School of Plant Sciences, The University of Reading, 2 Earley Gate, Whiteknights, Reading RG6 6AU, UK

Models permeate epidemiology, both seen and unseen. The concern of this symposium is with the visible models - visible because they use the language of mathematics. Such models can be classified in many ways. I wish to distinguish a number of uses. I do not intend these to cover all possibilities, but rather to indicate the breadth of the subject. This abstract indicates lines of thought but mostly omits examples.

It is very easy to construct a verbal hypothesis about some system of pathological interest whose consequences are too complicated or too subtle to be determined by purely verbal reasoning. The problem arises in many areas. Perhaps the very simplest example is the exponential curve, whose quantitative properties, however simple, are not intuitive. This is a very valid use of modelling: it bears the same relation to economic impact as fundamental solid-state physics does to electronics. In the same category of 'use' fails the role of mathematics in making clear that the possibilities of a system may be wider than intuition leads us to expect.

Assumptions and style
A common statement is that 'all assumptions must be listed'. If they are listed, and the mathematics is sound, then the conclusions will be sound. This syllogism is true but less useful than it seems. It is usually impossible to list all assumptions and few may be critical to the interesting aspects of the model. Many will be implicit in the mathematical formulation and many others in the common-sense and background knowledge of the modeller and their audience. One of the most useful functions of models, I believe, can be the process by which an absurd or surprising result can be traced back through the mathematics to the assumptions that produce it. When this is done, something has been learned. A more useful formulation, but less easy, would be 'list the assumptions responsible for the interesting aspect of your prediction'.

Empirical models and prediction
All scientific models are empirical in some sense. What is usually meant by 'empirical model' is a mathematical summary of an observed regularity, where the regularity does not fit seamlessly into another body of knowledge. These models have at least two uses: first, to indicate fruitful questions, and second, for prediction. For the second, the usual caveat that a model should not be used outside the setting in which it was developed is crucially important.

Models in decision support systems may use any information in any form. However, where quantitative decisions have to be made, models are extremely useful. The oddly named simulation models are an interesting category to examine more carefully. They have the primary characteristic of describing the dynamic behaviour of a large-scale system by linking dynamic models of its small-scale components. The difficulty comes in evaluating the results. A simulation is an extremely complex hypothesis, which certain to be wrong in detail and falsification of it is not simple. If the model works, this may actually be more surprising than if it does not. Two problems arise. If a model works one can ask, why? Is there actually a simpler, generic explanation? Furthermore. if the model does not work, one can again ask, why? Here, the difficulty is that in constructing any simplified model, one has to leave out many things, many of them aspects of the system of which one is unaware. How important are unmeasurable genetic changes in adaptation in pathogen population dynamics?

Modelling describes a suite of ideas and methods that are particularly useful for investigating and describing large, complex and non-intuitive objects, such as pathosystems, and use mathematical language. In all aspects of epidemiology it complements common-sense, simple logic, observation and experiment.