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The extinction and persistence of infective individuals are closely relatedto the random change of the environment. In this paper, via therandom/stochastic SIRS models, we analyze qualitatively and quantitatively theimpact caused by the random change of the environment. Our contributionsconsist in (i) giving some sufficient conditions on extinction (persistence) ofthe infectious individuals even though they are persistent (resp. extinct) incertain fixed environments; (ii) revealing the influence of random switching ofincidence functions on extinction for the infectious individuals, which has notbeen studied before; (iii) establishing a criterion to judge extinction of theinfectious individuals for a range of random/stochastic SIRS models withstate-dependent switching via a stochastic comparison for functionals of jumpprocesses. Moreover, some examples are set to illustrate the applications ofour theory.

Asymptotic Behavior of SIRS Models in State-dependent Random Environments