Dyad Models for Childhood

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 a disease. However it has long been recognised that spatial and social structure may be important if we are to fully understand the long-term persistence and detailed behaviour of a disease within a community. We will introduce three pair-wise (or dyad) models which attempt to capture the underlying heterogeneous structure by studying the connections and correlations between individuals. Although modelling the network of connections necessarily leads to more complex equations and an explosion of parameters it will be shown that these pair-wise models naturally incorporate the local dynamical behaviour. As is common (if not compulsory) with childhood diseases we will focus our attention on the most well documented disease, measles.