This model is over-specified as stated, so we arbitrarily assume S s = 1 throughout, meaning S c represents the ratio of cattle-tosheep susceptibility. For a constant (distance-independent) kernel this is just a mass-action closed epidemic model with heterogeneous susceptibility and infectiousness. This model assumes susceptibility and infectiousness parameters scale linearly with the number of animals of different species on the farm, a relatively strong assumption imposed for model parsimony reasons. The mixing matrix embedded in (2) quantifies the 4 species-specific mixing rates between animals on different farms: cattle-to-cattle (S c I c ) sheep-to-cattle (S c I s ), cattle-to-sheep (S s I c ) and sheep-to-sheep (S s I s ). This model formulation is identical to that used by Keeling et al. [5] , except for the functional form of kernel used. The force of infection on a susceptible farm i at time t depends on the whole history of events and is just