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Necessary and sufficient condition for extinction and persistence of SIRS system with random perturbation. (English) Zbl 1334.92412
Summary: This paper characterizes the qualitative dynamics of a stochastic SIRS epidemic model. The study shows that the dynamics of the model are determined by a certain threshold quantity $$\mathcal{R}_{\mathcal{S}}$$ expressed in terms of the model parameters and the intensity of the noise. If $$\mathcal{R}_{\mathcal{S}} < 1$$, the disease will be eliminated from the community; whereas an epidemic occurs if $$\mathcal{R}_{\mathcal{S}} > 1$$. Our results recover the known results in the earlier literature as special cases. The presented results are illustrated by numerical simulations.

##### MSC:
 92D30 Epidemiology 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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##### References:
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