We consider the problem of estimating the intensity functions for a continuous time 'illness-death' model with intermittently observed data. In such a case, it may happen that a subject becomes diseased between two visits and dies without being observed. Consequently, there is an uncertainty about the precise number of transitions. Estimating the intensity of transition from health to illness by survival analysis (treating death as censoring) is biased downwards. Furthermore, the dates of transitions between states are not known exactly. We propose to estimate the intensity functions by maximizing a penalized likelihood. The method yields smooth estimates without parametric assumptions. This is illustrated using data from a large cohort study on cerebral ageing. The age-specific incidence of dementia is estimated using an illness-death approach and a survival approach.