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Dynamics of hybrid switching diffusions SIRS model. (English) Zbl 1366.60098
Summary: The main aim of this paper is to study the effect of the environmental noises in the asymptotic properties of a stochastic version of the classical SIRS epidemic model. The model studied here include white noise and telegraph noise modeled by Markovian switching. We obtained conditions for extinction both in probability one and in pth moment. We also established the persistence of disease under different conditions on the intensities of noises, the parameters of the model and the stationary distribution of the Markov chain. The highlight point of our work is that our conditions are sufficient and almost necessary for extinction and persistence of the epidemic. The presented results are demonstrated by numerical simulations.

##### MSC:
 60J28 Applications of continuous-time Markov processes on discrete state spaces 60J60 Diffusion processes 34F05 Ordinary differential equations and systems with randomness 60H40 White noise theory 92D30 Epidemiology
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