GLOBAL STABILITY ANALYSIS AND PERMANENCE FOR AN HIV-1 DYNAMICS MODEL WITH DISTRIBUTED DELAYS∗

This paper mainly investigates the global asymptotic stabilities of two HIV dynamics models with two distributed intracellular delays incorporating Beddington-DeAngelis functional response infection rate. An eclipse stage of infected cells (i.e. latently infected cells), not yet producing virus, is included in our models. For the first model, it is proven that if the basic reproduction number R0 is less than unity, then the infection-free equilibrium is globally asymptotically stable, and if R0 is greater than unity, then the infected equilibrium is globally asymptotically stable. We also obtain that the disease is always present when R0 is greater than unity by using a permanence theorem for infinite dimensional systems. What is more, a n-stage-structured HIV model with two distributed intracellular delays, which is the extensions to the first model, is developed and analyzed. We also prove the global asymptotical stabilities of two equilibria by constructing suitable Lyapunov functionals.


Introduction
Human Immunodeficiency Virus (HIV) is a retrovirus that infects the human cells which deal with the immune system. HIV viruses destroy CD4 T cells in vivo and decreases the resistance of the immune system. HIV dynamics within a host depend on the HIV replication resulting into dynamics of the infected and uninfected cells. Many mathematical models have been developed to understand the interaction of the HIV infection and the immune system [4,7,15,19]. Nowak and his cooperators in [19] consider one basic HIV model which included populations of three components: normal CD4 T-cells, infected CD4 T-cells and free HIV viruses. Many studies have showed that upon infection and transcription of viral RNA into cell DNA, a fraction of CD4 T-cells fail to actively produce virus until they are activated(an eclipse stage) [4,5,20]. Therefore, it is necessary that the latent stage of infected CD4 T-cells should be considered during the interaction. Especially, Elaiw and Azoz consider one HIV infection model with latently infected cells and Beddington-DeAngelis functional response in [4]. Moreover, there is an intracellular time delay between infection of a cell and production of new virus particles [8,9,11]. The infected cells would not change into productively infected cells in the same time. Distributed intracellular delays can describe continuous time delays between viral infection and production. Many papers consider viral dynamics models with distributed intracellular delays [13,17,24,25,29]. Dynamic behaviors for the basic HIV model with infinite distributed intracellular delays are researched in [1,13,25].
Motivated by the above works and features of HIV infection, we propose one model with two distributed delays, which is described by the following equations , The state variables x(t), y(t), z(t) and v(t) represent the concentrations of uninfected CD4 T cells, productively infected cells, infected cells in the eclipse stage and HIV virus, respectively. The uninfected CD4 T cells are produced at a rate λ and die at the rate d (average life span of cells), and become latently infected cells at the rate 1+αx(t)+γv(t) , the Beddington-DeAngelis functional response [7,15,27]. Parameter µ is the rate at which latently infected cells become productively infected cells and these cells die at the rate δ (average life span of cells in the eclipse state). The productively infected cells are produced at rate µz(t) and are removed at rate ay(t). HIV viruses are produced by the actively infected cells with rate constant p and are removed with constant rate r. The intracellular delay τ , allowing to be infinite because of drugs resistent strains, describes the periods between HIV viruses to enter infected cells and the infected cells start to become latently infected cells or the periods between latently infected cells become actively infected cells and actively infected cells start to produce new viral particles. The two distributed intracellular delays are introduced with the kernel functions given by f (τ ) = f 1 (τ )e −µ1τ , g(τ ) = g 1 (τ )e −µ2τ , where µ 1 and µ 2 are the per capita loss rates for the uninfected CD4 T-cells due to viral infection and HIV viruses during time interval [t−τ, t]. f (τ ) accounts for the probability that target cells contacted by the virus particles at time t − τ and become latently infected cells at time t , and g(τ ) accounts for the probability that a cell productively infected at time t − τ starts to yield new infectious virus at time t. The nonnegative functions f 1 (τ ) and g 1 (τ ) are incorporated to describe the probability distribution for the two processes and are assumed to satisfy Suppose all the parameters are nonnegative and the initial conditions Because of existence of the infinite delay, we consider Banach phase space [9,24,25]: where ω > 0 is a constant and the norm is defined ||φ|| = sup θ≤0 |φ(θ)|e ωθ .
