# A simple influenza model with complicated dynamics

## Abstract

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.

## Keywords

Seasonal influenza Transmission model Discrete dynamics Dynamical systems## Mathematics Subject Classification

37N25 92B## Notes

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