In mother-to-child transmission of HIV, identifying infected infants relies on a diagnostic test with imperfect sensitivity that is administered at scheduled visits. Under this scenario, a participant's true state may be unknown at the start and end times of the study, and the detection of transitions into illness may be delayed or missed altogether. This could lead to biased estimates of the risk of transmission and covariate associations. When a test has imperfect sensitivity, but perfect specificity, the additional uncertainty can be captured as a random variable measuring delay in detection. The cumulative distribution then defines a time-dependent sensitivity function that increases over time. We present a maximum likelihood based illness-death model that accounts for imperfect sensitivity by including the delay as an exponential distribution. We specify transition rates as penalized B-splines to allow for nonhomogeneity of risk and discuss the model under Markov and semi-Markov assumptions. We apply this method to our motivating data set, a study of 1499 mother and infant pairs at three sites in Africa.