Journal of Mathematical Biology

, Volume 25, Issue 4, pp 359–380 | Cite as

Dynamical behavior of epidemiological models with nonlinear incidence rates

  • Wei-min Liu
  • Herbert W. Hethcote
  • Simon A. Levin


Epidemiological models with nonlinear incidence rates λIpSqshow a much wider range of dynamical behaviors than do those with bilinear incidence rates λIS. These behaviors are determined mainly by p and λ, and secondarily by q. For such models, there may exist multiple attractive basins in phase space; thus whether or not the disease will eventually die out may depend not only upon the parameters, but also upon the initial conditions. In some cases, periodic solutions may appear by Hopf bifurcation at critical parameter values.

Key words

Epidemiological models Nonlinear incidence rates Hopf bifurcation Homoclinic-loop bifurcation 


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Copyright information

© Springer-Verlag 1987

Authors and Affiliations

  • Wei-min Liu
    • 1
    • 2
  • Herbert W. Hethcote
    • 4
  • Simon A. Levin
    • 1
    • 2
    • 3
  1. 1.Center for Applied MathematicsIthacaUSA
  2. 2.Section of Ecology and Systematics, Corson HallCornell UniversityIthacaUSA
  3. 3.Ecosystems Research CenterIowa CityUSA
  4. 4.Department of MathematicsUniversity of IowaIowa CityUSA

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