Proper heart function results from the periodic execution of a series of coordinated interdependent mechanical, chemical, and electrical processes within the cardiac tissue. Central to these processes is the action potential - the electrochemical event that initiates contraction of the individual cardiac myocytes. Many models of the cardiac action potential exist with varying levels of complexity, but none account for the electrophysiological role played by caveolae - small invaginations of the cardiac cell plasma membrane. Recent electrophysiological studies regarding these microdomains reveal that cardiac caveolae function as reservoirs of 'recruitable' sodium ion channels. As such, caveolar channels constitute a substantial and previously unrecognized source of sodium current that can significantly influence action potential morphology. In this thesis, I formulate and analyze new models of cardiac action potential which account for these caveolar sodium currents and provide a computational venue in which to develop and test new hypotheses. My results provide insight into the role played by caveolar ionic currents in regulating the electrodynamics of cardiac myocytes and suggest that in certain pathological cases, caveolae may play an arrhythmogenic role.

<p>This report consider a system describing three competing species with populations <em>x</em>, <em>y</em> and <em>z</em>. Sufficient conditions for every positive equilibrium to be asymptotically stable have been found. First it is shown that conditions on the pairwise competitive interaction between the populations are needed. Actually, these conditions are equivalent to asymptotic stability for any two-dimensional competing system of the three species. It is also shown that these alone are not enough, and that a condition on the competitive interaction between all three populations is also needed. If all conditions are fulfilled, each population will survive on a long-term basis and there will be a stable coexistence.</p>

This report consider a system describing three competing species with populations x, y and z. Sufficient conditions for every positive equilibrium to be asymptotically stable have been found. First it is shown that conditions on the pairwise competitive interaction between the populations are needed. Actually, these conditions are equivalent to asymptotic stability for any two-dimensional competing system of the three species. It is also shown that these alone are not enough, and that a condition on the competitive interaction between all three populations is also needed. If all conditions are fulfilled, each population will survive on a long-term basis and there will be a stable coexistence.

Boundary value problems (BVPs) are systems of ordinary differential equations (ODEs) with boundary conditions imposed at two or more distinct points. Such problems arise within mathematical models in a wide variety of applications. Numerically solving BVPs for ODEs generally requires the use of a series of complex numerical algorithms. Fortunately, when users are required to solve a BVP, they have a variety of BVP software packages from which to choose. However, all BVP software packages currently available implement a specific set of numerical algorithms and therefore function quite differently from each other. Users must often try multiple software packages on a BVP to find the one that solves their problem most effectively. This creates two problems for users. First, they must learn how to specify the BVP for each software package. Second, because each package solves a BVP with specific numerical algorithms, it becomes difficult to determine why one BVP package outperforms another. With that in mind, this thesis offers two contributions. <p> First, this thesis describes the development of the BVP component to the fully featured problem-solving environment (PSE) for the numerical solution of ODEs called pythODE. This software allows users to select between multiple numerical algorithms to solve BVPs. As a consequence, they are able to determine the numerical algorithms that are effective at each step of the solution process. Users are also able to easily add new numerical algorithms to the PSE. The effect of adding a new algorithm can be measured by making use of an automated test suite. <p> Second, the BVP component of pythODE is used to perform two research studies. In the first study, four known global-error estimation algorithms are compared in pythODE. These algorithms are based on the use of Richardson extrapolation, higher-order formulas, deferred corrections, and a conditioning constant. Through numerical experimentation, the algorithms based on higher-order formulas and deferred corrections are shown to be computationally faster than Richardson extrapolation while having similar accuracy. In the second study, pythODE is used to solve a newly developed one-dimensional model of the agglomerate in the catalyst layer of a proton exchange membrane fuel cell.

