The threshold behavior and periodic solution of stochastic SIR epidemic model with saturated incidence

Volume 9, Issue 6, pp 4909--4923 http://dx.doi.org/10.22436/jnsa.009.06.126

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Authors

Zhongwei Cao - Department of Applied Mathematics, Jilin University of Finance and Economics, Changchun 130117, China. Wenjie Cao - School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China. Xiaojie Xu - School of Science, China University of Petroleum (East China), Qingdao 266580, China. Qixing Han - School of Mathematics, Changchun Normal University, Changchun 130032, China. Daqing Jiang - School of Science, China University of Petroleum (East China), Qingdao 266580, China. - Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia.

Abstract

We investigate degenerate stochastic SIR epidemic model with saturated incidence. For the constant coefficients case, we achieve a threshold which determines the extinction and persistence of the epidemic by utilizing Markov semigroup theory. Furthermore, we conclude that environmental white noise plays a positive effect in the control of infectious disease in some sense comparing to the corresponding deterministic system. For the stochastic non-autonomous system, we prove the existence of periodic solution.

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Keywords

• SIR epidemic model
• Markov semigroups
• asymptotic stability
• threshold
• periodic solution.

MSC

•  47D07
•  92D30

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