  We study an SIR epidemic model with a seasonal contact rate and saturated
treatment function. It is shown that the basic reproduction number $(\mathcal
{R}_0)$ cannot be acted as a threshold that predicts the spread of an
infection. Based on this finding, we establish a set of sufficient conditions
for the existence of multiple positive periodic solutions by using
Gaines-Mawhin's continuation theorem. This is very similar to backward
bifurcation for an autonomous model. Further, we investigate the stability of
positive periodic solution under small seasonal perturbation by means of
Floquet theory. Numerical simulations support our theoretical analysis.
