STOCHASTIC SURVIVAL ANALYSIS OF AN EPIDEMIC MODEL WITH INNATE IMMUNITY AND TREATMENT

The innate immunity helps the individuals in the exposed compartment return into the ones in the susceptible compartment when a pathogen or virus invades the local population of having four compartments: the susceptible, the exposed, the infected and the recovered. In this study, we propose a stochastic SEIR model with innate immunity and treatment. Here, Holling type Ⅱ functional responses are used to describe the saturated effects of the innate immunity and treatment. Then, we obtain the extinction of the exposed and the infected when the basic reproduction number $ \mathcal{R}_0<1 $ and the exponential decline rate $ \nu<0 $ are valid. Moreover, we conclude that when innate immunity and treatment increase, the time that the exposed and the infected approach zero reduces. We also find that the deterministic SEIR model reaches extinction a bit faster than the stochastic SEIR model. Further, the persistence in the mean and stationary distribution of stochastic SEIR model are obtained under suitable conditions. Finally, the numerical investigations with two methods and a case study of Fuzhou COVID-19 epidemic of 2022 are discussed.