The aim of this paper is to propose a deterministic model designed for the analysis of social services policies directed toward large populations. The model is a distributed parameter system representing the evolution over time of a multi-class age-structured population. The social services offered to the individuals in the various population classes act as controls which influence the population dynamics.

The model we propose in sections 2 and 3 could be viewed as a deterministic counterpart to a semi-Markov stochastic population model (SMSPM). Kao (1973,1974) and, more recently, Hershey et al. (1981) have advocated the use of SMSPMs for the planning of progressive care hospitals. In their models a patient admitted to the hospital is described as a semi-Markov process, since he will visit various units of the hospital as his health condition evolves. Collart and Haurie (1976,1980), Haurie et al. (1981) and Alj and Haurie (1980a,1980b) have considered the problem of optimally controlling a SMSPM. It appeared that, in a discrete time setting, the problem reduces to a Markovian decision process with a very large state set while, in a continuous time setting, the problem involves the control of a non-Markov jump process. For small populations (less than 10 individuals), exact stochastic dynamic programming techniques could be used. For populations of 30 to 50 individuals efficient heuristics could also be developed.

The present paper is concerned with the modeling of very large populations (1000 individuals and up). Our motivation is in the analysis of problems like the planning of social services offered to a given population (e.g. the elderly, the handicapped, or other dependents). When the size of the population becomes large, the curse of dimensionality makes a SMSPM almost useless, at least in an optimal control setting. When the population becomes very large one can attempt a modeling through the use of stochastic or even deterministic diffusion equations. It is the latter approach which is explored in this paper.