Globale Stability in a Viral Infection Model with Beddington-DeAngelis Functional Response

The stability of a mathematical model for viral infection with Beddington-DeAngelis functional response is considered in this paper. If the basic reproduction number 0 1 R  , by the Routh-Hurwitz criterion and Lyapunov function, the uninfected equilibrium 0 E is globally asymptotically stable. Then, the global stability of the infected equilibrium 1 E is obtained by the method of Lyapunov function.


INTRODUCTION
Human Immunodeficiency Virus and Acquired Immune Deficiency Syndrome (AIDS) have received much attention from the first case of AIDS was diagnosed on December 1st in 1981.It is proven to be valuable in understanding the population dynamics of viral load in vivo with mathematical models.In the last decade, many mathematical models have been developed to describe the infection with Human Immunodeficiency virus (HIV) (see [1][2][3][4][5][6][7][8][9]).Nowak et al. [1,3] proposed the following model: In model (1.1), it is also assumed that healthy The biological meanings of these parameters are the similar to those appearing parameters in model (1.1).

EQUILIBRIA AND GLOBAL STABILITY ANALYSIS
2), which describes the average number of newly infected T-cells generated from one infected T-cells.We can obtain that 0 0 ( ,0,0) Ex  is an uninfected equilibrium where 0 x d

 
, and Now, we begin to study the stabilities of these two equilibria.
Firstly, we begin to study the stability of the uninfected equilibrium 00 ( ,0,0) Ex  .Evaluating the Jacobian matrix of model (1.2) at 0 .
By simple computations, the characteristic equation is , and According to the Routh-Hurwitz criterion, it is obtained that uninfected equilibrium 0 E is locally asymptotically stable when 0 1 R  .When 0 1 R  it is easy to obtain that 3 0 A  , and 3 (0) 0, ( ) A        .That is to say the charac- terristic equation has positive solution.So 0 E is unstable when 0 1 R  .Moreover, we construct a Lyapunov function for studying the global stability.Let By the LaSalle invariance principle, when 0 1 R  the uninfected equilibrium 0 E is globally asymptotically stable.
Summarizing the discussion above, we obtained the following conclusion.

Theorem 2.1 The uninfected equilibrium 0
E is globally asymptotically stable when 0 1 R  and is unstable when 0 1 R  .
Then we begin to analysis the stability of infected equilibrium 1 E .Let Along the trajectories of model (1.2), we obtained x y z is the equilibrium point of (1.2), we have (1 )

Zhanwei and Xia
Moreover  E exists and is globally asymptotically stable when 0 1 R  .

CONCLUSION
In this paper, we have investigated a HIV-I mathematical model with Beddington-DeAngelis function response.According to the Routh-Hurwitz criterion and LaSalle invariance principle, we obtained the following conclusion: 1) when 0 1 R  the uninfected equilibrium 0 E is globally asymptotically stable; 2) when 0 1 R  the infected equili- brium 1 E is globally asymptotically stable.

4 CD  T-cells are infected at a rate xv  , infected 4 CD 4 CD
x t y t and () vt represent the numbers(densities) of healthy 4 CD  T-cells, infected 4 CD  T-cells and viral particles at time t respectively.The model assumed that healthy  T-cells are lost at a rate ay , and virus are produced by infected  T-cells at a rate ky and removed at a rate rv .

4
CD  T-cells are input at a constant rate  , and die at a rate dx .
2 According to LaSalle invariance principle, the infected equilibrium 1 E is globally asymptotically stable.Theorem 2.2The infected equilibrium 1 1 zz 