JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2017, Vol. 52 ›› Issue (4): 72-82.doi: 10.6040/j.issn.1671-9352.0.2016.614

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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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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

CLC Number: 

  • Q332
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