Stochastic modeling of SIS epidemics with logarithmic Ornstein-Uhlenbeck process and generalized nonlinear incidence.

In this paper, we investigate a stochastic SIS epidemic model with logarithmic Ornstein-Uhlenbeck process and generalized nonlinear incidence. Our study focuses on the construction of stochastic Lyapunov functions to establish the threshold condition for the extinction and the existence of the stationary distribution of the stochastic system. We also derive the exact expression of the density function around the quasi-endemic equilibrium, which provides valuable insight into the transmission and progression of the disease within a population. Our findings demonstrate the importance of considering the impact of stochasticity on the spread of epidemics, particularly in the presence of complex incidence mechanisms and stochastic environmental factors. Additionally, the stochastic threshold reveals that ordinary differential equation models and white noise models underestimate the severity of disease outbreaks, while our proposed the stochastic epidemic model with logarithmic Ornstein-Uhlenbeck process accurately captures real-world scenarios.