We present a regression modelling framework to analyse infectious disease transmission during a time period where extensive exposure data are available, but where the outcome data are sparse. A latent variable model is used for each exposure time, allowing a straight-forward accumulation of risk for a collection of exposures for which outcome data are available. We describe an analysis of HIV infection from blood products among a cohort of haemophiliacs in Ireland between 1980 and 1985. The analysis provides estimates of the time pattern and batch effects; we show how analytical complexity such as smoothly varying coefficients or random coefficient models can be accommodated by the model. Finally, we discuss other problems where the model is applicable.