Journal of Mathematical Biology

, Volume 29, Issue 3, pp 271–287

Some epidemiological models with nonlinear incidence

  • H. W. Hethcote
  • P. van den Driessche

DOI: 10.1007/BF00160539

Cite this article as:
Hethcote, H.W. & van den Driessche, P. J. Math. Biol. (1991) 29: 271. doi:10.1007/BF00160539


Epidemiological models with nonlinear incidence rates can have very different dynamic behaviors than those with the usual bilinear incidence rate. The first model considered here includes vital dynamics and a disease process where susceptibles become exposed, then infectious, then removed with temporary immunity and then susceptible again. When the equilibria and stability are investigated, it is found that multiple equilibria exist for some parameter values and periodic solutions can arise by Hopf bifurcation from the larger endemic equilibrium. Many results analogous to those in the first model are obtained for the second model which has a delay in the removed class but no exposed class.

Key words

Epidemiological modelNonlinear incidenceHopf bifurcationTime delay

Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • H. W. Hethcote
    • 1
  • P. van den Driessche
    • 2
  1. 1.Department of MathematicsUniversity of IowaIowa CityUSA
  2. 2.Department of Mathematics and StatisticsUniversity of VictoriaVictoriaCanada