SIR model with vaccination: bifurcation analysis

João P. S. Maurício de Carvalho, Alexandre A. Rodrigues

Published 2022-11-29Version 1

There are few adapted SIR models in the literature that combine vaccination and logistic growth. In this article, we study bifurcations of a SIR model where the class of \emph{Susceptible} individuals grows logistically and has been subject to constant vaccination. We explicitly prove that the endemic equilibrium is a \emph{singularity of codimension 2} in the parameter space $(\mathcal{R}_0, p)$, where $\mathcal{R}_0$ is the \emph{basic reproduction number} and $p$ is the proportion of \emph{Susceptible} individuals successfully vaccinated at birth. We exhibit explicitly the \emph{Hopf}, \emph{transcritical}, \emph{Belyakov}, \emph{heteroclinic} and \emph{saddle-node bifurcation} curves unfolding the \emph{singularity}. The two parameters $(\mathcal{R}_0, p)$ are very suitable to study the proportion of vaccinated individuals necessary to eliminate the disease and to conclude how the vaccination may affect the outcome of the epidemic. We also exhibit the region in the parameter space where the disease robustly persists and we illustrate our main result with numerical simulations, emphasizing the role of the parameters $(\mathcal{R}_0, p)$.