  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 $\mathcal{R}_0<1$.
