Many chronic diseases have a natural interpretation in terms of staged progression. Multistate models based on Markov processes are a well-established method of estimating rates of transition between stages of disease. However, diagnoses of disease stages are sometimes subject to error. The paper presents a general hidden Markov model for simultaneously estimating transition rates and probabilities of stage misclassification. Covariates can be fitted to both the transition rates and the misclassification probabilities. For example, in the study of abdominal aortic aneurysms by ultrasonography, the disease is staged by severity, according to successive ranges of aortic diameter. The model is illustrated on data from a trial of aortic aneurysm screening, in which the screening measurements are subject to error. General purpose software for model implementation has been developed in the form of an R package and is made freely available.