Pathogen evolution poses a significant challenge to public health, as efforts to control the spread of infectious diseases struggle to keep up with a shifting target. To better understand this adaptive process, we turn to mathematical modelling. Specifically, we use multi-type branching processes to describe a pathogen's stochastic spread among members of a host population or growth within a single host. In each case, there is potential for new pathogen strains with different characteristics to arise through mutation. We first develop a specific model to study the emergence of a newly introduced infectious disease, where the pathogen must adapt to its new host or face extinction in this population. In an extension of previous models, we separate the processes of host-to-host contacts and disease transmission, in order to consider each of their contributions in isolation. We also allow for an arbitrary distribution of host contacts and arbitrary mutational pathways/rates among strains. This framework enables us to assess the impact of these various factors on the chance that the process develops into a large-scale epidemic. We obtain some intriguing results when interpreted in a biological context. Secondly, motivated by a desire to investigate the time course of pathogen evolutionary processes more closely, we derive some novel theoretical results for multi-type branching processes. Specifically, we obtain equations for: (1) the distribution of waiting time for a particular type to arise; and (2) the distribution of population numbers over time, conditioned on a particular type not having yet appeared. A few numerical examples scratch the surface of potential applications for these results, which we hope to develop further. / Thesis (Master, Mathematics & Statistics) -- Queen's University, 2010-07-28 11:43:22.984

La théorie des processus de branchement multi-type en environnement i.i.d. est considérablement moins développée que dans le cas univarié, et les questions fondamentales ne sont pas résolues en totalité à ce jour. Les réponses exigent une compréhension profonde du comportement des produits des matrices i.i.d. à coefficients positifs. Sous des hypothèses assez générales et lorsque les fonctions génératrices de probabilité des lois de reproduction sont “linéaire fractionnaires”, nous montrons que la probabilité de survie à l’instant n du processus de branchement multi-type en environnement aléatoire est proportionnelle à 1/√n lorsque n → ∞. La démonstration de ce résultat suit l’approche développée pour étudier les processus de branchement uni-variés en environnement aléatoire i. i. d. Il utilise de façon cruciale des résultats récents portant sur les fluctuations des normes de produits de matrices aléatoires i.i.d. / The theory of multi-type branching process in i.i.d. environment is considerably less developed than for the univariate case, and fundamental questions are up to date unsolved. Answers demand a solid understanding of the behavior of products of i.i.d. matrices with non-negative entries. Under mild assumptions, when the probability generating functions of the reproduction laws are fractional-linear, the survival probability of the multi-type branching process in random environment up to moment n is proportional to 1/√n as n → ∞. Techniques for univariate branching process in random environment and methods from the theory of products of i.i.d. random matrices are required.