We came across this problem when looking at the ability of flies to move from one herd of cattle to another and thereby act as a disease vector. It was known that the average density of herds in the region was l/km2, but no information was available about the distribution of distance between these herds.
To solve the problem, we made a couple of assumptions. First, let's assume that the herds are randomly distributed over the area. That would mean that the herds are not concentrated around rivers, roads, in valleys, etc. For the area in question, this was a reasonable assumption [If it wasn't we could have looked at doing sub-models for different areas.] Then, we can say that for an area A, the actual number of herds is Poisson(l*A).
The second assumption is that each herd is essentially concentrated at a single point, i.e. its diameter is insignificant compared to the distances between herds. That too was a reasonable assumption in this case. If it hadn't been, the model could have been extended by associating a radius with each random herd.
A simplified version of our solution is provided in model Distance to Nearest Herd
[The actual model we produced took into account variations in herd concentration between regions.] The model creates a large square area (large meaning that there will be a large number of herds expected to exist within such an area, say 150 or so), and randomly places herds within that area. Assuming that the random herd is located in the center of the area, the distance to its nearest neighbor is calculated using the formula:
Distance =
Where {x,y} is the center of the area (position of a herd), and {x0,y0) is the position of the randomly selected herd. The formula can be extended to space rather than area very easily:
Distance =
Actually, after a bit of research we found a paper by Cliff and Ord (1975). Under the same assumptions we made, they note several results, which we convert here into Crystal Ball formulae, where l is the average concentration of herds/km2:
Distance to nearest neighbor (km) = Rayleigh(SQRT(1/(2*PI()*l)))
Distance to nearest neighbor (km) = SQRT(Exponential(PI()*l))
Distance to rth nearest neighbor (km) = SQRT(Gamma(0, 1/(PI()*l), r))
The model includes the first two results and shows that they all match.
The links to the Distance to Nearest Herd software specific models are provided here:
