Dynamics of a Stochastic Measles Model with General Incidence Rate and Black–Karasinski Process

In this paper, considering the inevitable impact of environmental perturbations on the spread of measles, we propose a stochastic SVEIR epidemic model in which the incidence rate is general and the transmission rate satisfies the Black–Karasinski process. For the stochastic system, we first prove that it has a unique positive global solution. The existence of stationary distribution is obtained by constructing several suitable Lyapunov functions and using the ergodicity of the Black–Karasinski process, which implies the persistence of the disease. Furthermore, sufficient conditions for disease extinction are also given. Then, we establish the exact expression of the probability density function of the stationary distribution. Finally, using the data from the corresponding deterministic model, numerical simulations are presented to illustrate our theoretical results.