  The SIRS model with constant vaccination and immunity waning rates is well
known to show a transition from a disease-free to an endemic equilibrium as the
basic reproduction number $r_0$ is raised above threshold. It is shown that
this model maps to Hethcote's classic endemic model originally published in
1973. In this way one obtains unifying formulas for a whole class of models
showing endemic bifurcation. In particular, if the vaccination rate is smaller
than the recovery rate and $r_- < r_0 < r_+$ for certain upper and lower bounds
$r_\pm$, then trajectories spiral into the endemic equilibrium via damped
infection waves. Latest data of the SARS-CoV-2 Omicron variant suggest that
according to this simplified model continuous vaccination programs will not be
capable to escape the oscillating endemic phase. However, in view of the strong
damping factors predicted by the model, in reality these oscillations will
certainly be overruled by time-dependent contact behaviors.
