AAM  >> Vol. 5 No. 3 (August 2016)

    Bifurcation and Chaos Analysis of a Class of Discrete SIR Epidemic Models

  • 全文下载: PDF(597KB) HTML   XML   PP.390-398   DOI: 10.12677/AAM.2016.53048  
  • 下载量: 979  浏览量: 3,189   国家自然科学基金支持


庞琴,张建刚,邓田,殷俊,卢加荣:兰州交通大学数学学院,甘肃 兰州

离散SIR模型Flip分岔Hopf分岔混沌随机参数Discrete-Time SIR System Flip Bifurcation Hopf Bifurcation Chaos Random Parameter



The paper discusses the dynamical behaviors of a discrete-time SI epidemic model. The local sta-bility of the disease-free equilibrium and endemic equilibrium is obtained. It is shown that the model undergoes Flip bifurcation and Hopf bifurcation by using center manifold theorem and bi-furcation theory. So it exhibits the complex dynamical behaviors. These results reveal far richer dynamical behaviors of the discrete epidemic model.

庞琴, 张建刚, 邓田, 殷俊, 卢加荣. 一类离散SIR流行病模型的分岔和混沌分析[J]. 应用数学进展, 2016, 5(3): 390-398. http://dx.doi.org/10.12677/AAM.2016.53048


[1] Morens, D.M., Folkers, G.K. and Fauci, A.S. (2004) The Challenge of Emerging and Re-Emerging Infectious Diseases. Nature, 430, 242-249.
[2] Kohn, G.C. (2004) The Wordsworth Encylo-pedia of Plague and Pestilence. Facts on File, New York, 25-26.
[3] Johnson, N.P.A.S. and Mueller, J. (2002) Up-dating the Accounts: Global Mortality of the 1918-1920 Spanish Influenza Pandemic. Bulletin of the History of Medicine, 76, 105-115.
[4] Li, X. and Wang, W. (2005) A Discrete Epidemic Model with Stage Structure. Chaos, Solitons & Fractals, 26, 947- 958.
[5] Bjørnstad, O.N., Finkenstaedt, B.F. and Greenfell, B.T. (2002) Dynamics of Measles Epidemics: Estimating Scaling of Transmission Rates Using a Time Series SIR Model. Ecological Monographs, 72, 169-184.
[6] Stone, L., Olinky, R. and Huppert, A. (2007) Seasonal Dynamics of Re-current Epidemics. Nature, 446, 533-536.
[7] Aron, J.L. and Schwartz, I.B. (1984) Seasonality and Period-Doubling Bifurcations in an Epidemic Model. Journal of Theoretical Biology, 110, 665-679.
[8] Olsen, L.F. and Schaffer, W.M. (1990) Chaos versus Noise Periodicity: Alternative Hypotheses for Childhood Epidemics. Science, 249, 499-504.
[9] Nowak, M. and May, R.M. (1993) AIDS Pathogenesis: Mathe-matical Models of HIV and SIV Infections. AIDS, 7, S3- S18.
[10] Lund, O., Mosekilde, E. and Hansen, J. (1993) Period Doubling Route to Chaos in a Model of HIV Infection of the Immune System. Simulation Modelling Practice and Theory, 1, 49-55.
[11] 崔岩, 刘素华, 葛晓陵. Langford系统Hopf分岔极限环幅值控制. 物理学报, 2012, 61(10): 6-14.
[12] Chen, X.W., Fu, X.L. and Jing, Z.J. (2013) Complex Dynamics in a Discrete-Time Predator-Prey System without Allee Effect. Acta Mathematicae Applicatae Sinica, 29, 355-376.
[13] 刘素华, 赵成刚, 唐驾时, 杨先林. Qi系统的Hopf分分析与幅值控制[J]. 动力学与控制学报, 2008, 6(2): 141-145.
[14] Ouyang, Q. (2000) Pattern Formation in Reac-tion-Diffusion Systems. Shanghai Sci-Tech Education Publishing House, Shanghai.
[15] 张美华. Lorenz-84系统的分岔与数值分析[J]. 科学技术与工程, 2010, 10(3): 0743-0746.
[16] Kuznetsov, Y.A. (1999) Elements of Applied Bi-furcation Theory. 2nd Edition, Springer, New York.
[17] Wiggins, S. (1990) Introduction to Applied Nonlinear Dy-namical Systems and Chaos. Texts in Applied Mathematics, Vol. 2, Springer, New York.
[18] Hsu, S.B. and Huang, T.W. (1995) Global Stability for a Class of Predator-Prey System. SIAM: SIAM Journal on Applied Mathematics, 55, 763-783.
[19] D’Innocenzo, A., Paladini, F. and Renna, L. (2006) A Nu-merical Investigation of Discrete Oscillating Epidemic Models. Physica A, 364, 497-512.
[20] Hassell, M.P., Comins, H.N. and May, R.M. (1991) Spatial Structure and Chaos in Insect Population Dynamics. Nature, 353, 255-258.
[21] Engbert, R. and Drepper, F.R. (1994) Chance and Chaos in Population Biology—Models of Recurrent Epidemics and Food Chain Dynamics. Chaos, Solitons & Fractals, 4, 1147-1169.