Structured population epidemic models
by Frank G Ball
Abstract: Standard deterministic models of epidemics implicitly assume that the population among which the disease is spreading is locally as well as globally large, in the sense that if the population is partitioned into groups, e.g. by age, sex and/or geographical location, then each of these groups, and not just the total population, is large. The same assumption is made when analysing the large-population behaviour of many stochastic epidemic models. However, this assumption is unrealistic for many human epidemics, since such populations contain small social groups, such as households, school classes and workplaces, in which transmission is likely to be enhanced. Thus, there has been a growing interest in models for epidemics among populations whose structure remains locally finite as the population becomes large. This talk is concerned with a general class of structured-population epidemic models, in which individuals mix at two levels: global and local. After some introductory comments, a stochastic model for SIR (susceptible - infected - removed) epidemics among a closed finite population is described, in which during its infectious period a typical infective makes both local and global contacts. Each local contact of a given infective is with an individual chosen independently according to a contact distribution centred on that infective and each global contact is with an individual chosen independently and uniformly from the whole population. The threshold behaviour of the model is determined, as is the asymptotic final outcome in the event of an epidemic taking off. The theory is specialised to (i) the households model, in which the population is partitioned into households and local contacts are chosen uniformly within an infective's household; (ii) the overlapping groups model, in which the population is partitioned in several ways, with local uniform mixing within the elements of the partitions; and (iii) the great circle model, in which individuals are equally spaced on a circle and local contacts are nearest-neighbour. A main use of epidemic models is to evaluate the efficacy of control measures, such as vaccination, and optimal vaccination policies for the households model are also considered.