Magister Educationis - MEd / The National Curriculum Statement introduced mathematical literacy officially in 2006. Learners in general perform poorly at mathematics in South Africa but there is strong belief that learners should graduate from schools sufficiently literate to deal with the mathematical issues they will encounter in out-of-school situations. Based on this, this study is an investigation of the contexts, which grades 8 - 10 learners would prefer to engage with mathematics. The aim of this study was thus to investigate mathematical literacy in relation to learners' motivations for the contexts they would prefer to deal with in mathematical literacy. The emphasis of the study is to concentrate on learners' written motivations for mathematical contexts. Data were collected using a questionnaire that deals with contexts for mathematics. / South Africa

Magister Educationis - MEd / The study explored the comparison of grade 8 to 10 urban and peri-urban learners' contexts preferences in mathematical literacy. There is currently a strong emphasis on the use of contexts for school mathematics. This has been also the case for South Africa when grade 10 learners have to make a choice between mathematics and mathematical literacy as one of their compulsory subjects for grade 10. This study focused more on the use of mathematics in real life situations. Data was collected by using questionnaires developed as part of the Relevance of School Mathematics Education (ROSME) project. The questionnaire dealt with contexts preferred by grade 10 learners from urban and peri-urban areas. The data were analysed using non-parametric statistical techniques. The findings radicate that there were contexts highly preferred by learners from both urban and peri-urban areas; least preferred by learners from both areas, highly preferred by learners from periurban areas but not by learners from urban areas and least preferred by learners from urban areas but not by those from peri-urban areas and vice versa. It is recommended that contexts highly preferred by learners should be incorporated in the learning experiences of learners. / South Africa

The cell cycle is an essential process in all living organisms that must be carefully regulated to ensure successful cell growth and division. Disregulation of the cell cycle is a key contributing factor towards the formation of cancerous cells. Understanding events at a cellular level is the first step towards comprehending how cancer manifests at an organismal level. Mathematical modelling can be used as a means of formalising and predicting the behaviour of the biological systems involved in cancer. In response, cell cycle models have been constructed to simulate and predict what happens to the mammalian cell over a time course in response to variable parameters.Current cell cycle models rarely account for certain precursors of cell growth such as energy usage and the need for non-essential amino acids as fundamental building blocks of macromolecules. Normal and cancer cell metabolism differ in the way they derive energy from glucose. In addition, normal and cancer cells also demonstrate different levels of gene expression. Two versions of a mammalian cell cycle and metabolism model, based on ordinary differential equations (ODEs) that respond to fluctuations in glucose concentration levels, have been developed here for the normal and cancer cell scenarios. Sensitivity analysis is performed for both normal and cancer cells using these cell cycle and metabolism models to investigate which kinetic reaction steps have a greater effect over the cell cycle period. Detailed analysis of the models and quantitatively assessing metabolite levels at various stages of the cell cycle may offer novel insights into how the glycolytic rate varies during the cell cycle for both normal and cancer cells.The results of the sensitivity analysis are used to identify potential drug targets in cancer therapy. Combinations of these individual targets are also investigated to compare the different effects of single and multiple drug compounds on the time it takes to complete a cell division cycle.

Background: Malaria causes over 400,000 estimated deaths annually worldwide, with about 90% in sub-Saharan Africa. Long-lasting insecticidal nets (LLINs) and indoor residual spraying (IRS) are two vector-control interventions proven to reduce malaria transmission, but their use together compared to separate has shown mixed results. Methodology: We used a mathematical model to examine the impact of LLINs and IRS on malaria transmission. Time-series analyses and basic reproductive numbers (R0) were developed using MATLAB. We also assessed IRS timing and performed a sensitivity analysis on R0. Results: Modelling scenarios combining LLINs with IRS were similar to those with LLINs alone. Shorter IRS impulses had greater reductions in mosquito populations. The LLIN feeding-inhibition rate was a key parameter with a negative correlation to R0. Discussion/Conclusion: We developed an understanding of the effect of vector-control strategies on malaria transmission. IRS, when paired with LLINs, showed only small improvements in reducing malaria transmission compared to LLINs alone. These results can assist vector-control programmes.

For centuries, pandemic influenza has emerged at irregular and unpredictable intervals to cause widespread illness, hospitalization and death. Uncertainty surrounding the timing and severity of future influenza pandemics present challenges for preparedness and response efforts. The objective of this dissertation is to advance pandemic influenza knowledge and preparedness, through a series of interrelated articles that address the follow research questions: 1. What are the likely consequences of a pandemic flu event in Canada? 2. What do mathematical models tell us about preparing for such an event? 3. What is the best way to mitigate the consequences of an influenza pandemic? Six articles were prepared for submission in scientific, peer-reviewed journals. The first is a historical review of the burden of pandemic influenza. The second and third are systematic reviews of the effectiveness of interventions to interrupt pandemic influenza transmission. The fourth and fifth are research papers presenting a novel mathematical model, assessing the preparedness of the Canadian hospital system to accommodate expected surges in patient demand and evaluating intervention strategies to mitigate impact. The sixth is a policy-oriented paper discussing pandemic policy options within the context of public health ethics and risk management principles. Pandemic vaccination, antiviral treatment, voluntary isolation and personal protective measures were identified as the most cost-effective interventions available. Antiviral prophylaxis, community-contact reduction, school closure and quarantine were less effective, and tended to be associated with higher associated economic burdens. The timely implementation of layered intervention strategies appears likely to protect hospital-resource adequacy, though areas of Southwestern Ontario appear to be more vulnerable to surges in patient demand. However, the potential for high health and economic burdens, coupled with the uncertain severity of future pandemics, necessitates a flexibility in preparedness and response plans.

