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

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.


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 ẋ t λ − dx t − 1 − u rt βx t v t , 1.1 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 u rt ∈ 0, 1 and u p ∈ 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 R 0 .If R 0 ≤ 1, then the uninfected steady state is globally asymptotically stable GAS and if R 0 > 1, then the infected steady state exists and it is GAS.

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: x 1 , the uninfected CD4 T cells; y 1 , the infected CD4 T cells; x 2 , the uninfected macrophages; y 2 , the infected macrophages; v, the free virus particles.Here, α i , i 1, 2 are positive constants, β i 1 − u rt β i , and p i 1 − u p p i , i 1, 2. The factors e −n i τ , 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 functions f i τ and g i τ are assumed to satisfy f i τ > 0 and g i τ > 0, i 1, 2 and where s is a positive number.Then

2.7
The initial conditions for system 2.1 -2.5 take the form

Nonnegativity and Boundedness of Solutions
In the following, we establish the nonnegativity and boundedness of solutions of Proof.From 2.1 and 2.3 we have which indicates that x i t ≥ 0, for all t ≥ 0. Now from 2.2 , 2.4 , and 2.5 we have where σ i min{d i , a i }.Hence lim sup t → ∞ X i t ≤ L i , where L i λ i /σ i , i 1, 2. On the other hand,

Steady States
It is clear that system 2.1 -2.5 has an uninfected steady state E 0 x 0 1 , 0, x 0 2 , 0, 0 , where In addition to E 0 , the system can also have a positive infected steady The coordinates of the infected steady state, if they exist, satisfy the following equalities: 2.14 where 2.17 Following van den Driessche and Watmough 32 , we define the basic reproduction number for system 2.1 -2.5 as where R 1 and R 2 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.
Proof.From 2.14 and 2.15 we have where δ i α i β i /d i .From 2.20 into 2.16 we get

2.21
Equation 2.21 can be written as If R 0 > 1, then the positive solution of 2.21 is given by:

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 ∈ {x 1 , y 1 , x 2 , y 2 , v}.We also define a function H : 0, ∞ → 0, ∞ as

2.24
It is clear that H z ≥ 0 for any z > 0 and H has the global minimum H 1 0.
Proof.Define a Lyapunov functional W 1 as follows:

2.25
where The time derivative of W 1 along the trajectories of 2.1 -2.5 satisfies

2.26
Collecting terms of 2.26 we get

2.27
If R 0 ≤ 1 then dW 1 /dt ≤ 0 for all x 1 , x 2 , 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 {dW 1 /dt 0}.Clearly, it follows from 2.27 that dW 1 /dt 0 if and only if x i x 0 i , 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
This yields y 1 y 2 0. Hence dW 1 /dt 0 if and only if x i x 0 i , y i 0, i 1, 2, and v 0. From La Salle's Invariance Principle, E 0 is GAS.
Proof.We construct the following Lyapunov functional:

2.29
Differentiating with respect to time yields

2.30
Collecting terms we obtain g i τ e −n i τ y i t − τ dτ cv * .

2.31
Using the infected steady state conditions 2.14 -2.16 , and the following equality: we obtain

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

2.35
Equation 2.35 can be rewritten as

2.36
Using the following equality: we can rewrite dW 2 /dt as

2.38
It is easy to see that if x * i , y * i , v * > 0, i 1, 2, then dW 2 /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 {dW 2 /dt 0}.It can be seen that dW 2 /dt 0 if and only if x i x * i , v v * , and H 0, that is,

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 R 0 is less than unity, then the uninfected steady state is GAS and if R 0 > 1, then the infected steady state exists and it is GAS.
2 τ y 2 η − τ dτ dη, 2.11 confirming that y 1 t , y 2 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 ẋi t ≤ λ Therefore, x 1 t , y 1 t , x 2 t , y 2 t , and v t are ultimately bounded.
then from 2.39 we have y i y * i , and hence dW 2 /dt equal to zero at E 1 .LaSalle's Invariance Principle implies global stability of E 1 .