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The study of cardiac action potentials has many medical applications. Dr. Dennis Noble first used mathematical models to study cardiac action potentials in the 1960s. We begin our study of cardiac action potentials with one form of the Fitzhugh-Nagumo partial differential equation. We use the non-classical method to produce a closed form solution for the decoupled Fitzhugh Nagumo equation. Using voltage recording data of action potentials in a cardiac myocyte and in purkinje fibers, we estimate parameter values for the closed form solution with standard linear and non-linear regression methods. Results are limited, thus leading us to perturb the solution to obtain a better fit. We turn to singular perturbation theory to justify our pole-based approach. Finally, we test our model on independent action potential data sets to evaluate our model and to draw conclusions on how our model can be applied.

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There is a growing interest in the dynamics of the co-infection of these two diseases. In this thesis, firstly we focus on studying the effect of a distributed delay representing the incubation period for the malaria parasite in the mosquito vector to possibly reduce the initial transmission and prevalence of malaria. This model can be regarded as a generalization of SEI models (with a class for the latently infected mosquitoes) and SI models with a discrete delay for the incubation period in mosquitoes. We study the possibility of occurrence of backward bifurcation. We then extend these ideas to study a full model of HIV and malaria co-infection. To get further inside into the dynamics of the model, we use the geometric singular perturbation theory to couple the fast and slow models from the full model. Finally, since the governing models are very complex, they cannot be solved analytically and hence we develop and analyze a special class of numerical methods to solve them.

There is a growing interest in the dynamics of the co-infection of these two diseases. In this thesis, firstly we focus on studying the effect of a distributed delay representing the incubation period for the malaria parasite in the mosquito vector to possibly reduce the initial transmission and prevalence of malaria. This model can be regarded as a generalization of SEI models (with a class for the latently infected mosquitoes) and SI models with a discrete delay for the incubation period in mosquitoes. We study the possibility of occurrence of backward bifurcation. We then extend these ideas to study a full model of HIV and malaria co-infection. To get further inside into the dynamics of the model, we use the geometric singular perturbation theory to couple the fast and slow models from the full model. Finally, since the governing models are very complex, they cannot be solved analytically and hence we develop and analyze a special class of numerical methods to solve them.

Philosophiae Doctor - PhD / There is a growing interest in the dynamics of the co-infection of these two diseases. In this thesis, firstly we focus on studying the effect of a distributed delay representing the incubation period for the malaria parasite in the mosquito vector to possibly reduce the initial transmission and prevalence of malaria. This model can be regarded as a generalization of SEI models (with a class for the latently infected mosquitoes) and SI models with a discrete delay for the incubation period in mosquitoes. We study the possibility of occurrence of backward bifurcation. We then extend these ideas to study a full model of HIV and malaria co-infection. To get further inside into the dynamics of the model, we use the geometric singular perturbation theory to couple the fast and slow models from the full model. Finally, since the governing models are very complex, they cannot be solved analytically and hence we develop and analyze a special class of numerical methods to solve them. / South Africa

Les lipides des microalgues et les glucides de cyanobactéries peuvent être transformés en biodiesel et en bioéthanol, respectivement. L'amélioration de la production de ces molécules doit prendre en compte les entrées périodiques (principalement la lumière) forçant le réseau métabolique de ces organismes photosynthétiques. Il est donc nécessaire de tenir compte de la dynamique du réseau métabolique en réduisant sa dimension pour assurer la maniabilité mathématique. Le but de ce travail est de concevoir une approche originale pour réduire les réseaux métaboliques dynamiques tout en conservant la dynamique de base. Cette méthode est basée sur une séparation en échelles de temps. Pour une classe de modèles de réseaux métaboliques décrits par des ODE, la dynamique des systèmes réduits est calculée à l'aide du théorème de Tikhonov pour les systèmes singulièrement perturbés. Cette approximation quasi-stationnaire coïncide avec la dynamique du réseau d'origine, avec une erreur bornée. L'approche est d'abord développée pour les systèmes de réaction pouvant être linéarisés autour d'un point de travail et forcés par des entrées continues. Ensuite, une généralisation de cette méthode est donnée pour les réseaux à réactions rapides de cinétiques de Michaelis-Menten et tout type de cinétiques lentes, prenant également en compte un nombre fini d'entrées continues externes. La méthode de réduction met en évidence une relation entre la grandeur de la concentration des métabolites et la gamme des vitesses de réaction : les métabolites consommés par les réactions rapides ont une concentration inférieure d'un ordre de grandeur à celle des métabolites consommés à faible vitesse. Cette propriété est satisfaite pour les métabolites à dynamique rapide ne se trouvant pas dans un piège de flux, concept introduit dans ce travail. Le système réduit peut être calibré avec des données expérimentales à l'aide d'une procédure d'identification dédiée basée sur la minimisation. L'approche est illustrée par un réseau métabolique de microalgues autotrophes, comprenant le métabolisme central et représentant la dynamique des glucides et des lipides. Cette approche permet de bien ajuster les données expérimentales de Lacour et al. (2012) avec la microalgue Tisochrysis lutea. Enfin, un schéma visant à optimiser la production de molécules cibles est proposé en utilisant le système réduit. / Lipids from microalgae and carbohydrates from cyanobacteria can be transformed into biodiesel and bioethanol, respectively. Enhancing the production of these molecules must account for the periodic inputs (mainly light) forcing the metabolic network of these photosynthetic organisms. It is therefore necessary to account for the dynamics of the metabolic network, while reducing its dimension to ensure mathematical tractability. The aim of this work is to design an original approach to reduce dynamic metabolic networks while keeping the core dynamics. This method is based on time-scale separation. For a class of metabolic network models described by ODE, the dynamics of the reduced systems are computed using the theorem of Tikhonov for singularly perturbed systems. This Quasi Steady State Approximation accurately coincides with the original network dynamics, with a bounded error. The approach is first developed for reaction systems that can be linearized around a working point and that are forced by external continuous inputs. Then, a generalization of this method is given for networks with fast reactions of Michaelis-Menten kinetics and any type of slow kinetics, also considering a finite number of external continuous inputs. The reduction method highlights a relation between the concentration magnitude of the metabolites and the range of the reaction rates: the metabolites that are consumed by fast reactions have concentration one order of magnitude lower than metabolites consumed at slow rates. This property is satisfied for metabolites with fast dynamics that are not in a flux trap, a concept introduced in this work. The reduced system can be calibrated with experimental data using a dedicated identification procedure based on minimization. The approach is illustrated with an autotrophic microalgae metabolic network, including the core metabolism and representing the carbohydrates and lipids dynamics. The approach efficiently fits the experimental data from Lacour et al. (2012) with the microalgae Tisochrysis lutea. Finally, a scheme to optimize the production of target molecules is proposed using the reduced system.

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