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.