Computational methods for Markov series with large state spaces, with application to AIDS modeling

AIDS models, and epidemiological models generally, are almost exclusively either differential equations or Markov processes. Of course, the phenomena are fundamentally random, so at best differential equations track the expectation of the process and variability is masked. There are few techniques in the literature for numerical analysis of Markov chains of any size, and so those wishing to analyze stochastic epidemics presently have little alternative to simulation. The contribution of the present paper is to propose numerical techniques capable of finding marginal probabilities of Markov chains having thousands and even millions of states. The ideas are illustrated by application to AIDS models in the literature which formerly had been investigated only through Monte Carlo. This introductory foray has not plumbed the depths of the computational methodology, which yet needs refinement and streamlining that comes through experience. Yet in its primitive form, it is shown herein to be adequate for a computation on the scale of a two-population partition of the San Francisco homosexual epidemic. The closing discussion compares the strengths and weaknesses of the present numerical techniques with the simulation approach to investigation of Markov epidemics.