Here we suppose the initial conditions for system (1.1-1.4) are given When f 1 (τ ) = g 1 (τ ) = δ(τ − 0) with δ(·) being the Dirac delta function, system (1.1-1.4) reduces to system in [4]. When f 1 (τ ) = δ(τ − τ 1 ) and g 1 (τ ) = δ(τ − 0), system (1.1-1.4) reduces to a similar model with a single discrete delay [27]. When f 1 (τ ) = δ(τ − τ 1 ) and g 2 (τ ) = δ(τ − τ 2 ), system (1.1-1.4) reduces to a similar model with two discrete delays in [28] where the authors established global stabilities of two equilibria by constructing Lyapunov functionals. It is obvious that system (1.1-1.4) includes a simple three-dimensional model (α = γ = 0) studied in works [13,25] where global stabilities of equilibria have been obtained as well. Moreover, our system includes another system with delay functions [18]. The main purpose of this paper is to analyse global properties for system (1.1-1.4) and explore the impact of distributed intracellular delays on the dynamical behavior of the system. In addition, we Study the conditions under which diseases will die out. We begin model analysis with proving the positivity of the solutions, boundedness of the solutions, and uniform persistence of the system. We also prove that global stabilities of two equilibria only depend on the basic reproduction number by constructing suitable Lyapunov functionals which are motivated by the recent works [7,12,13,16,23,25]. The paper is ordered as follows: equilibria, properties analysis and uniform persistence are studied in Section 2. Global asymptotic stabilities of two equilibria are discussed by constructing suitable Lyapunov functionals in Section 3. The n-stage-structured model for HIV infection is studied in Section 4. Discussion and conclusion in Section 5.

Properties analysis and uniform persistence
First, we establish the nonnegativity and boundedness of solutions of system (1.1-1.4) in the following analysis.
Proposition 2.1. Let (x(t), z(t), y(t), v(t)) ∈ R 4 be any solution of system (1.1-1.4) satisfying the initial conditions (1.5) and (1.6), then x(t), z(t), y(t), and v(t) are all nonnegative for t > 0 and are ultimately bounded. Proof. The methods are similar to those in the article [14]. Assume that there exists t > 0 at which x(t), z(t), y(t) or v(t) is equal to 0. Denote If x(t * ) = 0, it follows that That means x(t) < 0 for t ∈ (t * − ε, t * ), where ε > 0 is sufficiently small. That is in contradiction with x(t) > 0 for t ∈ (0, t * ). Thus x(t) > 0 for all t ≥ 0. If z(t * ) = 0, it follows that x(t * ) > 0, y(t * ) ≥ 0 and v(t * ) ≥ 0 when t ∈ [0, t * ]. Then for t ∈ [0, t * ], we can easily obtain dz(t) dt ≥ −δz(t) − µz(t). It follows that This is a contradiction to the assumption of z(t * ) = 0. Hence, z(t) > 0 for all t ≥ 0. Similarly, we can obtain y(t) > 0 and v(t) > 0 for all t ≥ 0. Solutions remain positive for positive initial conditions. It follows from the equation That means x(t) is ultimately bounded. For convenience of notations, we set From the equation (1.2), we can obtain Thus, we obtain lim sup t→∞ z(t) ≤ βλb1 dγ(µ+δ) . This implies that z(t) is also ultimately bounded. By a similar argument, we can obtain Therefore, y(t) and v(t) are both ultimately bounded. This completes the proof.
From biological considerations, we study system (1.1-1.4) in the closed set: where R 4 + denotes the non-negative cone. Proposition 2.1 implies that the omega limit sets of system (1.1-1.4) are contained in the region P . Obviously, solutions of system (1.1-1.4) remain nonnegative for nonnegative initial conditions. So P is a positively invariant set with respect to system (1.1-1.4). In the following, we will investigate dynamic behavior on P .
We note that the basic reproduction number R 0 for the system is given by The basic reproduction number R 0 represents the average number of secondary cases generated by an infected case throughout its infection period.
The steady states ( equilibria ) of system (1.1-1.4) satisfy the following equations: We obtain that the equations always have an infection-free equilibrium We will establish the uniform persistence of system (1.1-1.4) and start with the definition.
