Pulse vaccination in a SIR model: Global dynamics, bifurcations and seasonality

We analyze a periodically-forced dynamical system inspired by the SIR model with impulsive vaccination. We fully characterize its dynamics according to the proportion p of vaccinated individuals and the time T between doses. If the basic reproduction number is less than 1 (i.eRp1, then a globally stable T-periodic solution emerges with positive coordinates. We draw a bifurcation diagram (T,p) and we describe the associated bifurcations. We also find analytically and numerically chaotic dynamics by adding seasonality to the disease transmission rate. In a realistic context, low vaccination coverage and intense seasonality may result in unpredictable dynamics. Previous experiments have suggested chaos in periodically-forced biological impulsive models, but no analytic proof has been given. •Analyze pulse vaccination in a SIR model with and without seasonality.•Design an optimal vaccination program.•Relate the basic reproduction number with the uniform permanence of disease.•Study the bifurcations between the different types of behaviors.•Find chaotic dynamics under the presence of seasonality.