Determining the emergent behaviour of a population from the interactions of its individuals is an ongoing challenge in the modelling of biological phenomena. Many classical models assume that the spatial location of each individual is independent of the locations of all other individuals. This mean-field assumption is not always realistic; in biological systems we frequently see clusters of individuals develop from uniform initial conditions. In this thesis, we explore situations in which the mean-field approximation is no longer valid for volume-excluding processes on a regular lattice. We provide methods which take into account the spatial correlations between lattice sites, thus more accurately reflecting the system's behaviour, and discuss methods which can provide information as to the validity of mean-field and other approximations.

Although death due to lethal cardiac arrhythmias is the leading cause of mortality in Western Society, many of the fundamental mechanisms underlying their onset, maintenance and termination, still remain poorly understood. In recent years, experimental techniques such as optical mapping have provided useful high-resolution recordings of cardiac electrical dynamics during complex arrhythmias and defibrillation episodes, which have been combined with detailed computer simulations to further our understanding of these phenomena. However, mechanistic enquiry is severely restricted as the optical mapping technique suffers from a number of distortion effects which compromise the fidelity of the experimental measurements, presenting difficulties in the comparison of experimental data with computational simulations. This Thesis presents a thorough investigation into the distortion effects encountered in optical mapping experiments, guided by the development of a coherent series of computational models. The models presented successfully characterise the specific mechanisms of fluorescent signal distortion due to photon scattering. Photon transport in cardiac tissue is modelled using both continuous (reaction-diffusion) and discrete stochastic (Monte Carlo) approaches to simulate the effects of photon scattering within the myocardium upon the recorded fluorescent signal, which include differing levels of detail and associated computational complexity. Specifically, these models are used to investigate the important role played by the complex ventricular structural anatomy, as well as the specifics of the experimental set-up itself. In addition, a tightly coupled electromechanical model of a contracting cardiac fibre is developed which provides an important first-step towards the development of a model to quantitatively assess the distortion observed when recording from a freely contracting cardiac preparation. Simulation of these distortion effects using the models allows discrimination to be made between those parts of the experimental signal which are due to underlying tissue electrophysiology and those due to artifact, facilitating a more accurate interpretation of experimentally-obtained data. The models presented succeed in two main respects. Firstly, they provide a ‘post-processing’ tool which can be added on to computational simulations of electrical activation, allowing for a more accurate and faithful comparison between simulations and experiments, helping to validate predictions made by electrical models. Secondly, they provide a higher degree of mechanistic insight into the fundemental ways in which optical signals are distorted, showing how this distortion can be maximised or controlled. The understanding and quantification of the fundemental mechanisms of optical mapping signal distortion, provided by this Thesis, therefore fulfils an important role in the study of arrhythmia mechanisms.

This thesis considers aspects of modelling solid tumours. We begin by considering the common assumption that nutrient or drug concentrations in avascular tumour spheroids are radially symmetric. We derive a simple Poisson equation for biomolecular diffusion into an avascular tumour, but with highly oscillatory boundary conditions due to the surrounding capillary network. We find that the assumption of radial symmetry is legitimate for biomolecules that are taken up in sufficient quantities by proliferating cancer cells; however radially symmetric profiles need not be observed otherwise. We then investigate how the gap between an avascular tumour and the neighbouring vasculature varies as the tumour grows. This is explored by (i) using scaling arguments based on ordinary differential equations, (ii) coupling the rate of oxygen flux from the vasculature to oxygen evolution within the tumour, and (iii) deriving a system of six coupled non-linear partial differential equations modelling the tumour evolution. It is found that as the tumour grows any initial gap between the tumour and neighbouring vasculature closes since there is no mechanism which would sufficiently up-regulate non-cancerous cell proliferation. This is in contrast to the intra-cornea implantation observations, upon which several mathematical models are based. Finally, we study the growth and treatment of a vascular tumour subjected to chemotherapies, particularly when the therapies can exhibit an anti-angiogenic effect and resistance to the therapy is incorporated. A multi-compartment model is derived for the evolution of a tumour undergoing treatment and parameters are estimated, with extensions to incorporate numerous different therapy protocols in the literature. We find that anti-angiogens can be effective, though the appropriate scheduling is counter-intuative and contradicts many standard therapy rules. We conclude that chemotherapy protocol design is very sensitive to the mode of action of the drug and simple general strategies will, in many cases, not be the most effective.

A system of ordinary differential equations is developed to model the effect of fatty acids on chronic intestinal inflammation that is typical of a Crohn’s patient. Several murine studies have shown an anti-inflammatory response when specific polyunsaturated fatty acids are included regularly in the diet. It is believed that the fatty acids serve as a specific ligand that activates the Peroxisome Proliferator Activated Receptor (PPAR) which is located on multiple cell types that are active in the inflammatory response. The binding of the PPAR results in a suppression of the inflammatory pathway. Results of the model indicate a muted inflammatory response when fatty acids are added regularly to the diet in mild to moderate cases of Crohn’s. Results of mathematical analysis show a stable fixed point with decreased inflammatory markers and pathogen levels when fatty acids are added regularly to the diet.

In this thesis, basic concepts of populational models are studied from a theoretical point of view, especially the long term behaviours. All models are at least time dependent with additional age structure, spatial structure. The last model which is an extension of the von Foerster equation, is dependent on all o f these structures and have a long-term solution for large values of time. Modeling population is a frequent subject in modern biology. It is hard to create a model that appears as realistic as possible. First one might consider that a population size is governed by the current size of the population, along with rates of how each individual contributes (give birth), so that the population increases. and how frequent an individual dies, causing the population to decrease in size. However these sort of models can only describe the size of population in a shorter span of time.

