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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2015, Vol. 32 ›› Issue (6): 845-860.doi: 10.3969/j.issn.1005-3085.2015.06.006

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一类空间非齐次的SIR传染病模型的稳态解

蒋丹华,   聂  华   

  1. 陕西师范大学数学与信息科学学院，西安 710119
  • 收稿日期:2014-01-20 接受日期:2015-03-20 出版日期:2015-12-15 发布日期:2016-02-15
  • 基金资助:
    教育部新世纪优秀人才项目 (NCET-12-0894)；中央高校基本科研业务费专项项目 (GK201303008)；陕西省自然科学基础研究计划项目 (2015JM6273).

Steady-state Solutions of an SIR Epidemic Model with Spatial Heterogeneity

JIANG Dan-hua,   NIE Hua   

  1. College of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710119
  • Received:2014-01-20 Accepted:2015-03-20 Online:2015-12-15 Published:2016-02-15
  • Supported by:
    The Program of New Century Excellent Talents of Ministry of Education of China (NCET-12-0894); the Fundamental Research Funds for the Central Universities (GK201303008); the Natural Science Basic Research Plan in Shaanxi Province (2015JM6273).

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摘要： 为研究环境的空间异质性和种群扩散对传染病持续和消除的影响，本文提出了一类空间非齐次的SIR传染病模型．首先，构造模型的基本再生数$R_{0}$，并分析染病者的扩散对$R_{0}$的影响．若$R_{0}<1$，则无病平衡点全局渐近稳定；若$R_{0}>1$，则无病平衡点不稳定．其次，在低危险区域，我们运用分歧理论研究了地方病平衡点的存在性和稳定性．结果表明，减少染病者的扩散并不有利于传染病的消除，但地方病平衡点的不稳定性表明最终传染病可以得到控制．

关键词: SIR传染病模型, 基本再生数, 无病平衡点, 地方病平衡点, 分歧

Abstract:

To study the impact of spatial heterogeneity of environment and the movement of individuals on the persistence and extinction of a disease, we propose a spatial SIR reaction diffusion model. First, the basic reproduction number $R_{0}$ is defined and the effects of the diffusion of infected individuals on $R_{0}$ are analyzed. It is shown that if $R_{0}<1$, the disease-free equilibrium is globally asymptotically stable. If $R_{0}>1$, the disease-free equilibrium is unstable. Second, the existence and stability of endemic equilibrium are studied by the bifurcation theory for low risk domains. The results show that reducing the diffusion of the infected individuals is not the optimal strategy of eradicating diseases. But the instability of the endemic equilibrium implies that the disease can be controlled eventually.

Key words: SIR epidemic model, basic reproduction number, disease-free equilibrium, endemic equilibrium, bifurcation

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