  We analyze a multiparameter periodically-forced dynamical system inspired in
the SIR endemic model. We show that the condition on the \emph{basic
reproduction number} $\mathcal{R}_0 < 1$ is not sufficient to guarantee the
elimination of \emph{Infectious} individuals due to a \emph{backward
bifurcation}. Using the theory of rank-one attractors, for an open subset in
the space of parameters where $\mathcal{R}_0<1$, the flow exhibits
\emph{persistent strange attractors}. These sets are not confined to a tubular
neighbourhood in the phase space, are numerically observable and shadow the
ghost of a two-dimensional invariant torus. Although numerical experiments have
already suggested that periodically-forced biological models 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 work provides a preliminary investigation of the interplay between
seasonality, deterministic dynamics and the prevalence of strange attractors in
a nonlinear forced system inspired by biology.
