Purged and partial Markov chains

In multiple decrement theory, disability theory, nuptiality and fertility theory, and in other connections, actuaries and demographers often study what in effect is simple Markov chain models. In these models, the transition probabilities, are defined on the basis of the forces of transition (infinitesimal transition probabilities). For various reasons, new models are sometimes constructed by replacing some of the original forces of transition by 0. We shall call the transition probabilities of such a new model partial probabilities.