| Summary: | This thesis is mainly concerned with avian flu epidemic recurrence, its current paradigm, and further mathematical research. Generally, this thesis aims to characterise the recurrent pattern of epidemics simulated by stochastic avian flu models using mathematical techniques. Of particular interest here are the stochastic fluctuations observed in recurrent epidemics. This thesis has two main parts. The first part presents a thorough analysis of a simple stochastic avian flu model to provide insight into the role of different transmission routes in its recurrent dynamics. Recent modelling work on avian influenza in wild bird population takes into account demographic stochasticity and highlights the importance of environmental transmission in determining the outbreak periodicity, but only for a weak between-host transmission rates. A new analytic approach is used here to determine the relative contribution of environmental and direct transmission routes to the features of recurrent epidemics. Using an approximation method to describe noise-sustained oscillations, the recurrent epidemics simulated by the stochastic model is identified to be governed by the product of rotation and a slow-varying standard mean-reverting stochastic process, in a limiting sense. By analytically computing the intrinsic frequency and theoretical power spectral density, it can be shown that the outbreak periodicity can be explained by both types of transmission, and even by either one in the absence of the other. The final part of this thesis presents a novel approach to understanding the role of parametric (e.g. seasonal) forcing and stochasticity in the stochastic fluctuations around a cyclic solution. An approximate description about these stochastic fluctuations is developed, which paves the way for a new mathematical tool to be used for analysing oscillations generated from the interactions of non-linear terms and stochasticity. The theory developed here is used to explore a stochastic avian flu model with seasonally forced environmental transmission which may be applicable to other stochastic system with seasonal forcing. This thesis highlights the importance of approximation theory to analyse complex stochastic systems such as avian flu epidemic recurrence. Graduate Studies, College of (Okanagan) Graduate
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