Dynamic properties, probability density, and estimation of an SIRIS epidemic model with an Ornstein-Uhlenbeck process★ ★This research is supported by the Research Innovation Program for Postgraduates of Xinjiang Uygur Autonomous Region under Grant (No. XJ2023G017), the National Natural Science Foundation of China (No. 12061070) and the Natural Science Foundation of Xinjiang Uygur Autonomous Region of China (No. 2021D01E13).

Abstract

Considering the relapse and temporary immunity of some diseases, a stochastic SIRIS epidemic model with saturation incidence is developed in this paper, in which the Ornstein-Uhlenbeck process is introduced to describe the randomness of the model. The existence and uniqueness of the global solution are first proved to investigate the dynamic properties of the SIRIS model. By constructing a series of Lyapunov functions, the thresholds R0E and R0S are further derived to determine the extinction and persistence of the disease. Subsequently, we prove that the solution of the model admits a stationary distribution. The probability density function is obtained by the algebraic method. In addition, the unknown parameters of the density function are estimated by using the maximum likelihood estimation. Finally, numerical simulations verify the results of the above properties.