Computing marginal expectations for large compartmentalized models with application to AIDS evolution in a prison system.

The customary models for the AIDS epidemic are compartmentalized according to criteria such as risk factors, sexual habits, gender, race, age, and HIV status and stage. Hitherto, with very few exceptions, investigators have resorted to deterministic approximations or to simulation for the computational investigation of such models, which do not yield to purely analytic methods. The present paper describes a numerical technique, not dependent on Monte Carlo simulations, for such compartmentalized Markov population processes. Analytic error bounds and computational evidence suggest that this technique is quite accurate. The study is motivated and illustrated by a model for a prison system, with ten interrelated prisons, twenty compartments, and thousands of individuals. This model is of increasing interest in itself because the HIV/AIDS epidemic is particularly virulent among prison populations, where the environment offers special opportunities to investigate various prevention and educational programmes quantitatively. Our computational techniques are shown to be effective for the analysis of such a prison system, even though the resulting Markov process is an order of magnitude more complicated than other stochastic epidemic models currently being investigated. The modelling approach and numerical device appear to be applicable to a wide variety of population processes involving migration between population patches.