The semiparametric maximum likelihood estimation is considered in a three-state duration dependent Markov process when times of the intermediate transition (e.g., onset of a disease) are interval censored and the times of transitions to an absorbing state (e.g., death) are known exactly or are right censored. It is assumed that the intensity of the transition to an absorbing state depends both on chronological time and duration in the intermediate state. De Gruttola and Lagakos (1989, Biometrics 45, 1-11) and Frydman (1992, Journal of the Royal Statistical Society, Series B 54, 853-866) discussed non-parametric estimation from the same sampling scheme under the assumption that the intensity of transition to an absorbing state depends only on the duration in the intermediate state or only on the chronological time respectively. The approach taken here generalizes, but in discrete time framework, the results from Frydman (1992). The distribution of the time to the intermediate transition is modelled nonparametrically and the intensity of onset of terminal condition semiparametrically. The algorithm is developed for the computation of the estimators. The methods are illustrated with AIDS data.