  We consider the SIRWJS epidemiological model that includes the waning and
boosting of immunity via secondary infections. We carry out combined analytical
and numerical investigations of the dynamics. The formulae describing the
existence and stability of equilibria are derived. Combining this analysis with
numerical continuation techniques, we construct global bifurcation diagrams
with respect to several epidemiological parameters. The bifurcation analysis
reveals a very rich structure of possible global dynamics. We show that
backward bifurcation is possible at the critical value of the basic
reproduction number, $\mathcal{R}_0 = 1$. Furthermore, we find stability
switches and Hopf bifurcations from steady states forming multiple endemic
bubbles, and saddle-node bifurcations of periodic orbits. Regions of
bistability are also found, where either two stable steady states, or a stable
steady state and a stable periodic orbit coexist. This work provides an insight
to the rich and complicated infectious disease dynamics that can emerge from
the waning and boosting of immunity.
