Dynamical Behavior of a Stochastic SVI Infectious Disease Model with Ornstein-Uhlenbeck Process

Considering the great social benefits of vaccination and the unpredictability of changes in the natural environment, this paper introduces the Ornstein-Uhlenbeck process to characterize the stochastic fluctuations in the transmission rate, proposes and studies a stochastic SVI epidemic model. Firstly, the existence and uniqueness of the global positive solution for the model are analyzed. Secondly, by constructing appropriate Lyapunov functions, defining a compact set, and using tools such as Itô's formula, the existence of a stationary distribution for the model is proved. In addition, the sufficient condition for disease extinction is derived. Finally, some numerical simulations are used to explain the main theoretical results. The results show that, increasing vaccination rates can inhibit the rise in the number of infections and reduce the risk of transmission, but it can not completely eliminate the disease.