Correlation models for childhood epidemics

One of the simplest set of equations for the description of epidemics (the SEIR equations) has been much studied, and produces reasonable approximations to the dynamics of communicable disease. However, it has long been recognized that spatial and social structure are important if we are to understand the long-term persistence and detailed behaviour of disease. We will introduce three pair models which attempt to capture the underlying heterogeneous structure by studying the connections and correlations between individuals. Although modelling the correlations necessarily leads to more complex equations, this pair formulation naturally incorporates the local dynamical behaviour generating more realistic persistence. In common with other studies on childhood diseases we will focus our attention on measles, for which the case returns are particularly well documented and long running.