Le besoin actuel de comprendre les conséquences précises que l'administration d'une molécule va avoir sur un organisme et les organes qui le composent, est un enjeu majeur pour la recherche pharmaceutique et l'étude de la toxicité des xénobiotiques. Il n’est pas difficile de se rendre compte, quand les effets sont observables, qu’il existe un lien entre la dose administrée d’un xénobiotique et ses effets. La difficulté de les prédire, qu'ils soient bénéfiques ou délétères, réside principalement dans le fait qu'un nombre très important de mécanismes complexes sont mis en jeu, dès l'entrée de cette molécule dans l'organisme et jusqu'à son excrétion. Afin de comprendre et quantifier ce lien, et pour pouvoir faire des prédictions, il est nécessaire de connaître les principaux mécanismes biologiques impliqués et de proposer des modèles mathématiques les décrivant. Le travail présenté dans cette thèse montre que l'utilisation de la biologie systémique n'est pas facile et manque encore de maturité. Au-delà de la diversité des connaissances auxquelles elle fait appel, on se rend compte que la quantité de données et de paramètres à gérer est considérable. Pour un modèle ne prenant en compte qu'une seule voie de signalisation, comme celui présenté ici, plusieurs mois ont été nécessaires pour sa calibration. Cette durée est en grande partie imputable au temps de calculs nécessaire aux estimations des paramètres, et à celui nécessaire à la récolte et aux traitements des données très diverses (données PK, omiques, physiologiques, cellulaires, etc). Il est très important que le protocole de collecte des données soit défini en commun par l'ensemble des équipes les utilisant par la suite. / The current need to understand the consequences of the administration of a specific molecule to a given organism is a major issue for pharmaceutical and toxicological research. It is not difficult to realize, when the effects are observable, that there is a relationship between the dose of a xenobiotic and its effects. The difficulty in predicting such effects comes mainly from the fact that a large number of complex mechanisms are involved, from the entry of the molecule in the body to its excretion. To understand and quantify this relationship it is necessary to know the main biological mechanisms involved and to propose corresponding mathematical models. Pharmacokinetics, pharmacodynamics and systems biology are the scientific fields most appropriate to meet this need. The first examines the fate of xenobiotics in the body and the second, the evolution of their effects. Systems biology is a relatively new approach which combines different levels of information (experimental data, chemical and biological knowledge, assumptions, etc) with mathematical models to understand how complex biological systems work. Our work shows that the use of systems biology is not easy and still lacks maturity. The amount of data and parameters to manage is typically huge. For a model taking into account only one signaling pathway, several months were needed for its calibration. This length of time is largely due to computation time required for parameter estimates, but also to the time required for harvesting and processing of diverse data (PK data omics, physiological, cellular, etc.) It is important that data collection protocol be defined in common by all the teams involved.

The hair cycle represents a dynamic process during which a complex mini- organ, the hair follicle, rhythmically regresses and regenerates. The control mechanism that governs the hair cycle ('hair cycle clock') is thought to be an autonomous oscillator system, however, its exact nature is not known. This thesis aims to understand the human hair cycle as a systems biology problem using theoretical and experimental techniques in three distinct study approaches. Using mathematical modelling, a simple two-compartment model of the human hair cycle was developed. The model concentrates on the growth control of matrix keratinocytes, a key cell population responsible for hair growth, and bi-directional communication between these cells and the inductive fibroblasts of the dermal papilla. A bistable switch and feedback inhibition produces key characteristics of human hair cycle dynamics. This study represents the first mathematically formulated theory of the 'hair cycle clock'.A second chronobiological approach was adopted to explore the molecular control of the human hair follicle by a peripheral clock mechanism. The hypothesis was tested that selected circadian clock genes regulate the human hair cycle, namely the clinically crucial follicle transformation from organ growth (anagen) to organ regression (catagen). This revealed that intra- follicular expression of core clock and clock-controlled genes display a circadian rhythm and is hair cycle-dependent. Knock-down of Period1 and Clock promotes anagen maintenance, hair matrix keratinocyte proliferation and stimulates hair follicle pigmentation. This provides the first evidence that peripheral Period1 and Clock gene activity is a component of the human 'hair cycle clock' mechanism. Lastly, an unbiased gene expression profiling approach was adopted to establish important genes and signalling pathways that regulate the human hair cycle. This revealed that similar genes and pathways previously shown to control the murine hair cycle in vivo, such as Sgk3, Msx2 and the BMP pathway, are also differentially regulated during the anagen-catagen transformation of human hair follicles. In summary, by using a three-pronged systems biology approach, the thesis has shed new light on the control of human hair follicle cycling and has generated clinically relevant information: a) The hair cycle model may predict how hair cycle modulatory agents alter human hair growth. b) Period1 and Clock are new therapeutic targets for human hair growth manipulation. c) Gene expression profiling points to additional key players in human hair cycle control with potential for future therapeutic targets.

