Journal of Mathematical Biology

, Volume 78, Issue 3, pp 607–624 | Cite as

A simple influenza model with complicated dynamics

  • M. G. RobertsEmail author
  • R. I. Hickson
  • J. M. McCaw
  • L. Talarmain


We propose and analyse a model for the dynamics of a single strain of an influenza-like infection. The model incorporates waning acquired immunity to infection and punctuated antigenic drift of the virus, employing a set of differential equations within a season and a discrete map between seasons. We show that the between-season map displays a variety of qualitatively different dynamics: fixed points, periodic solutions, or more complicated behaviour suggestive of chaos. For some example parameters we demonstrate the existence of two distinct basins of attraction, that is the initial conditions determine the long term dynamics. Our results suggest that there is no reason to expect influenza dynamics to be regular, or to expect past epidemics to give a clear indication of future seasons’ behaviour.


Seasonal influenza Transmission model Discrete dynamics Dynamical systems 

Mathematics Subject Classification

37N25 92B 



The authors would like to thank Viggo Andreasen (Roskilde University) who pointed out the similarity with the results in Andreasen (2003), David Simpson (Massey University) who drew our attention to Simpson (2016), and two anonymous referees whose suggestions improved the paper.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Natural and Mathematical Sciences, New Zealand Institute for Advanced Study and the Infectious Disease Research CentreMassey UniversityAucklandNew Zealand
  2. 2.IBM Research AustraliaMelbourneAustralia
  3. 3.School of Mathematics and Statistics, Faculty of ScienceThe University of MelbourneMelbourneAustralia
  4. 4.Centre for Epidemiology and Biostatistics, Melbourne School of Population and Global Health, Faculty of Medicine, Dentistry and Health SciencesThe University of MelbourneMelbourneAustralia
  5. 5.Institut National des Sciences Appliquées LyonLyonFrance
  6. 6.Medical Research Council Cancer Unit, Hutchison/MRC Research CentreUniversity of CambridgeCambridgeUK

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