您的位置：山东大学 -> 科技期刊社 -> 《山东大学学报（理学版）》

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

• • 上一篇    下一篇

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

PDF (PC)

617

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
[1] 石栋梁.两类带有时滞的非线性传染病模型的定性分析[D].湖北:湖北师范大学,2016. SHI Dongliang. Stability analysis of two kinds of nonlinear epidemic model with time delay[D]. Hubei: Hubei Normal University, 2016.
[2] 王丽敏,刘熙娟.一类具有时滞和阶段结构的SIR流行病模型分析[J].云南民族大学学报(自然科学版),2015,24(3):211-216. WANG Limin, LIU Xijuan. Analysis of an SIR epidemic model with time delay and strage-structured characteristics[J]. Journal of Yunnan Nationalities University(Natural Sciences Edition), 2015, 24(3):211-216.
[3] 李甜甜.几类带有时滞的传染病模型稳定性分析[D].山西:中北大学,2014. LI Tiantian. Stability analysis of a few kind of epidemic model with Time Delay[D]. Shanxi: North University of China, 2014.
[4] LIN Qun, JIANG Daqing. Asymptotic behaviors of a stochastic delayed SIR epidemic model with nonlinear incidence[J]. Communications in Nonlinear Science and Numerical Simulation, 2016, 40(12):89-99.
[5] CHANG Zhengbo, MENG Xinzhu. Analysis of a novel stochastic SIRS epidemic model with two different saturated incidence rates[J]. Physica A, 2017, 472(4):103-116.
[6] 朱位秋.非线性随机动力学与控制-Hamilton理论体系框架[M].北京:科学出版社,2003. ZHU Weiqiu. Nonlinear stochastic dynamics and control-Hamilton theory system framework[M].Beijing:Science Press, 2003.
[7] JIA Wantao, ZHU Weiqiu. Stochastic averaging of quasi-partially integrable Hamiltonian systems under combined Gaussian and Poisson white noise excitations[J].Physica A, 2014, 398(3):125-144.
[8] LIU Weiyan, ZHU Weiqiu.Stochastic stability of quasi-integrable and non-resonant Hamiltonian systems under parametric excitations of combined Gaussian and Poisson white noises[J].International Journal of Nonlinear Mechanics, 2014, 58(1):191-198.
[9] LIU Weiyan, ZHU Weiqiu. Stochastic stability of quasi-integrable and resonant Hamiltonian systems under parametric excitations of combined Gaussian and Poisson white noises[J].International Journal of Nonlinear Mechanics, 2014, 67(12):52-62.
[10] 李会民,王洪礼.随机激励下藻类生态系统的分岔行为[J].天津大学学报,2007,40(12):1507-1510. LI Huimin, WANG Hongli. Bifurcation of the algal ecosystem with random excitation[J].Journal of Tianjin University, 2007, 40(12):1507-1510.
[11] HUANG Zaitang, YANG Qigui, CAO Junfei. The stochastic stability and bifurcation behavior of an internet congestion control model[J].Mathematical and Computer Modeling, 2011, 54(11):1954-1955.
[12] LIM Y, CAI G. Probabilistic structural dynamics[M]. McGraw-Hill: Mcgraw-hill Professional Publishing, 2004.
[13] ARNOLD L. Random dynamical systems[M]. New York: Springer, 1998.
[14] HAS'MINSKII R Z.On the principle of averaging for Ito stochastic differential equations[C]. Prague: Kybernetika, 1968: 260-279.
[15] 傅衣铭.非线性随机参数系统的动力学与控制研究[D].陕西:西北工业大学,2007. FU Yimin.The dynamics of nonlinear stochastic parameter system and control research[D]. Shanxi: Northwestern Polytechnical University, 2007.
[16] NAMACHCHIVAYA N.Stochastic bifurcation[J].Appl Math Comput, 1990, 38(2):101-159.
[1] 张道祥,胡伟,陶龙,周文. 一类具有不同发生率的双疾病随机SIS传染病模型的动力学研究[J]. 山东大学学报（理学版）, 2017, 52(5): 10-17.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
No Suggested Reading articles found!
版权所有 © 2018 《山东大学学报(理学版)》编辑部　
地址: 山东省济南市山大南路27号山东大学中心校区（250100）　电话: +86-0531-88366917　E-mail: xblxb@sdu.edu.cn
本系统由北京玛格泰克科技发展有限公司设计开发