Age-structure and transient dynamics in epidemiological systems
Mathematical models of childhood diseases date back to the early twentieth century. In several cases, models that make the simplifying assumption of homogeneous time-dependent transmission rates give good agreement with data in the absence of secular trends in population demography or transmission. The prime example is afforded by the dynamics of measles in industrialized countries in the pre-vaccine era. Accurate description of the transient dynamics following the introduction of routine vaccination has proved more challenging, however. This is true even in the case of measles which has a well-understood natural history and an effective vaccine that confers long-lasting protection against infection. Here, to shed light on the causes of this problem, we demonstrate that, while the dynamics of homogeneous and age-structured models can be qualitatively similar in the absence of vaccination, they diverge subsequent to vaccine roll-out. In particular, we show that immunization induces changes in transmission rates, which in turn reshapes the age distribution of infection prevalence, which effectively modulates the amplitude of seasonality in such systems. To examine this phenomenon empirically, we fit transmission models to measles notification data from London that span the introduction of the vaccine. We find that a simple age-structured model provides a much better fit to the data than does a homogeneous model, especially in the transition period from the pre-vaccine to the vaccine era. Thus, we propose that age structure and heterogeneities in contact rates are critical features needed to accurately capture transient dynamics in the presence of secular trends.