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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.

92D30 Epidemiology
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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