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arXiv · 2312.02306

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

João P. S. Maurício de Carvalho·Alexandre A. Rodrigues
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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 $p$ of vaccinated individuals and the time $T$ between doses. If the basic reproduction number is less than 1 (i.e. $\mathcal{R}_p<1$), then we obtain precise conditions for the existence and global stability of a disease-free it $T$-periodic solution. Otherwise, if $\mathcal{R}_p>1$, 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.

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João P. S. Maurício de Carvalho, Alexandre A. Rodrigues. 2023-12-04. Pulse vaccination in a SIR model: global dynamics, bifurcations and seasonality. https://arxiv.org/abs/2312.02306

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