Seasonally forced disease dynamics explored as switching between attractors

Matt J. Keeling, Pejman Rohani, Bryan T. Grenfell

Research output: Contribution to journalArticlepeer-review

212 Scopus citations


Biological phenomena offer a rich diversity of problems that can be understood using mathematical techniques. Three key features common to many biological systems are temporal forcing, stochasticity and nonlinearity. Here, using simple disease models compared to data, we examine how these three factors interact to produce a range of complicated dynamics. The study of disease dynamics has been amongst the most theoretically developed areas of mathematical biology; simple models have been highly successful in explaining the dynamics of a wide variety of diseases. Models of childhood diseases incorporate seasonal variation in contact rates due to the increased mixing during school terms compared to school holidays. This 'binary' nature of the seasonal forcing results in dynamics that can be explained as switching between two nonlinear spiral sinks. Finally, we consider the stability of the attractors to understand the interaction between the deterministic dynamics and demographic and environmental stochasticity. Throughout attention is focused on the behaviour of measles, whooping cough and rubella.

Original languageEnglish (US)
Pages (from-to)317-335
Number of pages19
JournalPhysica D: Nonlinear Phenomena
Issue number3-4
StatePublished - Jan 15 2001
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Condensed Matter Physics
  • Applied Mathematics


  • Childhood diseases
  • Nonlinearity
  • SIR models
  • Seasonal forcing
  • Stochasticity


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