This talk is divided in two parts.

Firstly, we analyze the effect of the seasonality in the SIR model, realized through the addition of a sinusoidal map in the transmission rate. Our major findings are the following:

(i) the condition on the basic reproduction number R0<1 is not enough to guarantee the elimination of the disease;

(ii) for R0<1, using the theory of Rank-one attractors developed by Wang and Young (2003), the flow exhibits persistent strange attractors.

Although numerical experiments have already suggested that periodically-forced SIR model may exhibit chaos, a rigorous proof (without computer-aided) was not given before. Our results are consistent with the empirical belief that intense seasonality induces chaos.

In the second part, we analyze the bifurcations of a classical SIR model with constant vaccination. We show that our system exhibits a Double-zero bifurcation by choosing the parameters (R0, p), where R0 is the basic reproduction number and p is the proportion of susceptible individuals successfully vaccinated. We give a precise estimation for the region in the parameter space where the disease robustly persists.