Simple epidemiological models with information dependent vaccination functions can generate sustained oscillations via Hopf bifurcation of the endemic state. The onset of these oscillations depend
on the shape of the vaccination function. A “global” approach is used to characterise the instability
condition and identify classes of functions that always lead to stability/instability. The analysis allows
the identification of an analytically determined “threshold vaccination function” having a simple interpretation: coverage functions lying always above the threshold always lead to oscillations, whereas
coverage functions always below never lead to instability.
