一类具有信息变量和等级治愈率的SIR传染病模型的研究

doi:10.6040/j.issn.1671-9352.0.2015.212

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

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^*

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^*

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/Abstract

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

史学伟,贾建文. 一类具有信息变量和等级治愈率的SIR传染病模型的研究[J]. 山东大学学报（理学版）, 2016, 51(3): 51-59.

SHI Xue-wei, JIA Jian-wen. Study on an SIR epidemic model with information variable and graded cure rate[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2016, 51(3): 51-59.

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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

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

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