  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 Susceptible individuals grows logistically and has been
subject to constant vaccination. We explicitly prove that the endemic
equilibrium is a codimension two singularity in the parameter space
$(\mathcal{R}_0, p)$, where $\mathcal{R}_0$ is the basic reproduction number
and $p$ is the proportion of Susceptible individuals successfully vaccinated at
birth.
  We exhibit explicitly the Hopf, transcritical, Belyakov, heteroclinic and
saddle-node bifurcation curves unfolding the singularity. The two parameters
$(\mathcal{R}_0, p)$ are written in a useful way to evaluate 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 persists and we illustrate our
main result with numerical simulations, emphasizing the role of the parameters.
