Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

Long-time behavior of an SIR model with perturbed disease transmission coefficient

Pages: 3429 - 3440, Volume 21, Issue 10, December 2016      doi:10.3934/dcdsb.2016105

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Nguyen Huu Du - Faculty of Mathematics, Mechanics, and Informatics, University of Science-VNU, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam (email)
Nguyen Thanh Dieu - Department of Mathematics, Vinh University, 182 Le Duan, Vinh, Nghe An, 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 disease-free $(\frac{\alpha}{\mu}, 0, 0)$ 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.
Mathematics Subject Classification:  34C12, 60H10, 92D25.

Received: December 2015;      Revised: June 2016;      Available Online: November 2016.