A generalized stochastic SIRS epidemic model incorporating mean-reverting Ornstein–Uhlenbeck process

The aim of this work is to study a new stochastic SIRS epidemic model that includes the mean-reverting Ornstein–Uhlenbeck process and a general incidence rate. First, we prove the global existence and positivity of the solution by using Lyapunov functions. Second, we analytically make out the stochastic epidemic threshold T̃0S which pilots the extinction and persistence in mean of the disease. We have proven that the disease extinguishes when T̃0S<1. Otherwise, if T̃0S>1, then disease is persistent in mean. For the critical case T̃0S=1, we have shown that the disease dies out by using an approach involving some appropriate stopping times. Finally, we present a series of numerical simulations to confirm the feasibility and correctness of the theoretical analysis results.