Analysis of a stochastic HIV/AIDS model with commercial heterosexual activity and Ornstein–Uhlenbeck process

Abstract

Considering the effects of commercial heterosexual behavior and environmental noise on the dynamics of HIV/AIDS disease, in this study, we establish a stochastic HIV/AIDS model that combines commercial heterosexual behavior and ln-type Ornstein–Uhlenbeck process. Firstly, for the deterministic model, i.e., the stochastic noise is not taken into account, the local and global asymptotic stability of the equilibria in terms of basic reproduction number $R_0$ are given. Then, for the stochastic model, the uniqueness and existence of global solutions, the existence of stationary distribution under $R_0^S>1$, and the exponential extinction of HIV-positive patients under $R_0^E<1$ can all be obtained by constructing suitable random Lyapunov functions. In particular, under certain conditions, specific expression of the probability density function near the quasi-endemic equilibrium of the stochastic system can be obtained. Finally, numerical simulations indicate that controlling noise intensity, as well as commercial heterosexual activities, can help control the transmission of HIV.