We derive explicit lower and upper bounds for the probability generating functional of a stationary locally stable Gibbs point process, which can be applied to summary statistics such as the F function. For pairwise interaction processes we obtain further estimates for the G and K functions, the intensity, and higher-order correlation functions. The proof of the main result is based on Stein's method for Poisson point process approximation.

The aim of this paper is to investigate local spatial dependency with regard to Salmonella seropositivity in data from the Danish swine salmonellosis control programme and its application in informing surveillance strategies. We applied inhomogeneous and observed-difference K-function estimation, and geo-statistical modelling to data from the Danish swine salmonellosis control programme. Slaughter-pig farm density showed large variation at both the country-wide and local level in Denmark (median 0·23, range 0·02–0·47 farms/km2). The spatial distribution of pig farms followed a random inhomogeneous Poisson process but was not aggregated. We found evidence for aggregation of Salmonella case farms over that of all farms at distances of up to 6 km and semivariogram analyses of Salmonella seropositivity revealed spatial dependency between pairs of farms up to 4 km apart. The strength of the spatial dependency was positively associated with slaughter-pig farm density. We proposed sampling more intensively those farms within a 4 km radius of farms that were identified with a high Salmonella status, and reduced sampling of farms that are within this radius of ‘Salmonella-free’ farms. Our approach has the potential to optimize sampling strategies while maintaining consumer confidence in food safety and also has potential to be used for other zoonotic disease surveillance systems.