Boundary value problems (BVPs) are systems of ordinary differential equations (ODEs) with boundary conditions imposed at two or more distinct points. Such problems arise within mathematical models in a wide variety of applications. Numerically solving BVPs for ODEs generally requires the use of a series of complex numerical algorithms. Fortunately, when users are required to solve a BVP, they have a variety of BVP software packages from which to choose. However, all BVP software packages currently available implement a specific set of numerical algorithms and therefore function quite differently from each other. Users must often try multiple software packages on a BVP to find the one that solves their problem most effectively. This creates two problems for users. First, they must learn how to specify the BVP for each software package. Second, because each package solves a BVP with specific numerical algorithms, it becomes difficult to determine why one BVP package outperforms another. With that in mind, this thesis offers two contributions. <p> First, this thesis describes the development of the BVP component to the fully featured problem-solving environment (PSE) for the numerical solution of ODEs called pythODE. This software allows users to select between multiple numerical algorithms to solve BVPs. As a consequence, they are able to determine the numerical algorithms that are effective at each step of the solution process. Users are also able to easily add new numerical algorithms to the PSE. The effect of adding a new algorithm can be measured by making use of an automated test suite. <p> Second, the BVP component of pythODE is used to perform two research studies. In the first study, four known global-error estimation algorithms are compared in pythODE. These algorithms are based on the use of Richardson extrapolation, higher-order formulas, deferred corrections, and a conditioning constant. Through numerical experimentation, the algorithms based on higher-order formulas and deferred corrections are shown to be computationally faster than Richardson extrapolation while having similar accuracy. In the second study, pythODE is used to solve a newly developed one-dimensional model of the agglomerate in the catalyst layer of a proton exchange membrane fuel cell.

Efficient and robust numerical simulation of multiscale problems encountered in science and engineering is a formidable challenge. Full resolution of multiscale problems using direct numerical simulations requires enormous amounts of computational time and resources. This thesis develops seamless multiscale methods for ordinary and partial differential equations under the framework of the heterogeneous multiscale method (HMM). The first part of the thesis is devoted to the development of seamless multiscale integrators for ordinary differential equations. The first method, which we call backward-forward HMM (BFHMM), uses splitting and on-the-fly filtering techniques to capture slow variables of highly oscillatory problems without any a priori information. The second method, denoted by variable step size HMM (VSHMM), as the name implies, uses variable mesoscopic step sizes for the unperturbed equation, which gives computational efficiency and higher accuracy. VSHMM can be applied to dissipative problems as well as highly oscillatory problems, while BFHMM has some difficulties when applied to the dissipative case. The effect of variable time stepping is analyzed and the two methods are tested numerically. Multi-spatial problems and numerical methods are discussed in the second part. Seamless heterogeneous multiscale methods (SHMM) for partial differential equations, especially the parabolic case without scale separation are proposed. SHMM is developed first for the multiscale heat equation with a continuum of scales in the diffusion coefficient. This seamless method uses a hierarchy of local grids to capture effects from each scale and uses filtering in Fourier space to impose an artificial scale gap. SHMM is then applied to advection enhanced diffusion problems under incompressible turbulent velocity fields. / text

Malaria is an infectious disease, transmitted between humans through mosquito bites, that kills about two million people a year. We derive and analyze a mathematical model to better understand the transmission and spread of this disease. Our main goal is to use this model to compare intervention strategies for malaria control for two representative areas of high and low transmission. We model malaria using ordinary differential equations. We analyze the existence and stability of disease-free and endemic (malaria persisting in the population) equilibria. Key to our analysis is the definition of a reproductive number, R₀ (the number of new infections caused by one individual in an otherwise fully susceptible population through the duration of the infectious period). We prove the loss of stability of the disease-free equilibrium as R0 increases through R₀ = 1. Using global bifurcation theory developed by Rabinowitz, we show the bifurcation of endemic equilibria at R₀ = 1. This bifurcation can be either supercritical (leading to stable endemic equilibria for R₀ > 1) or subcritical (leading to stable endemic equilibria for R₀ < 1 in the presence of hysteresis). We compile two reasonable sets of values for the parameters in the model: for areas of high and low transmission. We compute sensitivity indices of R₀ and the endemic equilibrium to the parameters around the baseline values. R₀ is most sensitive to the mosquito biting rate in both high and low transmission areas. The fraction of infectious humans at the endemic equilibrium is most sensitive to the mosquito biting rate in low transmission areas, and to the human recovery rate in high transmission areas. This sensitivity analysis allows us to compare the effectiveness of different control strategies. According to our model, the most effective methods for malaria control are the use of insecticide-treated bed nets and the prompt diagnosis and treatment of infected individuals.

This thesis deals with the numerical solution of ordinary differential equations (ODEs) using finite difference (FD) methods. In particular, boundary summation equation (BSE) preconditioning for FD approximations for ODEs with constant coefficients on locally refined meshes is studied. Firstly, the BSE for FD approximations of ODEs with constant coefficients is derived on a locally refined mesh. Secondly, the obtained linear system of equations are solved by the iterative method GMRES. Then, the arithmetic complexity and convergence rate of the iterative solution of the BSE formulation are discussed. Finally, numerical experiments are performed to compare the new approach with the FD approach. The results show that the BSE formulation has low arithmetic complexity and the convergence rate of the iterative solvers is fast and independent of the number of grid points.

