Dynamics and numerical simulations of a generalized mosquito-borne epidemic model using the Ornstein-Uhlenbeck process: Stability, stationary distribution, and probability density function

In this paper, we proposed a generalized mosquito-borne epidemic model with a general nonlinear incidence rate, which was studied from both deterministic and stochastic insights. In the deterministic model, we proved that the endemic equilibrium was globally asymptotically stable when the basic reproduction number $R_0$ was greater than unity and the disease free equilibrium was globally asymptotically stable when $R_0$ was lower than unity. In addition, considering the effect of environmental noise on the spread of infectious diseases, we developed a stochastic model in which the infection rates were assumed to satisfy the mean-reverting log-normal Ornstein-Uhlenbeck process. For this stochastic model, two critical values, known as $R_0^s$ and $R_0^E$, were introduced to determine whether the disease will persist or die out. Additionally, the exact probability density function of the stationary distribution near the quasi-equilibrium point was obtained. Numerical simulations were conducted to validate the results obtained and to examine the impact of stochastic perturbations on the model.