  We consider an epidemiological model that includes waning and boosting of
immunity. Assuming that repeated exposure to the pathogen fully restores
immunity, we derive an SIRS-type model with discrete and distributed delays.
First we prove usual results, namely that if the basic reproduction number,
$\mathcal{R}_0$, is less or equal than $1$, then the disease free equilibrium
is globally asymptotically stable, whereas for $\mathcal{R}_0>1$ the disease
persists in the population. The interesting features of boosting appear with
respect to the endemic equilibrium, which can go through multiple stability
switches by changing the key model parameters. We construct two-parameter
stability charts, showing that increasing the delay can stabilize the positive
equilibrium. Increasing $\mathcal{R}_0$, the endemic equilibrium can cross two
distinct regions of instability, separated by Hopf-bifurcations. Our results
show that the dynamics of infectious diseases with boosting of immunity can be
more complex than most epidemiological models, and calls for careful
mathematical analysis.
