Global Properties of an Improved Hepatitis B Virus Model with Beddington-deangelis Infection Rate and Ctl Immune Response

This paper investigates the global stability of an improved hepatitis B virus model with Beddington-DeAngelis infection rate. CTL immune response is studied by constructing Lyapunov functions. If the basic production number is less than or equal to one, the uninfected steady state is globally asymptotically stable. If the basic production number is more than one, the immune-free equilibrium is globally asymptotically stable. If the immune response reproduction number is more than one, the endemic equilibrium is globally asymptotically stable.


Introduction
In the past decades, there has been much interest in mathematical modeling of HIV dynamics [11][12].Clearly, we are now able to understand the dynamics of infections at the cellular level.Of the many different mechanisms of the immune system, defenses against viral infections are of interest because many of the diseases caused by them, e.g.hepatitis B and AIDS, are chronic and incurable Zhengzhi Cao [10].
To model the immune response during a viral infection, researchers first consider the basic interactions between the immune system and the virus using the following system of differential equations [1,3].Then we introduce the model constructed by Nowak and Bangham [9].where susceptible cells x are produced at a constant rate λ , die at a density-dependent rate dx , and become infected with a rate xv β ; infected cells y are produced at rate xv β and die at a density-dependent rate ay ; free virus particles v are released from infected cells at a rate ky and die at a rate uv .z is the concentration of CTLs.Infected cells y are removed at a rate pz by the CTL immune response and the virus-specific CTL cells proliferate at a rate cy by contact with infected cells, and die at a rate bz .
In addition, we take the non cytolytic mechanisms of CTL cells into consideration, and build the following model: the bilinear term qyz represents the CTLcells cure the infected hepatocytes by a nonlyriceffectors mechanism [8].
It is true that the rate of infection in most virus dynamics models is assumed to be bilinear in the virus v and susceptible cells x .However, the actual incidence rate is probably not linear over the entire range of v and x .Thus, it is reasonable to assume that the infection rateof HIV-1 is given by Beddington-DeAngelis functional response, . The functional response nv mx xv + + 1 β was introduced by Beddington [2] and DeAngelis et al. [4].
In general we incorporate a Beddingto-DeAngelis into the model (1.2) and construct the following model:

Global asymptotical stability
Note that the basic reproductive ratio of the system (1.3) is ( ) , then the uninfected steady state ( ) is the unique steady state, called the infection-free equilibrium. (ii , then in addition to the uninfected steady state, there exists an immune-free equilibrium ) , , , ( the infected cell for per unit time is ( ) , in spite of CTL immune response.CTL cells reproduced by infected cells stimulating per unit time is 1 cy .The CTL load of a cell during the life cycle is , corresponding to the survival of the virus and CTL cells, there is an endemic equilibrium ) In this section, we consider the global asymptotical stability of these three equilibria.

Theorem 2.1 The infection-free equilibrium 0
E is globally asymptotically stable if L as follows: ( ) where d x λ = 0 , along the positive solutions of model ( 1.3), we calculating the time derivative of ( ) ,the infection-free equilibrium 0 E is stable.For . Set M be the largest invariant set in the set . The global asymptotical stability of 0 E follows from LaSalle invariance principle [5].
Proof.Define a Lyapunov function 2 L as follows: ( ) Obviously, ( ) is positive define with respect to ( ) . Along the positive solutions of model (1.3), we calculating the time derivative of ( ) ).
is a positive equilibrium point of (1.3), we have Then, we get Since the arithmetic mean is greater than or equal to the geometric mean, it is clear that Zhengzhi Cao and the equality holds only for . Thus, the immune-free equilibrium 1 E is stable.And we have Along the positive solutions of model (1.3), we calculating the time derivative of ( ) Since the arithmetic mean is greater than or equal to the geometric mean, it is clear that 0 1

Discussion
Korobeinikov [6][7] constructed a class of Lyapunov function.In [13], they proved global stability of the virus model with the incidence rate ton { } * E .Therefore, the endemic equilibrium * E is globally asymptotically stable by the LaSalle invariance principle[5].The theorems are proved.1204ZhengzhiCao presented a global analysis of model (1.1) by Lyapunov functions.In present paper, a class of more general HIV-1 infection model with Beddington-DeAngelis incidence rate and CTL immune response is considered.