Journal of Mathematical Biology

, Volume 23, Issue 2, pp 187–204

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

Authors

  • Wei-min Liu
    • Section of Ecology and SystematicsCorson Hall, Cornell University
    • Center for Applied MathematicsCornell University
    • Shanghai Institute of BiochemistryChinese Academy of Sciences
  • Simon A. Levin
    • Section of Ecology and SystematicsCorson Hall, Cornell University
    • Ecosystems Research CenterCornell University
    • Center for Applied MathematicsCornell University
  • Yoh Iwasa
    • Ecosystems Research CenterCornell University
Article

DOI: 10.1007/BF00276956

Cite this article as:
Liu, W., Levin, S.A. & Iwasa, Y. J. Math. Biology (1986) 23: 187. doi:10.1007/BF00276956

Abstract

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

Epidemiologynonlinear incidence ratesHopf bifurcationperiodicitiesinfectious diseases

Copyright information

© Springer-Verlag 1986