Global insights into a stochastic SIRS epidemic model with Beddington–DeAngelis incidence rate

Abstract

This study develops a stochastic SIRS compartmental model for exploring the transmission dynamics of infectious diseases, integrating the Beddington–DeAngelis incidence rate and vaccination. In the deterministic case, the reproduction number [Formula: see text] is derived, and the global dynamics is analyzed using the Lyapunov function with respect to [Formula: see text]. The outcomes underscore that [Formula: see text] completely governs the overall dynamics of the system. In the stochastic case, the primary challenge arises from the two-dimensional boundary system, preventing the Fokker–Planck equation from obtaining the density function of the invariant measure. To address the weak convergence property regarding the invariant measure for both the stochastic system and its corresponding two-dimensional boundary system, the concept of limit measures is introduced. The theoretical results indicate that the persistence and extinction of the infectious disease are entirely determined by the Lyapunov exponent [Formula: see text], representing the long-term growth rate. Numerical simulations further support these findings.