Stationary distribution and extinction of a stochastic HLIV model with viral production and Ornstein–Uhlenbeck process

Abstract

In the past few decades, the human immunodeficiency virus (HIV), as a deadly pathogen, has imposed great harmful effects on human health. In addition, from the viewpoint of the microscopic, the interference of random factor exists in the process of virus replication. However, there are few works devoted to the theoretical studies of viral infection models with biologically reasonable stochastic effects yet. To study the effects of stochastic perturbations on the pathogenesis process of HIV/AIDS, in this paper, we develop and study a stochastic HLIV model with viral production and Ornstein–Uhlenbeck process, in which we assume that the conversion rate from an eclipse-phase cell to an infected cell satisfies the Ornstein–Uhlenbeck process. Firstly, by using the existence and uniqueness theory of solutions, we prove that there is a unique global solution to the stochastic model for any initial value. Then we use stochastic Lyapunov function methods and Fatou lemma to establish sufficient conditions for the existence of a stationary distribution of positive solutions to the stochastic system, which reflects the strong persistence of the disease. In addition, by adopting the comparison theorem of one-dimensional stochastic differential equation and the theory of ergodicity, we obtain sufficient conditions for complete eradication of the infected cells and free viruses. Our theoretical results show that the bigger speed of reversion or the smaller intensity of volatility is conducive to clearing infected cells and free viruses while the larger intensity of volatility or the smaller speed of reversion is not. We also find that this way of introducing stochastic perturbations is reasonable from the perspective of mathematics and biology. Finally, numerical simulations are carried out to demonstrate our main results. The modeling and methods can be applied to study other multigroup viral infection models or epidemic model with Ornstein–Uhlenbeck process, such as S-DI-A model, DS-I-A model, DS-DI-A model, SEI epidemic model, et al.