Department of Agricultural Botany, The University of Reading, 2 Earley Gate, Whiteknights, Reading RG6 6AU, UK

Background and objectives
The probability that a wind-dispersed pathogen will move a given distance is expected to be distributed according to a function which behaves as a power-law at long distances. This is confirmed by empirical studies, although it is very difficult to obtain data at the long distances required to reject clearly any particular model. The power-law arises from several phenomena, including the eddy structure of air movements, which contain structure on all scales. However, most modelling investigations of the spatial structure of disease epidemics have assumed that dispersal distances followed a probability distribution which decreased at least as fast as an exponential at long distances. The objectives of the continuing work described here are to model and describe the distribution patterns produced by diverse dispersal patterns from single or multiple foci.

Materials and methods
The starting point is an individual-based and probabilistic simulation method based on a spatial index of the individuals in the population. Because the simulation is individual based, details are kept only of those places where individuals exist and sparse patterns over large areas can be handled with reasonable memory. The index makes it simple to determine whether a new individual is adjacent to others and therefore implement density-dependence or sexual reproduction efficiently. The spatial patterns produced have been studied using a hierarchy of grid sizes and the concept of conditional incidence. This is the incidence in a grid of a given size within a square of the next larger grid size, conditional on at least one individual occurring in the larger square. A graph of conditional incidence against scale appears characteristic of simulations with particular parameters, even though simulations vary wildly in the exact patterns arising. If conditional incidence is constant, the pattern is fractal. The same analytic technique can be applied to maps of disease in homogeneous fields, such as are given in aerial photographs, and the patterns compared with the simulated ones.

Results and conclusions
Dispersal probability distributions with exponential tails give rise in the simulations to smoothly expanding foci, as expected from a large body of theory. Distributions with power-law tails give more dispersed patterns, with secondary and higher-order foci apparent, in which a clear wavefront cannot be identified. With a single focus, conditional incidence changes with scale in a way that is characteristic of the exponent in the power-law. At power-law exponents close to those reported in the literature [1,2], the patterns can be nearly fractal over wide ranges. The patterns measured from aerial photographs and distribution maps often show distinctive features not seen in the simulations so far. The clear ability of the incidence graphs to reject certain models suggests that they are a useful way to summarize certain aspects of pattern and how pattern varies with scale. They also offer one way to escape the restrictive dichotomy of descriptions in terms of aggregated/regular. However, as with most analytical methods for spatial data, there are also difficulties in working with imperfect samples rather than complete maps.

1. Shaw MW, 1995. Proceedings of the Royal Society B 259, 249-257.
2. Shaw MW, 1996. In: White EM , et al.<1i>, eds. Modelling in Applied Biology: Spatial Aspects. Warwick: AAB, pp.165-172.