When studying the effect of a transient exposure on the risk of a rare illness, for time and cost effectiveness it is desirable to follow a cohort of individuals who are 'prone' to the illness over an observation period. In this paper, we present a method of analysis for data arising from such a study. The proposed method can be used to estimate the relative risk of an exposure triggering the illness and the distribution of the time delay from exposure to the onset of illness. The model is extended to include covariate effects and to the situation where there are two types of exposure. For the two types of exposures situation, a model to handle a possible synergism of the exposures is proposed. Finally, the method is applied to study the potential triggers of attacks of Menière's disease.