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

Abstract

We analyze a periodically-forced dynamical system inspired by the SIR model with impulsive vaccination. We fully characterize its dynamics according to the proportion of vaccinated individuals and the time between doses. If the basic reproduction number is less than 1 (i.e ), then we obtain precise conditions for the existence and global stability of a disease-free -periodic solution. Otherwise, if , then a globally stable -periodic solution emerges with positive coordinates. We draw a bifurcation diagram 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.