In order to study the permanence of system (1.1-1.4), we present the persistence theory on infinite dimensional systems from [6]. The persistence theory and techniques we are using have been recently employed in [10,22,25]. First, we introduce some concepts, notations and terminology. Denote by T (t), t ≥ 0, the family of solution operators corresponding to system (1.1-1.4). We also denote X = (C * + ) 4 and the positive orbit Lemma 2.1 ( [10,22,25]). Suppose that we have the following conditions: Then, T (t) is a uniform repeller with respect to X 0 , i.e. there is an η > 0 such that for any Proposition 2.2. Assume that R 0 > 1, then the disease is endemic and system hence, by integration for t ≥ 0 and the initial condition x(0), we have This leads to x(t) is uniformly bounded away from zero and has positive ultimate lower boundary. That is, there exists a constant η 1 = λ Thus we will prove that other three variables have positive ultimate lower boundary. Let In the following, we verify that the conditions in Lemma 2.1 are satisfied. It is straightforward to see that the conditions (1) and (2) are satisfied. From Proposition 2.1, we obtain that T (t) is point dissipative in X. So the condition (3) is satisfied. Let U be a bounded set of X, and B > 0 be such that for any (σ, σ 2 , σ 3 , σ 4 ) ∈ U , σ < B and σ i ≤ B, i = 2, 3, 4. Denote ϕ(s) := Be ωs , s ≤ 0. Consider the solutionx(t),z(t),ȳ(t), andv(t) with initial condition x(0) = B, z 0 = y 0 = v 0 = ϕ. For any solution x(t), z(t), y(t), and v(t) with initial solution from U , we can obtain that This is a contradiction to y(t) ≤ȳ(t). So we have v(t) <v(t) for t ≥ 0. Similarly, we have x(t) <x(t), z(t) <z(t) and y(t) <ȳ(t) for t ≥ 0. As a result of all this, we get that T (t) is monotone. Using the arguments of Proposition 2.1, we obtainx(t),z(t),ȳ(t) andv(t) are bounded and dominate every solution with initial data from U . Hence, we obtain the condition (4). Next we show that T (t) is asymptotically smooth. Let We define a set Υ by We obtain that Υ is compact set in C * from Lemma 3.2 of the paper [2]. Let x t , z t , y t and v t be the segment of solution with initial condition lying in an arbitrary bounded set U ∈ X. We will show that lim t→∞ d(x t , Υ) = 0. z t , y t , and v t can be derived analogously. we can obtain that there exists a T > 0 such that x(t) ≤ M for t ≥ T and x(t) = M or x(t) < M for t > 0 from Proposition 2.1. We first consider the case x(t) = M for t > 0 and denote K be the maximum of x(t) on [0, T ] and define the function ψ t (s) for t > T Thus, we have ψ t (s) ∈ Υ and Separating to the intervals [ Thus, we get that lim t→∞ d(x t , Υ) = 0 and T (t) is asymptotically smooth. The case x(t) < M for t > 0 can be treated similarly. Thus, we verified the condition (5). Regarding (6), we can obtain that Λ = E 0 and isolated. The covering is simple N = E 0 , which is acyclic (there is no orbit that connects E 0 to itself in X 0 ). At last, we prove that W s (E 0 ) X 0 = Φ. Suppose it is not true, i. e. there exists a solution u t ∈ X 0 such that Then for any sufficient small > 0, there is T 1 ( ) large enough, such that x(t) > λ d − , z(t) < ε, y(t) < ε, and v(t) < ε for all t ≥ T 1 ( ). When R 0 > 1, for the given > 0, we have There exists a T 2 such that For the given T 2 , there exists a T 3 > max{T 1 , T 2 } such that x(t − τ ) ≥ x 0 − , for any t > T 3 and τ ∈ [0, T 2 ], we have What is more, we get By the mean value theorem for integrals we obtain that there is a ξ t such that Meanwhile we also consider If z(t), y(t), v(t) → 0, as t → +∞, then by a standard comparison argument and the nonnegativity, the solution (z c (t), y c (t), v c (t)) of , has to convergence to 0 as well. However, we define the function Therefore, we can obtain lim t→∞ G(t) = 0. On the other hand, differentiating with respect to time gives Therefore, G(t) goes to infinity or approaches a positive limit as t → ∞ which is a contradictions to lim t→∞ G(t) = 0. Thus, W s (E 0 ) X 0 = Φ is satisfied. By Lemma 2.1, T (t) is a uniform repeller with respect to X 0 . That is, we obtain that there exists η 2 > 0 for initial condition φ ∈ X 0 such that So, we denote η = min{η 1 , η 2 } and obtain lim inf That shows that system (1.1-1.4) is uniformly persistent. This completes the proof.
