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山东大学学报(理学版) ›› 2016, Vol. 51 ›› Issue (3): 51-59.doi: 10.6040/j.issn.1671-9352.0.2015.212

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一类具有信息变量和等级治愈率的SIR传染病模型的研究

史学伟,贾建文*   

  1. 山西师范大学数学与计算机科学学院, 山西 临汾 041004
  • 收稿日期:2015-05-05 出版日期:2016-03-20 发布日期:2016-04-07
  • 通讯作者: 贾建文(1963— ), 男, 硕士, 教授, 研究方向为微分方程及其在生物数学中的应用. E-mail:jiajw.2008@163.com E-mail:434780667@qq.com
  • 作者简介:史学伟(1990— ), 女, 硕士研究生, 研究方向为生物数学. E-mail:434780667@qq.com
  • 基金资助:
    山西省自然科学基金资助项目(2013011002-2)

Study on an SIR epidemic model with information variable and graded cure rate

SHI Xue-wei, JIA Jian-wen*   

  1. School of Mathematics and Computer Science, Shanxi Normal University, Linfen 041004, Shanxi, China
  • Received:2015-05-05 Online:2016-03-20 Published:2016-04-07

摘要: 研究了一类具有信息变量,饱和发生率和等级治愈率的SIR传染病模型。首先给出基本再生数R0,利用Routh-Hurwitz 判据和特征根方法得到当R0<1时,无病平衡点局部渐近稳定;当R0>1时,地方病平衡点局部渐近稳定; 模型出现两种分支, 分别为跨临界分支和Hopf分支。 其次通过构造Lyapunov函数证明了无病平衡点的全局稳定性, 利用自治收敛定理证明了地方病平衡点的全局稳定性。 最后用数值模拟验证了理论结果的正确性。

关键词: SIR传染病模型, 模型分支, 信息变量, 全局渐近稳定, 等级治愈率

Abstract: The stability of the SIR epidemic model with information variable, saturated incidence rate and graded cure rate was studied. We obtain the basic reproduction number R0, The local asymptotical stability of equilibria is verified by analyzing the eigenvalues and using the Routh-Hurwitz criterion,this model exhibits two bifurcations, one is transcritical bifurcation and the other is Hopf bifurcation. We also discuss the global asymptotical stability of the disease free equilibrium by constructing a Lyapunov function and the endemic equilibrium by autonomous convergence theorem. A numerical analysis is given to show the effectiveness of the main results.

Key words: information variable, bifurcation of model, global stability, SIR epidemic model, graded cure rate

中图分类号: 

  • O175.12
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