Interest lies in understanding the mechanisms that underlie temporal oscillations of diseases and their associated mathematical models. Here, we study the oscillations reported in Heffernan and Keeling (Proc. Biol Sci R Soc 276:2071–80, 2009) that use a model of \({>}500\) ordinary differential equations to model measles infection and immunity. While very large, the system is sparse and lends itself to numerical bifurcation analysis. Bifurcation analysis uncovers a Hopf bifurcation with small amplitude. Medium- and large-amplitude cycles are also found to exist. The existence of all three generators of cyclic dynamics and the changes of cycles related to global bifurcations are described using a corresponding Poincaré return map.