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.e R p < 1 ), then we obtain precise conditions for the existence and global stability of a disease-free T -periodic solution. Otherwise, if R p > 1 , 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.