Journal of Mathematical Biology

, Volume 31, Issue 5, pp 495–512

A disease transmission model in a nonconstant population

  • W. R. Derrick
  • P. van den Driessche

DOI: 10.1007/BF00173889

Cite this article as:
Derrick, W.R. & van den Driessche, P. J. Math. Biol. (1993) 31: 495. doi:10.1007/BF00173889


A general SIRS disease transmission model is formulated under assumptions that the size of the population varies, the incidence rate is nonlinear, and the recovered (removed) class may also be directly reinfected. For a class of incidence functions it is shown that the model has no periodic solutions. By contrast, for a particular incidence function, a combination of analytical and numerical techniques are used to show that (for some parameters) periodic solutions can arise through homoclinic loops or saddle connections and disappear through Hopf bifurcations.

Key words

Epidemiological modelNonlinear incidence functionHopf bifurcationHomoclinic loopSaddle connection

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • W. R. Derrick
    • 1
  • P. van den Driessche
    • 2
    • 3
  1. 1.Department of MathematicsUniversity of MontanaMissoulaUSA
  2. 2.Department of MathematicsUniversity of VictoriaVictoriaCanada
  3. 3.Mathematisches InstitutUniversität MünchenMünchen 2Germany