In the well known SIR endemic model, the infection-free steady state is globally stable for R0 < 1 and unstable for R0 > 1. Hence, we have a forward bifurcation when R0 = 1. When R0 > 1, an asymptotically stable endemic steady state exists. The basic reproduction number R0 is the main threshold bifurcation parameter used to determine the stability of steady states of SIR endemic models. In this thesis we study extensions of the SIR endemic model for which a backward bifurcation may occur at R0 = 1. We investigate the biologically reasonable conditions for the change of stability. We also analyse the impact of di erent factors that lead to a backward bifurcation both numerically and analytically. A backward bifurcation leads to sub-critical endemic steady states and hysteresis. We also provide a general classi cation of such models, using a small amplitude expansion near the bifurcation. Additionally, we present a procedure for projecting three dimensional models onto two dimensional models by applying some linear algebraic techniques. The four extensions examined are: the SIR model with a susceptible recovered class; nonlinear transmission; exogenous infection; and with a carrier class. Numerous writers have mentioned that a nonlinear transmission function in relation to the infective class, can only lead to a system with an unstable endemic steady state. In spite of this we show that in a nonlinear transmission model, we have a function depending on the infectives and satisfying certain biological conditions, and leading to a sub-critical endemic equilibriums.

In the well known SIR endemic model, the infection-free steady state is globally stable for R0 < 1 and unstable for R0 > 1. Hence, we have a forward bifurcation when R0 = 1. When R0 > 1, an asymptotically stable endemic steady state exists. The basic reproduction number R0 is the main threshold bifurcation parameter used to determine the stability of steady states of SIR endemic models. In this thesis we study extensions of the SIR endemic model for which a backward bifurcation may occur at R0 = 1. We investigate the biologically reasonable conditions for the change of stability. We also analyse the impact of di erent factors that lead to a backward bifurcation both numerically and analytically. A backward bifurcation leads to sub-critical endemic steady states and hysteresis. We also provide a general classi cation of such models, using a small amplitude expansion near the bifurcation. Additionally, we present a procedure for projecting three dimensional models onto two dimensional models by applying some linear algebraic techniques. The four extensions examined are: the SIR model with a susceptible recovered class; nonlinear transmission; exogenous infection; and with a carrier class. Numerous writers have mentioned that a nonlinear transmission function in relation to the infective class, can only lead to a system with an unstable endemic steady state. In spite of this we show that in a nonlinear transmission model, we have a function depending on the infectives and satisfying certain biological conditions, and leading to a sub-critical endemic equilibriums.

In the well known SIR endemic model, the infection-free steady state is globally stable for R0 < 1 and unstable for R0 > 1. Hence, we have a forward bifurcation when R0 = 1. When R0 > 1, an asymptotically stable endemic steady state exists. The basic reproduction number R0 is the main threshold bifurcation parameter used to determine the stability of steady states of SIR endemic models. In this thesis we study extensions of the SIR endemic model for which a backward bifurcation may occur at R0 = 1. We investigate the biologically reasonable conditions for the change of stability. We also analyse the impact of di erent factors that lead to a backward bifurcation both numerically and analytically. A backward bifurcation leads to sub-critical endemic steady states and hysteresis. We also provide a general classi cation of such models, using a small amplitude expansion near the bifurcation. Additionally, we present a procedure for projecting three dimensional models onto two dimensional models by applying some linear algebraic techniques. The four extensions examined are: the SIR model with a susceptible recovered class; nonlinear transmission; exogenous infection; and with a carrier class. Numerous writers have mentioned that a nonlinear transmission function in relation to the infective class, can only lead to a system with an unstable endemic steady state. In spite of this we show that in a nonlinear transmission model, we have a function depending on the infectives and satisfying certain biological conditions, and leading to a sub-critical endemic equilibriums.

Many models in the physical and life sciences formulated as dynamical systems have a positive steady state, with the local behavior of this steady state determined by the eigenvalues of its Jacobian matrix. The first part of this thesis is concerned with analyzing the linear stability of the steady state by using sign patterns, which are matrices with entries from the set {+,-,0}. The linear stability is related to the allowed refined inertias of the sign pattern of the Jacobian matrix of the system, where the refined inertia of a matrix is a 4-tuple (n+, n_-, ; nz; 2np) with n+ (n_) equal to the number of eigenvalues with positive (negative) real part, nz equal to the number of zero eigenvalues, and 2np equal to the number of nonzero pure imaginary eigenvalues. This type of analysis is useful when the parameters of the model are of known sign but unknown magnitude. The usefulness of sign pattern analysis is illustrated with several biological examples, including biochemical reaction networks, predator{prey models, and an infectious disease model. The refined inertias allowed by sign patterns with specific digraph structures have been studied, for example, for tree sign patterns. In the second part of this thesis, such results on refined inertias are extended by considering sign and zero-nonzero patterns with digraphs isomorphic to strongly connected orientations of the Petersen graph. / Graduate

Computational and modelling skills are vital to most fields of biological research, yet traditional biology majors have no or little opportunity to develop these skills during their undergraduate education. We describe an approach, which can address this issue by a synergy of online resources called MathBench modules and Stock and Flow modelling. Using a step-by-step method starting with a MathBench ‘bootcamp’, we were able to achieve a significant gain in quantitative skills of students with no previous experience with model building. At the end of the course, the students were able to construct and analyse complex models and gained confidence in mathematical skills.