Acidic pH is a common characteristic of human tumours. It has a significant impact on tumour progression and response to therapies. In this thesis, we utilise mathematical modelling to examine the role of acidosis in the interaction between normal and tumour cell populations. In the first section we investigate the cell–microenvironmental interactions that mediate somatic evolution of cancer cells. The model predicts that selective forces in premalignant lesions act to favour cells whose metabolism is best suited to respond to local changes in oxygen, glucose and pH levels. In particular the emergent cellular phenotype, displaying increased acid production and resistance to acid-induced toxicity, has a significant proliferative advantage because it will consistently acidify the local environment in a way that is toxic to its competitors but harmless to itself. In the second section we analyse the role of acidity in tumour growth. Both vascular and avascular tumour dynamics are investigated, and a number of different behaviours are observed. Whilst an avascular tumour always proceeds to a benign steady state, a vascular tumour may display either benign or invasive dynamics, depending on the value of a critical parameter. Extensions of the model show that cellular quiescence, or non-proliferation, may provide an explanation for experimentally observed cycles of acidity within tumour tissue. Analysis of both models allows assessment of novel therapies directed towards changing the level of acidity within the tumour. Finally we undertake a comparison between experimental tumour pH images and the models of acid dynamics set out in previous chapters. This analysis will allow us to assess and verify the previous modelling work, giving the mathematics a firm biological foundation. Moreover, it provides a methodology of calculating important diagnostic parameters from pH images.

Este trabalho, ao apresentar os conceitos básicos da Epidemiologia Matemática e da modelagem de populações por classes etárias, tem por objetivo desenvolver e implementar três modelos epidemiológicos de transmissão do dengue, a fim de avaliar teoricamente os efeitos da aplicação de inseticidas em populações de Aedes aegypti, em relação às epidemias de dengue. Uma variedade de métodos têm sido empregados no controle do vetor, sendo o Aedes aegypti a principal espécie envolvida na transmissão do dengue. A· aplicação de inseticidas de ultrabaixo volume (UL V) é uma das técnicas amplamente utilizada, particularmente durante epidemias. Tal técnica tem como objetivo matar os mosquitos adultos (adulticida). Há muita controvérsia em tomo destas aplicações, no que diz respeito ao impacto no controle da transmissão do dengue. Desta forma, através deste trabalho, procuramos observar a influência do uso de inseticida na dinâmica populacional do vetor transmissor e na dinâmica da epidemia, e analisar as circunstâncias em que o inseticida pode ser utilizado a fim de agir eficientemente no controle da transmissão do dengue. O emprego de larvicidas também é abordado, a fim de que se possa observar a influência deste na dinâmica populacional do vetor transmissor e na dinâmica da epidemia. Neste trabalho são detalhadas as hipóteses utilizadas na construção de cada modelo de transmissão do dengue apresentado. Apresentados os modelos, são considerados os aspectos relativos à implementação. Assim, mediante os aspectos teóricos envolvidos na modelagem e implementação, resultados numéricos são obtidos, através das simulações, as quais nos auxiliam a avaliar os efeitos da aplicação de inseticidas em populações de Aedes aegypti no controle da transmissão do dengue. / In this work the basic Mathematical Epidemiology ideas are presented and three models of Dengue Fever transrnission are developed in order to measure the effects of the use o f insecticides on the populations o f the mosquito Aedes aegypti that are related to the dengue epidemics. There is a good variety of control methods on the Aedes aegypti populations. The use of ultra-low volume (UL V) insecticides is widely employed specially during epidemics. The goal of such method is to eliminate a fraction of the adult mosquito population. There is a great deal o f controversy on the effectiveness o f insecticide use during a dengue epidemic. In this way we propose to investigate the impact o f UL V on the mosquito dynarnics and on the epidemics dynarnics as well in order to determine in which circumstances the use o f UL V can truly effective on the course o f a epidemic. On the same line, we also propose a study on the use of larvicides as a control method. In this study we detail the hypotheses that are used to construct the dynarnic models. Once the models are presented we consider the implementation details. The numerical results are obtained after various simulations which provide the data that allow us to measure the impact ofthe control technique on the dengue epidemics.