Dynamical behaviors of a stochastic HIV/AIDS epidemic model with Ornstein-Uhlenbeck process

AIDS is a chronic infectious disease that has been having a major impact on human health and social stability. In this paper, based on the deterministic SIATR model, a stochastic SIATR model is developed to account for the spread of AIDS by introducing the Ornstein-Uhlenbeck process. First, the existence and uniqueness of the global positive solution of the model are investigated. Second, a threshold R-0(E) is set: when R-0(E) < 1, the disease will become extinct; when R-0(E )> 1, the disease will persist. Then, it is shown that there exists a stationary distribution for the model when R-0(E) > 1. On this basis, we derive the exact expression for the probability density function of the model in the neighborhood of the quasi-equilibrium state. Finally, the results of the previous proof are verified by several numerical simulations.