The Threshold Behavior Of Stochastic Autonomous And Non-autonomous SIS Epidemic Model With Standard Incidence

In this article, we extend SIS deterministic epidemic model with standard incidence by introducing a white noise in natural death rates, and get a autonomous stochastic SIS epidemic model and non-autonomous stochastic SIS epidemic modelFirst, for the autonomous system, we investigate the extinction and use Markov semigroup theory proving that the system is asymptotic stable and exists a invariant density under some conditions, furthermore, we obtain the threshold which determines the epidemic prevalence or extinction. Then, for the non-autonomous system with periodic coefficients, we prove the existence of positive periodic solution by periodic Markov process theory, and then verify the condition for extinction, finally, we get the threshold of non-autonomous system.