An algorithmic synthesis of the deterministic and stochastic paradigms via computer intensive methods

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In this paper, an approach to synthesizing the deterministic and stochastic paradigms, via computer intensive methods, is presented within the framework of a stochastic model of a HIV/AIDS epidemic in a population of homosexuals. Because of dependence among members of a population, the problem of determining threshold conditions was approached by systematically embedding a system of differential equations in a stochastic process and determining if the Jacobian matrix of this system is stable or not stable, when evaluated at a disease free equilibrium. It has been shown in numerous Monte Carlo simulation experiments that if this matrix is not stable, then an epidemic will develop in a population with positive probability, following the introduction of infectives into a population of susceptibles. This technique was used to search for points in the parameter space such that an epidemic would develop in a population of susceptibles, following the entrance of one or more infectious recruits during any time interval with small probability. Such recurrent rare events are of interest in the studying the emergence of new diseases, involving the transmission of a virus from a species that has evolved resistance to it to another species that lacks resistance.