Sometimes certain short-term risk exposures are postulated to act as a trigger for the onset of a specific acute illness. When the incidence of the illness is low it is desirable to investigate this possible association using only data on cases detected during a specific observation period. Here we propose an analysis for such a study based on a model expressed in terms of the probability that the exposure triggers the illness and a random delay from a triggered illness until its diagnosis. Both the natural hazard rate for the illness and the probability that the exposure triggers the illness are assumed to be small and possibly dependent on age and covariates such as sex and duration or severity of the exposure. The method of analysis is illustrated with a study of the association between long flights and hospitalization for venous thromboembolism.