Dynamics and density function of a stochastic differential infectivity epidemic model with Ornstein-Uhlenbeck process

This paper considers a stochastic hybrid differential infectivity epidemic model with standard incidence perturbed by mean-reverting Ornstein-Uhlenbeck process. Applying Lyapunov method, we first show existence and uniqueness of the global solution. Then sufficient conditions for persistence in the mean and exponential extinction of the infectious disease are obtained. Furthermore, by solving the corresponding Fokker-Planck equation, we derive that the global solution around the endemic equilibrium follows a unique probability density function. Finally, numerical simulations are employed to demonstrate the analytical results.