Stationary Distribution and Extinction of a Stochastic HIV-1 Infection Model with Distributed Delay and Logistic Growth

In this paper, we propose a stochastic HIV-1 infection model with distributed delay and logistic growth. Firstly, we transfer the stochastic model with weak kernel case into an equivalent system through the linear chain technique. Then, we establish sufficient conditions for the existence of a stationary distribution of the model by constructing a suitable stochastic Lyapunov function. Moreover, we obtain sufficient criteria for extinction of the infected cells; that is, the uninfected cells are survival and the infected cells are extinct. Our results show that the smaller white noise can ensure the existence of a stationary distribution when the basic reproduction number R0S of the stochastic system is bigger than one, while the larger white noise can lead to the extinction of the infected cells when the basic reproduction number R of the deterministic system is smaller than one.