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Long-time behaviour of a perturbed SIR model by white noise

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  • In this paper, we consider a stochastic SIR model with perturbed disease transmission coefficient. We present sufficient conditions for the disease to extinct exponentially. In the case of persistence, we analyze long-time behaviour of densities of the distributions of the solution. We will prove that the densities of the solution can converge in $L^1$ to an invariant density under appropriate conditions. Also we find the support of the invariant density. Specially, when the intensity of white noise is relatively small, we find a new threshold for an epidemic to occur.
    Mathematics Subject Classification: Primary: 47D07, 60H10; Secondary: 60J60, 92D30.


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