Stability analysis of multi-group deterministic and stochastic epidemic models with vaccination rate

doi:10.1088/1674-1056/23/9/090201

›› 2014, Vol. 23 ›› Issue (9): 90201-090201.doi: 10.1088/1674-1056/23/9/090201

• GENERAL • 下一篇

Stability analysis of multi-group deterministic and stochastic epidemic models with vaccination rate

王志刚^{a b}, 高瑞梅^c, 樊晓明^b, 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

收稿日期:2014-01-01 修回日期:2014-02-24 出版日期:2014-09-15 发布日期:2014-09-15
基金资助:
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).

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

Received:2014-01-01 Revised:2014-02-24 Online:2014-09-15 Published:2014-09-15
Contact: Fan Xiao-Ming, Han Qi-Xing E-mail:fanxm093@163.com;hanqixing123@163.com
Supported by:
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).

摘要/Abstract

摘要： 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.

关键词: MSIR epidemic model, equilibrium, graph theory, Brownian motion

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.

Key words: MSIR epidemic model, equilibrium, graph theory, Brownian motion

中图分类号: (Global analysis and analysis on manifolds)

02.40.Vh

89.20.Ff (Computer science and technology) 02.60.Lj (Ordinary and partial differential equations; boundary value problems)

王志刚, 高瑞梅, 樊晓明, 韩七星. Stability analysis of multi-group deterministic and stochastic epidemic models with vaccination rate[J]. , 2014, 23(9): 90201-090201.

Wang Zhi-Gang (王志刚), Gao Rui-Mei (高瑞梅), Fan Xiao-Ming (樊晓明), Han Qi-Xing (韩七星). Stability analysis of multi-group deterministic and stochastic epidemic models with vaccination rate[J]. Chin. Phys. B, 2014, 23(9): 90201-090201.

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Stability analysis of multi-group deterministic and stochastic epidemic models with vaccination rate

Stability analysis of multi-group deterministic and stochastic epidemic models with vaccination rate

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