Thresholds for Stochastic SIRS with Switching Exponents

This work examines a nonlinear stochastic SIRS epidemic model evolving in a randomly changing environment described by a finite-state Markov chain. The transmission mechanism incorporates regime-dependent nonlinear incidence rates, where the contact interaction between susceptible and infectious individuals is modeled by the term [Formula: see text] . This switching nonlinearity allows the model to capture varying environmental effects and heterogeneous transmission patterns more accurately, thereby providing a more realistic description of epidemic dynamics. To the best of our knowledge, the sufficient criteria governing the persistence and extinction of stochastic SIRS models with transmission rate exponents governed by Markovian switching have not yet been established in the existing literature. The principal contribution of the present study is the derivation of the rigorous sufficient conditions that characterize both the extinction and the long-term persistence of disease dynamics. Specifically, a threshold parameter Λ, expressed in terms of the switching exponents ρ_ξ(t) and ζ_ξ(t), is derived. That is, if Λ > 0, the disease exhibits strong stochastic persistence; conversely, if Λ < 0, the disease-free equilibrium state becomes globally asymptotically stable in probability, leading to eventual disease extinction. In the special case where there is no regime switching and ρ_ξ(t)=ζ_ξ(t)=1, our model recovers the classical threshold found in the literature. To support and validate the theoretical findings, numerical simulations are provided to demonstrate the dynamical behavior of the model under different environmental regimes.