### Abstract

Key problems for models of disease spread relate to threshold, velocity of spread, final size and control. All these aspects depend crucially on the network structure of individual interactions.

Networks of interest range from the highly localised case, where interactions are only between near neighbours, to the opposite global extreme where all interact equally with all, so that a disease can spread much more quickly through the population. Understandably, there has been much recent interest in `small-world' and meta-population models, in which a relatively small number of long-distance connections can change a network from local to effectively global. Such models seem particularly relevant to the changed patterns of human and animal diseases in a world whose connectivity, in terms of both travel and trade, has increased hugely in recent decades.

In consequence, a number of different mathematical and statistical approaches have been developed recently that focus on networks. I shall discuss the strengths and weaknesses of some of these approaches, with examples drawn from both human and animal diseases, susch as SARS, Foot and Mouth disease and avian flu. I shall also discuss the wider implications, as illustrating what mathematics can and cannot do in helping us predict and control disease outbreaks.

### Related Links

- http://www.ma.hw.ac.uk/~denis/ - web page, with links to papers etc