Periodic solutions in a seasonally forced epidemic model

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.