The structure of epidemic models

This paper reviews the basic components of epidemic models, and discusses some of the different ways of combining them, and relations between the resulting models. The fundamental aim is to help understanding of the relation between assumptions and the resulting dynamics: because without such understanding even a model which fits data perfectly can be of no scientific value. Analysis of the structure of epidemic models is vital because of (1) the scarcity of good data and (2) the sensitive dependence of results on assumptions. In evaluating model dynamics, we need to look carefully at their dependence, not only on parameters, but also on the structure of the model: for instance, whether the population is treated as stochastic or deterministic, discrete or continuous, and how the timing and distribution of infectious contacts within the population is modelled. The practical target is to identify those parts of models that have most effect on dynamics: a few key parameters can drive a model (see e.g. Mollison 1984, 1985, Cairns, this volume). The approach taken here is to analyse models in terms of their elements: expressing them in terms of simple key parameters that reflect individual lifehistories, flows between states, and contact relationships. Basic definitions must be in terms of what one individual does to another; this implies that discrete models are basic, and that the stochastic aspect is usually important, if only in formulating and interpreting models. Although more complex, stochastic models can have advantages in showing structure more clearly, as for instance in the technique of coupling which allows elegant comparisons of related models (see Ball, this volume). Some results can be very sensitive to model assumptions, including hidden assumptions implicit in seemingly innocent parts of our model structure. Other results are so robust that they can be derived by ‘pre-model’ arguments, that is, by considering relations between basic components without choosing a specific model. As an illustration of this approach, the final section (Section 3) tries to express some basic results on epidemic models in their simplest and most general form, so as to analyse the range and limits of their applicability.