Analysis of a stochastic. S E Iu Ir R epidemic model incorporating the Ornstein-Uhlenbeck process

This article aims to analyze a stochastic epidemic model S E I-u I-r R(Susceptible-exposed- undetected infected-detected infected (reported-recovered) assuming that the transmission rate at which people undetected become detected is perturbed by the Ornstein-Uhlenbeck process. Our first objective is to prove that the stochastic model has a unique positive global solution by constructing a nonnegative Lyapunov function. Afterward, we provide a sufficient criterion to prove the existence of an ergodic stationary distribution of the mode by constructing a suitable series of Lyapunov functions. Subsequently, we establish sufficient conditions for the extinction of the disease. Finally, a series of numerical simulations are carried out to illustrate the theoretical results.