Stability Analysis for an HIV Infection Model with Immune Response and Cure Rate

The dynamics of HIV infection model with immune response and cure rate is investigated. The explicit expression for the basic reproduction number of the model which determines where the virus dies out or not is obtained. With characteristic equation and Hurwitz criterion, the local stability of the equilibriums is analyzed.


Introduction
The human immuno-deficiency virus (HIV) infection, which can cause acquired immuno-deficiency syndrome(AIDS), has become an important infectious disease in both the developed and the developing nations. It causes mortality of millions of people and expenditure of a huge amount of money in disease control and health care. In recent years, some scholars have achieved many significant results by establishing the mathematical model of HIV pathogenesis which is used to study the HIV virus concentration and the change of CD4 + T cells concentration in the body [1][2][3][4][5][6][7]. The research [8] shows that, for a chemotherapy of HBV infection, under the effect of drugs, the infection of target cells can overflow from the nucleus into uninfected state covalently closed circular DNA (cccDNA). Zhou et al. [6], consider a HIV pathological model with the cure rate, and the results show that it will be able to effectively extend the duration of HIV infection if the cure rate is improved in a certain extent. The immune response following viral infection is universal and necessary in controlling or even eliminating the disease [9]. Under the inspiration of these references, we will consider a HIV infection model with immune response and cure rate in this paper as follows: x t d x x y y y t xy ay pyz y z t cyz bz x t is the concentration of uninfected cells at time t ; ( ) y t is the concentration of infected cells that produce virus at time t ; ( ) z t is the concentration of antigen-specific CTLs at time t .  is the growth rate of new healthy cells. a and d are the death rate of infected cells and uninfected cells, respectively .  is the rate constant characterizing infection of the cells. p is the death rate of infected cells due to the immune system.  is the rate at which infected cells return to healthy cells after treatment. The immune response is supposed to decay exponentially at a rate bz and get stronger at a rate cyz . All parameters in the model are positive.
Model (1) needs to be analyzed with the following initial conditions: (2) It can be proved that the solution of system (1) -(2) exists and is positive.

Equilibrium States
By solving the equations (3), we can obtain the three types of nonnegative equilibrium of model (1). 0, 0, 0.
dx xy y xy ay pyz y cyz bz We denote: R and 1 R are defined as the basic reproductive number and the immune reproduction number of model (1), respectively.

Local Stability
Next, with characteristic equation and Hurwitz criterion, we analyze the local stability of model (1). Theorem 1 Consider model (1) (i) If 0 1 R  , the infection-free equilibrium 0 E is locally asymptotically stable; (ii) If 0 1 R  and 1 1 R  , the immune-absence equilibrium 1 E is locally asymptotically stable; (iii) If 1 1 R  and 0 b c     , the immune-present equilibrium * E is locally asymptotically stable. Proof. (i) The Jacobian matrix of model (1) at 0 E is obtained as follows: By solving the third order determinant, equation (4) can be simplified as follows: The equation (5) The characteristic equation of   1 J E takes the form: By solving the third order determinant, equation (6) can be simplified as follows: The equation (7) a characteristic root: The rest of the characteristic roots satisfy the equation (8): , the equation (8) is simplified as follows: All coefficients of quadratic equation (9) are positive, and hence its roots have negative real parts. That is 1 1 R  ensures that all eigenvalues have negative real parts of equation (7) and hence the immune-absence equilibrium 1 E is locally asymptotically stable if 0 1 R  and 1 1 R  . (iii) The Jacobian matrix of model (1) at * E is obtained as follows: The By solving the third order determinant, the equation (10) can be simplified as follows:

Conclusion
In this paper, the dynamics of HIV infection model with immune response and cure rate is investigated. The explicit expression for the basic reproduction number of the model which determines where the virus dies out or not is obtained. The sufficient condition of local asymptotic stability of equilibrium is obtained by using the characteristic equation and Hurwitz criterion. The infection-free equilibrium 0 E of model (1)  x is a monotone increasing function of cure rate  . Therefore, increasing the cure rate  will further improve the concentration of CD4 + T cells in the equilibrium point of the disease, thereby prolonging the duration of HIV infection and thus effectively controlling the effect of HIV infection.
In this paper, the immune reproduction number 1 R is the monotone decreasing function of cure rate  , which means the increase of cure rate reduces the production of immune cells in the body. The mechanism is: the cure rate is to vaccinate measures can be considered in this study. Because vaccination is to replace the function of immune cells, the infection to the stimulation of immune becomes smaller. Thus, vaccination can reduce the generation of immune cells in the body.