Stability, dynamical crises and chaos characterise the SIRWS system as the duration of immunity is varied between 1 and 20 years. The peak prevalence (a) and the period of oscillation (b), each plotted as a function of the duration of immunity (1/κ) are shown. Crises, quasiperiodicity, and chaos are all evident as the duration of immunity is varied. Note the sudden jump in the maximum of I nearby a duration of immunity of 10 years. This value happens to be that previously chosen in the literature[11, 12] to model pertussis dynamics indicating the presence of a crisis nearby an epidemiologically favoured value for the duration of immunity. The figures were produced by integrating the system for sufficient time such that the transient dynamics had passed (t
= 1000yr) and then examining the time traces over the next 50 years of simulation. The peak prevalence was calculated as the maximum of the oscillation of I and the period of oscillation as the time between successive maxima of I.