Weak persistence and extinction of a stochastic epidemic model with distributed delay and Ornstein-Uhlenbeck process.

A stochastic distributed delay epidemic model with Markovian switching and Allee effect is constructed, where the infectious disease transmission rate follows a mean-reverting Ornstein-Uhlenbeck process. Hybrid dynamic effects of Ornstein-Uhlenbeck process and Lévy jumps on infectious disease transmission are discussed. Stochastically, the ultimate boundedness of the positive solution is investigated. The existence of a unique global positive solution is studied. By constructing appropriate stochastic Lyapunov functionals, sufficient conditions for weak persistence of the infected population are investigated. The existence of a unique ergodic stationary distribution is discussed based on Hasminskii's ergodic theory. Sufficient conditions for the extinction of infectious disease are discussed. Numerical simulations are carried out to show consistency with the theoretical analysis.