Chin. Phys. B, 2014, Vol. 23(9): 090201    DOI: 10.1088/1674-1056/23/9/090201
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# Stability analysis of multi-group deterministic and stochastic epidemic models with vaccination rate

Wang Zhi-Gang (王志刚)a b, Gao Rui-Mei (高瑞梅)c, Fan Xiao-Ming (樊晓明)b, Han Qi-Xing (韩七星)d
a College of Mathematics, Jilin University, Changchun 130012, China;
b School of Mathematical Sciences, Harbin Normal University, Harbin 150500, China;
c College of Science, Changchun University of Science and Technology, Changchun 130022, China;
d School of Mathematics, Changchun Normal University, Changchun 130032, China
Abstract  We discuss in this paper a deterministic multi-group MSIR epidemic model with a vaccination rate, the basic reproduction number R0, a key parameter in epidemiology, is a threshold which determines the persistence or extinction of the disease. By using Lyapunov function techniques, we show if R0 is greater than 1 and the deterministic model obeys some conditions, then the disease will prevail, the infective persists and the endemic state is asymptotically stable in a feasible region. If R0 is less than or equal to 1, then the infective disappear so the disease dies out. In addition, stochastic noises around the endemic equilibrium will be added to the deterministic MSIR model in order that the deterministic model is extended to a system of stochastic ordinary differential equations. In the stochastic version, we carry out a detailed analysis on the asymptotic behavior of the stochastic model. In addition, regarding the value of R0, when the stochastic system obeys some conditions and R0 is greater than 1, we deduce the stochastic system is stochastically asymptotically stable. Finally, the deterministic and stochastic model dynamics are illustrated through computer simulations.
Keywords:  MSIR epidemic model      equilibrium      graph theory      Brownian motion
Received:  01 January 2014      Revised:  24 February 2014      Accepted manuscript online:
 PACS: 02.40.Vh (Global analysis and analysis on manifolds) 89.20.Ff (Computer science and technology) 02.60.Lj (Ordinary and partial differential equations; boundary value problems)
Fund: Project supported by the National Natural Science Foundation of China (Grant No. 11326078) and the Project of Science and Technology of Heilongjiang Province of China (Grant No. 12531187).
Corresponding Authors:  Fan Xiao-Ming, Han Qi-Xing     E-mail:  fanxm093@163.com;hanqixing123@163.com