CHARLES MODE
	"DESTOBIO 2000" August 23-27, 2000 West Lafayette, Indiana, USA

"An Overview on Synthesizing Stochastic and Deterministic Paradigms Using Computer Intensive Methods."

ABSTRACT: In recent years, the author and his associates have been working on classes of non-linear stochastic models that arise in modelling epidemics of infectious diseases. Particular attention has been paid to sexually transmitted diseases with stages such as HIV/AIDS. Like branching process, a basic ingredient of these stochastic systems has been models for the life cycles of individuals as well as life cycle models for couples in those cases where couple formation and dissolution are included in the formulation. The mathematical structures used in the construction of these life cycle models are semi-Markov processes.

Life cycle models for individuals and couples were used to construct a population process, which, in most cases studied, may be viewed as a Markov jump process in continuous time whose state space consists of vectors of non-negative integers of high dimension. Such multidimensional processes are notoriously difficult to analyze mathematically. But, fortunately, they may be approximated by a class of discrete time processes commonly referred to as chain multinomial models, which have been used to compute samples of Monte Carlo realizations of the population process.

By operating on conditional expectations, given the past, in the population process, it is possible to embed non-linear difference equations in the stochastic process in such a way that the parameters of these deterministic systems are precisely those of the stochastic population process. By solving the non-linear difference equations numerically, it becomes possible to compare the trajectories of the embedded deterministic models with statistically summarized samples of Monte Carlo realizations of the process. Numerous examples have been observed in which the trajectories of the embedded deterministic model may diverge quite significantly from the trajectory of a mean of a Monte Carlo sample.

As an aid to understanding such divergences, it has been helpful to derive a systems of non-linear differential equations from the embedded non-linear difference equations by letting the length of each interval in the discrete time approximation go to zero. By using MAPLE to derive symbolic forms of the elements of the Jacobian matrices of these in these high dimensional systems, it has been relatively easy to write computer code to determine numerically whether these matrices are stable or not stable at some point in the parameter space. It has been found experimentally that such determinations can be useful indicators for understanding why the trajectories of the embedded deterministic and those of the population process may diverge significantly. Numerous computer generated graphs illustrating these divergences will be presented.

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