Endemic infectious diseases constantly circulate in human populations, with prevalence fluctuating about a (theoretical and unobserved) time-independent equilibrium. For diseases for which acquired immunity is not lifelong, the classic susceptible–infectious–recovered–susceptible (SIRS) model provides a framework within which to consider temporal trends in the observed epidemiology. However, in some cases (notably pertussis), sustained multiannual fluctuations are observed, whereas the SIRS model is characterized by damped oscillatory dynamics for all biologically meaningful choices of model parameters.
Using a range of techniques in dynamical systems analysis, this project studies the dynamical properties of generalised transmission models, including the SIRWS model in which immunity may be boosted through exposure, exploring the conditions under which sustained oscillatory dynamics, multistability and chaos may be supported. Numerical simulation, bifurcation analysis and steady-state stability analysis are used to study the properties of these systems. Applications to pertussis, measles and other vaccine preventable diseases are explored.