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山东大学学报（理学版） ›› 2017, Vol. 52 ›› Issue (4): 72-82.doi: 10.6040/j.issn.1671-9352.0.2016.614

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一类随机的SIR流行病模型的动力学行为分析

白宝丽1,张建刚1*,杜文举2,闫宏明3   

  1. 1. 兰州交通大学数理学院, 甘肃 兰州 730070;2. 兰州交通大学交通运输学院, 甘肃 兰州 730070; 3. 太原理工大学矿业工程学院, 山西 太原 030000
  • 收稿日期:2016-12-30 出版日期:2017-04-20 发布日期:2017-04-11
  • 通讯作者: 张建刚(1978— ),男,博士,副教授, 研究方向为非线性动力系统.E-mail: zhangjg7715776@163.com E-mail:2283961467@qq.com
  • 作者简介:白宝丽(1989— ),女,硕士,研究方向为非线性动力系统. E-mail: 2283961467@qq.com
  • 基金资助:
    国家自然科学基金资助项目(61364001)

Dynamic behavior analysis of a stochastic SIR epidemic model

BAI Bao-li1, ZHANG Jian-gang1*, DU Wen-ju2, YAN Hong-ming3   

  1. 1. School Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou 730070, Gansu, China;
    2. School of Traffic and Transportation, Lanzhou Jiaotong University, Lanzhou 730070, Gansu, China;
    3. School of Mining and Engineering, Taiyuan University of Technology, Taiyuan 030000, Shanxi, China
  • Received:2016-12-30 Online:2017-04-20 Published:2017-04-11

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摘要： 首先对受参数扰动的具有阶段结构的SIR流行病模型引入随机项,建立了具有阶段结构的随机SIR流行病模型的非线性微分方程,应用随机中心流形定理和随机平均法相关定理将其化为Ito微分方程。然后,基于Oseledec乘性遍历理论,应用最大Lyapunov指数和奇异边界理论分别分析了该随机系统的局部随机稳定性和全局随机稳定性;利用拟不可积Hamilton系统随机平均法对系统的随机Hopf分岔行为作了分析。最后,选取其中的某些参数作为分叉参数得到相应的平稳概率密度函数图和联合概率密度函数图,对发生分岔的概率和位置进行了验证。

关键词: 随机Hopf分岔, 随机SIR流行病模型, 随机稳定性, Hamilton理论

Abstract: Taking the random factors into account,we introduced the randomness into the SIR epidemic model and established the nonlinear differential equation of the random SIR epidemic model with stage structure. Then by applying stochastic center manifold and stochastic average method,the stochastic differential equation was reduced order and we got the corresponding Ito differential equation. Based on the Oseledec multiplicative ergodic theorem,the conditions of local and global stability of the system were discussed by using the largest Lyapunov exponent and boundary category. Besides,we selected some of these parameters as the bifurcate parameter,and the stochastic Hopf bifurcation behavior of the system were analyzed by the stochastic averaging method of the quasi-non-integrable Hamiltonian system. Finally,the functional image of stationary probability density and jointly stationary probability density were simulated,and the bifurcate point from the probability and location was verified.

Key words: stochastic SIR epidemic model, Hamilton theory, stochastic Hopf bifurcation, stochastic stability

中图分类号: 

  • Q332
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