On the inclusion of self regulating branching processes in the working paradigm of evolutionary and population genetics.

The principal goal of this methodological paper is to suggest to a general audience in the genetics community that the consideration of recent developments of self regulating branching processes may lead to the possibility of including this class of stochastic processes as part of working paradigm of evolutionary and population genetics. This class of branching processes is self regulating in the sense that an evolving population will grow only to a total population size that can be sustained by the environment. From the mathematical point of view the class processes under consideration belongs to a subfield of probability and statistics sometimes referred to as computational applied probability and stochastic processes.

Estimating the transmission potential of supercritical processes based on the final size distribution of minor outbreaks.

Use of the final size distribution of minor outbreaks for the estimation of the reproduction numbers of supercritical epidemic processes has yet to be considered. We used a branching process model to derive the final size distribution of minor outbreaks, assuming a reproduction number above unity, and applying the method to final size data for pneumonic plague. Pneumonic plague is a rare disease with only one documented major epidemic in a spatially limited setting.

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

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.

Determination of threshold conditions for a non-linear stochastic partnership model for heterosexually transmitted diseases with stages.

When comparing the performance of a stochastic model of an epidemic at two points in a parameter space, a threshold is said to have been crossed when at one point an epidemic develops with positive probability; while at the other there is a tendency for an epidemic to become extinct. The approach used to find thresholds in this paper was to embed a system of ordinary non-linear differential equations in a stochastic process, accommodating the formation and dissolution of marital partnerships in a heterosexual population, extra-marital sexual contacts, and diseases such as HIV/AIDS with stages. A symbolic representation of the Jacobian matrix of this system was derived.