Dynamical behavior of a stochastic epidemic model with general incidence rate and Black-Karasinski process

Epidemics pose a serious threat to public health, and effective disease control measures are necessary. Vaccination is one of the most effective strategies. Considering the huge benefits of vaccination and the unpredictability of changes in the natural environment, we propose and investigate a stochastic susceptible-vaccinated-infected-recovered epidemic model with general incidence rate and mean-reversion process by incorporating the Black-Karasinski process into the vaccination strategy model. Firstly, the existence and uniqueness of the global solution of the model are proved theoretically. Then, by constructing several suitable Lyapunov functions and a compact set, the existence of the stationary distribution for the model is obtained. In addition, by solving the corresponding Fokker-Planck equation and using the related algebraic equation theory, the probability density function of the model around the quasi-endemic equilibrium is derived. Finally, some numerical simulations are employed to explain our theoretical results.