Threshold dynamics of a stochastic seqirs model with semi-markov switching

This paper studies the threshold dynamics of a stochastic SEQIRS epidemic model that is subject to two sources of randomness: A white-noise perturbation of the transmission rate and a semi-Markov switching of the environment. The incidence rate has the nonlinear form $ \beta S^{p}f(E) $ with $ p\in\mathbb{N} $, and the sojourn time in each environmental regime is allowed to follow an arbitrary distribution rather than an exponential one. A threshold $ \mathcal{R}_{0}^{s} $ is derived from the stationary distribution of the embedded chain and from the mean sojourn times. The disease dies out almost surely under one of two different extinction mechanisms. The first one combines the threshold requirement $ \mathcal{R}_{0}^{s} < 1 $ with an upper bound on the noise intensity. The second one imposes only a sufficiently large lower bound on the noise intensity and needs no separate threshold assumption, because this lower bound already forces $ \mathcal{R}_{0}^{s} < 1 $. It persists, and the time averages of all five compartments admit positive lower bounds, when $ \mathcal{R}_{0}^{s} > 1 $, the noise intensity lies in a prescribed interval, and (only for $ p\ge2 $) the susceptible class is bounded away from zero for all large times. The threshold tends to the basic reproduction number of the corresponding deterministic system as the noise vanishes, and is never larger than it. Simulations carried out for the Holling type Ⅱ incidence confirm both alternatives. They show in particular that a subsystem with $ \mathcal{R}_{0}^{k} > 1 $ may coexist with extinction of the switched system, so that the sojourn-time distributions cannot be ignored when the fate of an epidemic is assessed.