考虑部分免疫和环境传播的麻疹传染病模型的全局稳定性

数学物理学报 ›› 2019, Vol. 39 ›› Issue (4): 909-917.

考虑部分免疫和环境传播的麻疹传染病模型的全局稳定性

靖晓洁,赵爱民*(),刘桂荣

山西大学数学科学学院太原 030006

收稿日期:2018-04-12 出版日期:2019-08-26 发布日期:2019-09-11
通讯作者: 赵爱民 E-mail:lgr5791@sxu.edu.cn
基金资助:
国家自然科学基金;山西省自然科学基金;the NSFC;the Natural Science Foundation Program of ShanxiProvince

Global Stability of a Measles Epidemic Model with Partial Immunity and Environmental Transmission

Xiaojie Jing,Aimin Zhao*(),Guirong Liu

School of Mathematical Sciences, Shanxi University, Taiyuan 030006

Received:2018-04-12 Online:2019-08-26 Published:2019-09-11
Contact: Aimin Zhao E-mail:lgr5791@sxu.edu.cn
Supported by:
国家自然科学基金;山西省自然科学基金;the NSFC;the Natural Science Foundation Program of ShanxiProvince

摘要/Abstract

摘要：

该文考虑一个具有部分免疫和环境传播的麻疹传染病模型，得到基本再生数R₀，并通过构造Lyapunov函数，研究了该模型的无病平衡点和地方病平衡点的全局稳定性.当R₀ < 1时，无病平衡点是全局渐近稳定的，即麻疹不会传播开；当R₀ > 1时，模型存在唯一的地方病平衡点，且是全局渐近稳定的，即麻疹的传播保持在一个稳定的状态.最后，通过数值分析说明了这些结果的合理性.该文工作对于预防和控制麻疹病毒的传播具有实际意义.

关键词: 部分免疫, 环境传播, 基本再生数, Lyapunov函数, 全局稳定性

Abstract:

In this paper, a measles epidemic model with partial immunity and environmental transmission is considered, and the basic reproduction number R₀ is obtained. By constructing Lyapunov functions, we prove the global asymptotic stability of the infection-free equilibrium and the endemic equilibrium. When R₀ < 1, the infection-free equilibrium is globally asymptotically stable, which implies that measles dies out eventually; when R₀ > 1, the model has a unique endemic equilibrium, which is globally asymptotically stable, that is the transmission of measles keeps a steady state. Finally, the simulations are carried to verify the rationality of the results. This work has practical significance for guiding us to prevent and control the measles spread.

Key words: Partial immunity, Environmental transmission, The basic reproduction number, Lyapunov function, Global stability

中图分类号:

O175.1

靖晓洁, 赵爱民, 刘桂荣. 考虑部分免疫和环境传播的麻疹传染病模型的全局稳定性[J]. 数学物理学报, 2019, 39(4): 909-917.

Xiaojie Jing, Aimin Zhao, Guirong Liu. Global Stability of a Measles Epidemic Model with Partial Immunity and Environmental Transmission[J]. Acta mathematica scientia,Series A, 2019, 39(4): 909-917.

参考文献

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