A Stochastic Model for Transmission Dynamics of AIDS with Protection Consciousness and Log-normal Ornstein-Uhlenbeck Process

In this study, a log-normal Ornstein-Uhlenbeck process and protection consciousness are included in a stochastic pandemic model of AIDS. For the 5-dimensional deterministic system, the local asymptotic stability of endemic equilibrium point is proved by Lyapunov function method instead of Routh-Hurwitz criterion. For stochastic system, we firstly verify the existence and uniqueness of global positive solution. Next, we give the sufficient condition for the presence of stationary distribution by constructing suitable Lyapunov function, and the sufficient condition for disease extinction is also given. Furthermore, the precise expression of the probability density function near the quasi-equilibrium is derived. Finally, the theoretical results are verified by numerical simulations, and the impact of log-normal Ornstein-Uhlenbeck process on the dynamic behavior of the stochastic model is also examined.