Strange attractors in the seasonally forced epidemic SIR model

Carvalho, Joao P. S. Maur'icio de; Rodrigues, Alexandre A.

We analyse a periodically-forced SIR model to investigate the influence of seasonality on the disease dynamics and we show that the condition on the basic reproduction number $\mathcal{R}_0<1$ is not enough to guarantee the elimination of the disease. Using the theory of rank-one attractors, for an open subset in the space of parameters of the model for which $\mathcal{R}_0<1$, the flow exhibits persistent strange attractors, producing infinitely many periodic and aperiodic patterns. Although numerical experiments have already suggested that periodically-forced SIR model may exhibit observable chaos, a rigorous proof was not given before. Our results agree well with the empirical belief that intense seasonality induces chaos. This should serve as a warning to all doing numerics (on epidemiological models) who deduce that the disease disappears merely because $\mathcal{R}_0<1$.