Analysis of three stochastic models for discrete populations.

摘要

Stochastic models of discrete populations appear in a broad range of contexts. In this thesis, three of these modelling settings are considered: epidemics which spread through both blood transfusion and surgery, stochastic gene expression, and the accumulation of error in velocity statistics in Molecular Dynamics simulations. We examine specific models in each case, illustrating methods for extracting quantitative statistical information from the systems, and discuss how each approach fits into the more general mathematical framework for discrete populations evolving under stochastic dynamics.;Stochastic models of gene expression are developed in a similar manner as the epidemic models, with the addition of spatial diffusion of the molecules. The presence of molecular motility leads to quantitative questions not present in the non-spatial model. We pose several of these questions and analytically calculate quantities related to the correlation of mRNA and protein molecules in the steady state. These quantities give a sense of the spatial scale of protein and mRNA clusters around a DNA point source.;Molecular Dynamics simulations typically involve the numerical integration of Newtonian dynamics governing the motion of systems of particles. Often the particles interact only through collisions, and a fixed time step integrator is used to move the system forward in time. In this case, small errors accrue in both position and velocity coordinates. By statistical mechanical considerations, we discard the positional information of the system and model it as a population of particles, distinguished only by their velocities, undergoing evolution via random collisions. Employing additional assumptions about the size of the velocity errors and the distribution of particle velocities, we construct a diffusion model for the drift in energy. This model is verified with simulations and is shown to accurately predict the energy drift in dilute systems for moderate time scales.;The stochastic epidemics are approximated by branching processes. Generating function techniques are used to derive distributions of the final number of infected individuals and contaminated blood or surgery products in the case that the epidemic eventually dies out (subcritical). This simple model provides a framework for extension to more complicated systems such as those that incorporate populations stratified by age or risk of exposure. To our knowledge, this mathematical approach has not been applied to these kinds of epidemic models elsewhere in the literature.