Journal of Mathematical Biology

, Volume 23, Issue 2, pp 187–204 | Cite as

Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models

  • Wei-min Liu
  • Simon A. Levin
  • Yoh Iwasa


When the traditional assumption that the incidence rate is proportional to the product of the numbers of infectives and susceptibles is dropped, the SIRS model can exhibit qualitatively different dynamical behaviors, including Hopf bifurcations, saddle-node bifurcations, and homoclinic loop bifurcations. These may be important epidemiologically in that they demonstrate the possibility of infection outbreak and collapse, or autonomous periodic coexistence of disease and host. The possible mechanisms leading to nonlinear incidence rates are discussed. Finally, a modified general criterion for supercritical or subcritical Hopf bifurcation of 2-dimensional systems is presented.

Key words

Epidemiology nonlinear incidence rates Hopf bifurcation periodicities infectious diseases 


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Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • Wei-min Liu
    • 1
    • 3
    • 4
  • Simon A. Levin
    • 1
    • 2
    • 3
  • Yoh Iwasa
    • 2
  1. 1.Section of Ecology and SystematicsCorson Hall, Cornell UniversityIthacaUSA
  2. 2.Ecosystems Research CenterCornell UniversityUSA
  3. 3.Center for Applied MathematicsCornell UniversityUSA
  4. 4.Shanghai Institute of BiochemistryChinese Academy of SciencesShanghaiChina

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