Insights into infectious diseases with horizontal and environmental transmission: A stochastic model with logarithmic Ornstein–Uhlenbeck process and nonlinear incidence

A novel stochastic susceptible-infected-recovered-susceptible-environment (SIRSW) epidemic model with logarithmic Ornstein–Uhlenbeck process and nonlinear incidence rate is proposed based on the diversity of transmission routes of some infectious diseases and the prevalence of stochastic disturbances. We show, firstly, the global dynamics of the according deterministic model. Secondly, the existence and uniqueness of the global positive solution of the stochastic model is discussed and some sufficient conditions for the extinction of this disease are also obtained. And then, the existence of the stationary distribution of our model is obtained by constructing suitable Lyapunov function and applying the Itô’s formula. In addition, by solving the Fokker–Planck equation, the specific form of the density function near the quasi-endemic equilibrium is given. And then, the main theoretical results are explained by some numerical simulations. Finally, as an application, our stochastic model fits the Ethiopian COVID-19 data well, which not only validates the model and identifies the main pathways of local disease transmission, but also gives reasonable control strategies for the spread of the disease.