The uniform persistence of system (1.1-1.4) shows that the disease will not only exist in every subpopulation but in fact the number of individuals in each group will always remain beyond a certain positive level.

Global stability of the equilibria
In this section, we would consider the global asymptotic stability of the two equilibria of system (1.1-1.4). We would apply the Lyapunov functional and the LaSalle invariance principle to establish global dynamical properties for the delay differential equations.
Proof. Define a Lyapunov functional V 1 (t) as follows: It is easily seen that V 1 (t) ≥ 0 for all (x(t), z(t), y(t), v(t)) > 0 and V 1 (t) = 0 if and only if x(t) = x 0 and y(t) = z(t) = v(t) = 0. Calculating the derivative of V 1 (t) with respect to time t along positive solutions of system (1.1-1.4), it follows that Therefore, we obtain Using integration by parts, we calculate the derivative of U 1 (t) , Noting that Similarly, we have Thus, we get From system (1.1-1.4), we obtain that the largest compact invariant set is the singleton {E 0 }. Accordingly, it follows from Lyapunov-LaSalle invariance principle that E 0 (x 0 , 0, 0, 0) is globally asymptotically stable when R 0 ≤ 1. Theorem 3.2. If R 0 > 1, the infected equilibrium E * (x * , z * , y * , v * ) of system (1.1-1.4) is globally asymptotically stable. Proof. Define a Lyapunov functional V 2 (t) as follows: where Calculating the derivative of V 2 (t) with respect to time t along positive solutions of system (1.1-1.4), it follows that We calculate the derivative of U 3 (t), Using integration by parts, we obtain Using integration by parts, meanwhile we calculate the derivative of U 4 (t), At the infected equilibrium E * , we have Thus, we obtain Note that: Therefore, we obtain ).
By calculating we have As we know function f (x) = 1 − x + lnx is always non-positive for x > 0, and f (0) = 0 if and only if x = 1. Noting that x * , z * , y * , v * > 0, it is easy to see that dV2(t) dt ≤ 0 and dV2(t) dt = 0 if and only if x(t) = x * , z(t) = z * , y(t) = y * , v(t) = v * . From Lyapunov-LaSalle invariance principle, it shows that E * is globally asymptotically stable when R 0 > 1. This completes the proof.

N-stage-structured model for HIV infection
In this section, we consider a n-stage-structured model of latently infected cells with two distributed delays. The model is described by where i = 2, · · ·, n. y i (t)(i = 2, · · ·, n − 1) denote a class of infected cells that are not yet producing virus, i.e. cells in the eclipse phase (stage-structured model).
Proof. Define a Lyapunov functional V 3 (t) as follows: Similarly, we have Calculating the derivative of V 3 (t), it follows that Then, we have So, it follows that dV3(t) dt ≤ 0 if R 1 ≤ 1. It is clear that E 00 (x 0 , 0, · · ·, 0, 0) is globally asymptotically stable when R 1 ≤ 1 from Lyapunov-LaSalle invariance principle.

Discussion and conclusion
In the present paper, we have proposed two mathematical models which includes two distributed intracellular delays. Meanwhile, we use the Beddington-DeAngelis functional response as nonlinear incidence rate. Here we introduce two general distributed delays with infinite delays. For the first model, we first derive that the solutions are all nonnegative and ultimately bounded for all t > 0. We also obtain the basic reproduction number R 0 > 1 which can determine global dynamics of the model and investigate uniform persistence of the model. What is more, we obtain global asymptotical stabilities of two equilibria by using the Lyapunov functional method. Our results imply that HIV virus can be eliminated at last if R 0 ≤ 1. HIV virus will persist in vivo if R 0 > 1. Our conclusions show that time delay has no effect on the global asymptotic stability of the model. In addition, for the n-stage-structured model of latently infected cells, we calculate the basic reproduction number and derive that two equilibria are globally asymptotically stable by constructing proper Lyapunov functionals. It shows that multistage delays cannot change global asymptotic stability under some conditions.