A stochastic SIRS epidemic model with a saturation incidence under semi-Markovian switching

The saturation incidence is of vital importance in the spread of large-scale epidemics. Although many achievements have been concerning deterministic epidemic models with saturated incidence, little is known about a stochastic version with a saturation incidence under semi-Markovian switching. Based on the above, this paper aims to propose such a stochastic model. Via constructing suitable Lyapunov functions, we prove the existence and uniqueness of a positive solution and provide a threshold value for disease tending to be extinct. Further, the bounded and persistent properties are provided for the model. Under sufficient conditions, the stochastic model has a unique ergodic stationary distribution. As illustrations, several examples are presented to show the theoretical results. The results show that the density function of the solution depends on regime switching. As an application, we study COVID-19 data in Iceland. The SMC-ABC algorithm is used to estimate unknown parameters of the model with semi-Markovian switching.