Stochastic HIV Infection Model with CTLs Immune Response Driven by Lévy Jumps

This paper mainly investigates the effect of the lévy jumps on the stochastic HIV infection model with cytotoxic T lymphocytes (CTLs) immune response. First, we prove that there is a unique global positive solution in any population dynamics, then we find sufficient conditions for the extinction of the disease. For proofing the persistence in mean, a special Lyapunov function be established, we obtain that if 1 1 R >  the infected CD4 + T-cells and virus particles will persistence in mean. Finally, numerical simulations are carried out to illustrate the theoretical results.

During the process of viral infection, the host response is induced, stimulating the production of CTLs, which kill the infected CD4 + T-cells by CTLs immune responses when the body is infected with HIV. CTLs are the major immune response factors in the human body that limit the virus's in vivo replication and determine the number of viruses. Recently, the dynamics of HIV infection with CTLs response have attracted much attention from scholars to conduct research.
Such as, Elaiw et al. showed the global stability of HIV/HTLV [4]. Koenig et al. pointed out, there is clearest evidence for the active CTLs selection of viral variants could contribute to the pathogenesis of AIDS and that clinical progression can occur despite high levels of circulating HIV-1-specific CTLs [5]. Therefore, CTLs play a critical role in inhibiting HIV by killing virus-infected T cells, much research has been done on the dynamics of HIV infection with CTLs response [6] [7] [8], and so on. The earlier models for viral dynamics with immune response are the general form: In ref [8], Nowak  From the studies of [8], we know that the CTLs response will increase when cy b > . If 0 1 R < , the model has an infection-free equilibrium 0 , 0, 0, 0 E d and it is asymptotically stable; while if 0 1 R > , the CTLs response may become only transiently activated, but eventually, the system will converge to the equilibrium ( ) 1 , , ,0 E x y v = , and when 1 1 R < is asymptotically stable without an active CTLs response; finally, if 1 [14] take into account the effect of randomly fluctuating environment and incorporate the white noise with two parameters of system (1.1) because epidemic systems are often subject to environmental noise, and the deterministic models do not incorporate the effect of fluctuating environment. They replaced the parameters ( ) Hence, the stochastic version corresponding to system (1.1) takes the following form: The infection-free equilibrium is stochastically asymptotically stable in the large, when 0 1 R < ; when the white noise is small, while if 0 1 R > and 1 1 R ≤ and satisfy ( ) X t represent the solution of system (1.2) and ⋅ is 2 l norm) is small, we have CTLs response is only transiently activated, when 1 1 R < , the CTLs response is activated. But when the white noise is large, the CTLs response is still transiently activated even is small, which never hap- Due to the inherently stochastic nature of biochemical processes, the dynamic process of HIV viral infection may suffer strong fluctuation [15], such that the classical stochastic model (1.2) cannot explain the strong, occasional fluctuations of the biological environment. It is reasonable to further consider another random noise, namely the lévy jump noise, into HIV viral dynamical model. Therefore, the aim of this paper is to present a comprehensive study for stochastic system with lévy jump process to describe this strong fluctuation. The main points and novel contributions of the paper are as follows: 2) Using the Khasminskii-Mao theorem and appropriate Lyapunov functions, we show that the model has a unique global positive solution.
3) By applying Itŏ's formula and the large number theorem for martingales, we established sufficient conditions for the extinction and persistence in mean of infected CD4 + T-cells and virus particles.
In this paper, we aim to introduce Lévy noise into above ecological epidemic model, as a result, model (1.1) becomes: , N is a Poisson counting measure with characteristic measure π on a measurable subset Z of ( ) are standard Brownian motions, and independent with N throughout the paper. This paper is organized as follows. In Section 2, we will give some preliminaries and show there exist a global and positive solution under appropriate conditions. In Section 3, we will investigate the conditions for the extinction of infected cells and CTLs extinction. In Section 4, conditions will be derived for the persistence in mean. In Section 5, numerical simulations are carried out to illustrate the theoretical results. Finally, we give some conclusions.

Preliminaries and Global Positive Solution
First, we introduce the following notations, throughout this paper, let is an Itŏ's-Lévy process of the form .
Then the generalized Itŏ's formula with Lévy jumps is given by For convenience, we introduce following notations and the assumption.
Assumption 1. We assume that ( ) 3) has the following properties: The following proof is similar with [16].    Proof. Since the coefficients of the equations is locally Lipschitz continuous, for any given initial value , where e τ is the explosion time.
To show this solution is global, we need to show that e τ = ∞ a.s. At first, we , , , ln ln 1 ln 1 ln , where a is a positive constant to be defined later.
The proof of the remainder is similar to the proof [18], so we omitted it.

Extinction of the Disease
For simplicity, we introduce the following notations: Integrating both sides of (3.1) from 0 to t yields,     )   2  2  2 2  2  3  2  2  2  2   3  3  2  3   1  d  d  2  2   d  ln 1  ln 1 Integrating from 0 to t on both sides of (3.4), we obtain  That is to say, the disease will die out with probability one.

Numerical Results
In this section, we present some numerical simulations to discuss the effect of lévy noise on the viral dynamics. Euler scheme is an order 1 2 approximation, be used to investigate the dynamics of stochastic differential equations driven by lévy process in some papers [23] [24] [25] [26].
In Figure 1, fix parameters as follows: , the intensity of noise 1 5 σ = , We compared the process of deterministic model, stochastic model with white noise and stochastic model with Lévy jump, the infected CD4 + T-cells, virus particles and CTLs cells extinction in all cases, but strong fluctuation will accelerate the extinction of all.
In Figure 2, fix parameters as follows: , the intensity of noise 1 1.5 σ = ,      Figure 3(a) and Figure 3(b), we can see that strong fluctuation will result in the decrease of the infected CD4 + T-cells, virus particles.

Conclusions
This Due to the inherently stochastic nature of HIV infection, some interesting topics deserve further discussion, such as considering the effects of time delay on the stochastic system, we will go about this case subsequently.