Longtime behavior of an SIR model with perturbed disease transmission coefficient
Nguyen Huu Du  Faculty of Mathematics, Mechanics, and Informatics, University of ScienceVNU, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam (email) Abstract: In this paper, we consider a stochastic SIR model with the perturbed disease transmission coefficient. We determine the threshold $\lambda$ that is used to classify the extinction and permanence of the disease. Precisely, $\lambda<0$ implies that the diseasefree $(\frac{\alpha}{\mu}, 0, 0)$ is globally asymptotic stable, i.e., the disease will disappear and the entire population will become susceptible individuals. If $\lambda>0$ the epidemic takes place. In this case, we derive that the Markov process $(S(t), I(t))$ has a unique invariant probability measure. We also characterize the support of a unique invariant probability measure and prove that the transition probability converges to this invariant measures in total variation norm. Our result is considered as an significant improvement over the results in [6,7,11,18].
Keywords: SIR model, extinction, permanence, stationary distribution, ergodicity.
Received: December 2015; Revised: June 2016; Available Online: November 2016. 
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