Global Stability of an HIV-1 Infection Model with General Incidence Rate and Distributed Delays

In this work an HIV-1 infection model with nonlinear incidence rate and distributed intracellular delays and with humoral immunity is investigated. The disease transmission function is assumed to be governed by general incidence rate f(T, V)V. The intracellular delays describe the time between viral entry into a target cell and the production of new virus particles and the time between infection of a cell and the emission of viral particle. Lyapunov functionals are constructed and LaSalle invariant principle for delay differential equation is used to establish the global asymptotic stability of the infection-free equilibrium, infected equilibrium without B cells response, and infected equilibrium with B cells response. The results obtained show that the global dynamics of the system depend on both the properties of the general incidence function and the value of certain threshold parameters R 0 and R 1 which depends on the delays.


Introduction
Immunity can be broadly categorized into adaptive immunity and innate immunity. Adaptive immunity is mediated by clonally distributed and lymphocytes, namely, humoral and cellular immunity, and is characterized by specificity and memory. The humoral immunity plays an important role in antiviral defence by attacking virus. A basic mathematical model describing HIV-1 infection dynamics model with humoral immunity was introduced by Murase et al.
where ( ), ( ), ( ), and ( ) represent the densities of uninfected cells, infected cells, virus, and cells at time , respectively. Λ and are the birth and death rate constants of uninfected cells. is the infection rate, is the average number of virus particles produced over the lifetime of a single infected cells, and is the death rate of infected cells; is the death rate constant of the virus, and are the recruited rate and death rate constants of cells, and is the cells neutralization rate. Mathematical models for virus dynamics with antibody immune response has drawn much attention of researchers (see, e.g., [1][2][3][4][5][6][7][8][9][10][11][12][13] and the reference therein). Recently many studies have been done to improve the model (1) by introducing delays and changing the incidence rate according to different practical background. These studies used different delayed models with different forms of incidence rate; see, for example, [6,[9][10][11] for discrete delays and [5,13] for distributed delays.
In the present paper, motivated by the works of [1,5,13], we propose the following model with a general incidence rate and distributed delays and humoral immunity: where the parameters in system (2) have the same meanings as in system (1). ( , ) is the general incidence rate. It is assumed in (2) (2), the probability distribution functions ( ), = 1, 2, are assumed to satisfy ( ) > 0, = 1, 2 and The function is assumed to be continuously differentiable in the interior of 2 + and satisfies the following hypotheses: (H 4 ) ( ( , ) )/ ≥ 0, for all , > 0.
The biological meaning of hypothesis (H 1 ) to (H 4 ) is given in [10].
The distributed delay is more general than the discrete one and it is more adapted to biological phenomena. ℎ 1 or ℎ 2 can be infinity. The present paper is organized as follows. In Section 2, we establish the nonnegativity and boundedness of solutions and we derived the basic reproduction ratios for viral infection and humoral immune response 0 and 1 , respectively. In Section 3, the existence of a possible three positive equilibria, an infection-free equilibrium * 0 , an infected equilibrium without cells response * 1 , and an infected equilibrium with cells response * 2 , is established. In Sections 4 and 5, we show that the global asymptotic stability of these equilibria depend only on the basic reproduction numbers under some hypotheses on the incidence function. In Section 6, some examples are given. A brief discussion is given in the last section to conclude this paper.
To simplify the notations we note that Global behaviour of system (2) may depend on the basic reproduction numbers 0 and 1 given by where * 0 = Λ/ and with, = * 0 − ( / 1 2 ). Here, 0 and 1 are the basic reproduction ratios for viral infection and humoral immune response of system (2), respectively. Based on the hypotheses (H 2 ) and (H 3 ) it is clear that 1 < 0 .

The Existence of Positive Equilibria
In this section we prove the existence of positive equilibrium. The system (2) always has an infection-free equilibrium * 0 = ( * 0 , 0, 0, 0). For other possible equilibriums, we have the following theorem.

Global Stability of the Infection-Free Equilibrium
In this section, we study the global stability of the infectionfree equilibrium * 0 of system (2). Proof. Define a Lyapunov functional: where 1 and 2 are given in (11).
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Global Stability of the Infected Equilibria
In this section, we study the global stability of the infected equilibrium without cells response * 1 and the infected equilibrium with cells response * 2 of system (2) by the Lyapunov direct method.
The function : .

Proof. Define a Lyapunov functional
where 1 and 2 are given in (11).

Conclusion
In the current paper, we have studied an HIV-1 infection model with humoral immune response and intracellular distributed delays and general incidence rate. The model has two distributed time delays describing time needed for infection of cell and virus replication. The global stability of our model is studied by employing the method of Lyapunov functionals which are motivated by McCluskey [16] for delayed epidemic models. This general incidence represents a variety of possible incidence functions that could be used in virus dynamics model as well as epidemic models. We establish that the global dynamics are determined by two threshold parameters, the basic reproduction ratios for viral infection and humoral immune response 0 and 1 , respectively, which depend on the incidence function and the delay. We have proved that the infection-free equilibrium * 0 is globally asymptotically stable if the basic reproduction ratios viral infection 0 ≤ 1. In this case, the virus is cleared up. The hypotheses on the general incidence function are used to assure the existence of infected equilibrium without cells response * 1 and infected equilibrium with cells response * 2 . We prove that if 1 ≤ 1 < 0 , the infected equilibrium without cells response * 1 is globally asymptotically stable and if 1 > 1, the infected equilibrium with cells response * 2 is globally asymptotically stable.