Bifurcations of an sirs epidemic model with nonlinear incidence rate

Zhixing Hu, Ping Bi, Wanbiao Ma, Shigui Ruan

Research output: Contribution to journalArticlepeer-review

41 Scopus citations


The main purpose of this paper is to explore the dynamics of an epidemic model with a general nonlinear incidence βSip (1 +αI q). The ex-istence and stability of multiple endemic equilibria of the epidemic model are analyzed. Local bifurcation theory is applied to explore the rich dynamical be-havior of the model. Normal forms of the model are derived for different types of bifurcations, including Hopf and Bogdanov-Takens bifurcations. Concretely speaking, the first Lyapunov coefficient is computed to determine various types of Hopf bifurcations. Next, with the help of the Bogdanov-Takens normal form, a family of homoclinic orbits is arising when a Hopf and a saddle-node bifur- cation merge. Finally, some numerical results and simulations are presented to illustrate these theoretical results.

Original languageEnglish (US)
Pages (from-to)93-112
Number of pages20
JournalDiscrete and Continuous Dynamical Systems - Series B
Issue number1
StatePublished - Jan 2011


  • Bogdanov-Takens bifurcation
  • Hopf bifur-cation
  • Nonlinear incidence rate
  • SIRS epidemic model
  • Stability

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics


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