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The evolution of cooperative behaviours has received a large amount of attention in the literature. A recurrent result is that a spatial population structure often aids the evolution of cooperation. One such possible structure is a graph. Members of the population reside on vertices and interact with those connected by edges. The population changes over time via births and deaths and these changes are manifest in changing gene frequencies. I am interested in the change in frequency of a cooperative allele and one way to calculate this is with the inclusive fitness effect. The inclusive fitness effect is the sum of the effects of a behaviour on the members of a population, each effect weighted by a measure of genetic relatedness. In this thesis, I derive inclusive fitness theory in the context of evolutionary graphs. I provide new ways of calculating components of the inclusive fitness effect and high- light remaining challenges posed by graph-structured population models. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2012-10-02 17:55:12.267

Fungi generally exist as unicellular organisms (yeasts) or in a vegetative state in which a mycelium, i.e. an interconnected network of tubes (hyphae) is formed. The mycelium can operate over a very large range of scales (each hypha is only a few microns in diameter, yet mycelia can be kilometres across). Fungi are of fundamental importance to many natural processes: certain species have major roles in decomposition and nutrient cycling in the soil; some form vital links with plant roots allowing nutrient transfer. Other species are essential to industrial processes: citric acid production for use in soft drinks; brewing and baking; treatment of industrial effluent and ground toxins. Unfortunately, certain species can cause devastating damage to crops, serious disease in humans or can damage building materials. In this thesis we constructed new models for the development of fungal mycelia. At this scale, partial differential equations representing the interaction of biomass with the underlying substrate is the appropriate choice. Models are essentially based on those derived by Davidson and co workers (see e.g. Boswell et al.(2007)). These models are of a complex mathematical structure, comprising both parabolic and hyperbolic parts. Thus, their analytic and numerical properties are nontrivial. The objectives of this thesis are to: (i) obtain a solid understanding of the physiology of growth and function and the varying mathematical techniques used in model construction. (ii) revisit existing models to reinterpret the various model components in a simple form. (iii) construct models to compare the growth dynamics of different phenotype for new species to see if these "scale " appropriately.

Many infectious pathogens, and in particular viruses, have an extremely high rate of mutation. This can lead to rapid evolution driven by selection pressures operating at both the within- and between-host levels, as strains compete for resources within their chosen host while also competing to effectively transmit to new hosts. In the case of chronic viral infections, such as the human immunodeficiency virus (HIV) or hepatitis C, substantial viral evolution may take place within a single infected host. The fitness of a pathogen has been studied at the between-host level and at the within-host level, but linking the two levels of selection pressure is a difficult problem that has yet to be studied satisfactorily. We modify a simple model describing the within host dynamics of HIV infection by including multiple pathogen strains with different properties and allowing these strains to mutate. Within the host we observe different strategies for pathogen success during different stages of infection, which often leads to different strains predominating within the host over the course of infection. We then embed our within-host model into a Monte Carlo simulation that models the interactions between infected individuals. This approach allows us to combine selective pressure at the within-host level with pressures at the between-host level and helps us to predict which strains are most likely to be present within the population. We show that under our model assumptions the co-existence of multiple strains is possible and we explore the factors leading to the success of a pathogen.

Many infectious pathogens, and in particular viruses, have an extremely high rate of mutation. This can lead to rapid evolution driven by selection pressures operating at both the within- and between-host levels, as strains compete for resources within their chosen host while also competing to effectively transmit to new hosts. In the case of chronic viral infections, such as the human immunodeficiency virus (HIV) or hepatitis C, substantial viral evolution may take place within a single infected host. The fitness of a pathogen has been studied at the between-host level and at the within-host level, but linking the two levels of selection pressure is a difficult problem that has yet to be studied satisfactorily. We modify a simple model describing the within host dynamics of HIV infection by including multiple pathogen strains with different properties and allowing these strains to mutate. Within the host we observe different strategies for pathogen success during different stages of infection, which often leads to different strains predominating within the host over the course of infection. We then embed our within-host model into a Monte Carlo simulation that models the interactions between infected individuals. This approach allows us to combine selective pressure at the within-host level with pressures at the between-host level and helps us to predict which strains are most likely to be present within the population. We show that under our model assumptions the co-existence of multiple strains is possible and we explore the factors leading to the success of a pathogen.

This thesis investigates the effect of coinfection on the emergence of resistant pathogens. Firstly, a multiple infection model with treatment is derived and the conditions for invasion are established. The invasion condition is then related to an equivalent and easier to obtain condition, R0, by applying the Next-Generation Theorem. Due to its biological interpretation, a heuristic derivation of R0 as the invasion condition is also given. Then assuming that resistance comes at a cost to the pathogen, and using a very simple within-host model, we establish under which specific set of biological assumptions we should expect coinfection to increase or decrease R0. Specifically, we obtain that in the no cost of resistance case, reduced transmission case, and increased mortality case, that coinfection will increase the R0 value and that in the reduced growth and poor competitor case that the effect is indeterminate. We also introduced a method for approximating the intrinsic growth rate when the coinfection efficiency is assumed to be small. Using this method, we show that we obtain the same trend for the cost of resistance cases when comparing our estimate for the intrinsic growth rate for the coinfection case versus the intrinsic growth rate for the single infection case. We also use this approximation to estimate the percentage of resistance as a function of time. Finally, we analyze how both the intrinsic growth rate and R0 respond to a changing treatment rate, compared to the intrinsic growth rate and R0 value in the single infection case. We found that the change in R0 and the intrinsic growth rate can be greater or smaller than the change in R0 or the intrinsic growth rate for the single infection case. / Thesis (Master, Mathematics & Statistics) -- Queen's University, 2011-05-05 16:36:40.865

The thesis uses mathematical modeling and analysis to provide insights into the transmission dynamics of Human papillomavirus (HPV), and associated cancers and warts, in a community. A new deterministic model is designed and used to assess the community-wide impact of mass vaccination of new sexually-active susceptible females with the anti-HPV Gardasil vaccine. Conditions for the existence and asymptotic stability of the associated equilibria are derived. Numerical simulations show that the use of Gardasil vaccine could lead to the effective control of the spread of HPV in the community if the vaccine coverage is at least 78%. The model is extended to include the dynamics of the low- and high-risk HPV types and the combined use of the Gardasil and Cervarix anti-HPV vaccines. Overall, this study shows that the prospect of the effective community-wide control of HPV using the currently-available anti-HPV vaccines are encouraging.