  This is an epidemiological SIRV model based study that is designed to analyze
the impact of vaccination in containing infection spread, in a 4-tiered
population compartment comprised of susceptible, infected, recovered and
vaccinated agents. While many models assume a lifelong protection through
vaccination, we focus on the impact of waning immunization due to conversion of
vaccinated and recovered agents back to susceptible ones. Two asymptotic states
exist, the "disease-free equilibrium" and the "endemic equilibrium"; we express
the transitions between these states as function of the vaccination and
conversion rates using the basic reproduction number as a descriptor. We find
that the vaccination of newborns and adults have different consequences in
controlling epidemics. We also find that a decaying disease protection within
the recovered sub-population is not sufficient to trigger an epidemic at the
linear level. Our simulations focus on parameter sets that could model a
disease with waning immunization like pertussis. For a diffusively coupled
population, a transition to the endemic state can be initiated via the
propagation of a traveling infection wave, described successfully within a
Fisher-Kolmogorov framework.
