov functional:

Differentiating with respect to time yields

Collecting terms we obtain

Using the infected steady state conditions (2.14)–(2.16), and the following equality:

we obtain

Then collecting terms of (2.33) and using the following equalities:

we obtain

Equation (2.35) can be rewritten as

Using the following equality:

we can rewrite as

It is easy to see that if ,  , then . By Theorem 5.3.1 in [31], the solutions of system (2.1)–(2.5) limit to , the largest invariant subset of . It can be seen that if and only if ,  , and , that is,

If then from (2.39) we have , and hence equal to zero at . LaSalle's Invariance Principle implies global stability of .

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 is less than unity, then the uninfected steady state is GAS and if , then the infected steady state exists and it is GAS.

Acknowledgment

This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The author, therefore, acknowledges with thanks DSR technical and financial support.

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