We expand an SIR epidemic model with vertical and nonlinear incidence rates from a deterministic frame to a stochastic one. The existence of a positive global analytical solution of the proposed stochastic model is shown, and conditions for the extinction and persistence of the disease are established. The presented results are demonstrated by numerical simulations.

In recent years, the world lived a horrible nightmare named the Covid-19 pandemic. It’s changing lifestyle caused the closure of critical establishments like schools, sports halls, companies, etc., as well as causing damage and menace for humanity. Mathematical models are helpful tools for modeling and analysing the infectious disease transmission This article presents a stochastic model of the Covid-19 epidemic for a population with five compartments. We give a numerical analysis of the proposed stochastic model. Also, we compare it with results of the corresponding deterministic model.

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In this paper, we consider a stochastic SIRS epidemic model with nonlinear incidence and Markovian switching. By using the stochastic calculus background, we establish that the stochastic threshold R_{ swt}  can be used to determine the compartment dynamics of the stochastic system. Some examples and numerical simulations are presented to confirm the theoretical results established in this paper.