Journal of Mathematical Biology

, Volume 32, Issue 2, pp 109–121

Bifurcation analysis of periodic SEIR and SIR epidemic models

  • Yu. A. Kuznetsov
  • C. Piccardi

DOI: 10.1007/BF00163027

Cite this article as:
Kuznetsov, Y.A. & Piccardi, C. J. Math. Biol. (1994) 32: 109. doi:10.1007/BF00163027


The bifurcations of the periodic solutions of SEIR and SIR epidemic models with sinusoidally varying contact rate are investigated. The analysis is carried out with respect to two parameters: the mean value and the degree of seasonality of the contact rate. The corresponding portraits in the two-parameter space are obtained by means of a numerical continuation method. Codimension two bifurcations (degenerate flips and cusps) are detected, and multiple stable modes of behavior are identified in various regions of the parameter space. Finally, it is shown how the parametric portrait of the SEIR model tends to that of the SIR model when the latent period tends to zero.

Key words

Periodic epidemic models Subharmonic solutions Bifurcations Continuation method 

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • Yu. A. Kuznetsov
    • 1
  • C. Piccardi
    • 2
  1. 1.Russian Academy of Sciences, PushchinoResearch Computing CentreMoscow RegionRussia
  2. 2.C.N.R., Politecnico di MilanoCentro Teoria dei SistemiMilanoItaly