On some formulas in a partnership model from the perspective of a semi-Markov process

Many deterministic models of sexually transmitted diseases, as well as population models in general, contain elements of stochastic or statistical reasoning. An example of such a model is that of Dietz and Hadeler (1988) concerning sexually transmitted diseases in which there is partnership formation and dissolution. Among the interesting formulas in this paper, which enter into the analysis of the model, are those for the expected number of partners a male or female has during a lifetime. To a probabilist such formulas suggest the possibility that some stochastic process may be constructed so as to yield these formulas as well as others that may be of interest. The principal purpose of this paper is to demonstrate that such a stochastic process does indeed exist in the form of a three state semi-Markov process in continuous time with stationary laws of evolution and with a one-step density matrix determined by four parameters which were interpreted as constant latent risk functions in the classical theory of competing risks. This construction of a semi-Markov process not only provides a framework for the systematic derivation of the formulas of Dietz and Hadeler but also suggests pathways for extensions to the age-dependent case.