An approach to investigating cancer that has recently seen resurgence of interest is the “Warburg effect”. Otto Warburg originally described the altered metabolism of cancer cells and identified that they exhibit an increase in glucose uptake and lactate production. This up-regulation of glycolytic flux and glucose transport is now associated with 90% of cancers. In order to improve the overall understanding of the “Warburg effect” two forms of systems biology have been implemented - comparative in vitro analysis of kinetic activities and dynamic modelling. In this analysis, human breast cancer cell lines MCF-7, MDA-MB-231 and T47D and a non transformed breast cell line MCF-10A were used to identify key similarities and differences in kinetic activities across the glycolytic pathway. Additionally, activities of key glycolytic enzymes hexokinase, pyruvate kinase and lactate dehydrogenase were compared under hypoxic conditions to further understand regulation of cancer cells. The most prominent feature that arose from comparing the kinetic activities of the three malignant and one non-malignant cell line is that each cell line has its own specific set of activities for glycolysis. This indicates that there are differences in regulation across the glycolytic pathway for each of these cell lines. This is of specific interest in the search for therapeutic targets. Further, we determined that despite the prominence of oncogenic HIF signalling activities of hexokinase, pyruvate kinase and lactate dehydrogenase were further modulated by growth under hypoxic conditions. Despite the lack of obvious distinct kinetic differences between the non-cancerous and cancerous cells lines some discernible differences are apparent when modelled in silico.

In this thesis we investigate the mechanisms underlying the generation of abnormal EEG rhythms in epilepsy, which is a crucial step towards better treatment of this disorder in the future. To this end, macroscopic scale mathematical models of the interactions between neuronal populations are examined. In particular, the role of interactions between neural masses that are spatially distributed in cortical networks are explored. In addition, two other important aspects of the modelling process are addressed, namely the conversion of macroscopic model variables into EEG output and the comparison of multivariate, spatio-temporal data. For the latter, we adopt a vectorisation of the correlation matrix of windowed data and subsequent comparison of data by vector distance measures. Our modelling studies indicate that excitatory connectivity between neural masses facilitates self-organised dynamics. In particular, we report for the first time the production of complex rhythmic transients and the generation of intermittent periods of 'abnormal' rhythmic activity in two different models of epileptogenic tissue. These models therefore provide novel accounts of the spontaneous, intermittent transition between normal and pathological rhythms in primarily generalised epilepsies and the evocation of complex, self-terminating, spatio-temporal dynamics by brief stimulation in focal epilepsies. Two key properties of these models are excitability at the macroscopic level and the presence of spatial heterogeneities. The identification of neural mass excitability as an important processes in spatially extended brain networks is a step towards uncovering the multi-scale nature of the pathological mechanisms of epilepsy. A direct consequence of this work is therefore that novel experimental investigations are proposed, which in itself is a validation of our modelling approach. In addition, new considerations regarding the nature of dynamical systems as applied to problems of transitions between rhythmic states are proposed and will prompt future investigations of complex transients in spatio-temporal excitable systems.

Cette thèse s'intéresse à l'application d'un modèle mathématique unidimensionnel de formation et de croissance de biofilms multi-espèces. Le modèle se compose d'un système d'équations non linéaires aux dérivées partielles hyperboliques, décrivant la croissance d'espèces microbiennes dans le biofilm, et un système d'équations semi-linéaires aux dérivées partielles paraboliques, qui régit la diffusion de substrat de la phase aqueuse vers la matrice du biofilm. L'ensemble conduit à un problème de valeur limite libre, essentiellement hyperbolique. Dans une première étude, l'analyse et la simulation de la phase initiale de croissance du biofilm ont été examinées. Le problème mathématique résultant a été discuté en utilisant la méthode des caractéristiques et le théorème du point fixe a été utilisé pour déterminer l'existence et l'unicité des solutions mathématiques. Un deuxième aspect de la thèse porte sur l'analyse et la prévision de la dynamique des populations microbienne dans plusieurs types biofilms pour le traitement des eaux usées. Le modèle a été appliqué pour simuler la compétition bactérienne et évaluer l'influence de la diffusion du substrat sur la stratification microbienne des biofilms multi-espèces, en incluant les bactéries nitrifiantes, Anammox et bactéries sulfato-réductrices. Dans les deux cas, la méthode des caractéristiques a été utilisée à des fins numériques et l'équation de conservation de masse joue un rôle crucial pour vérifier l'exactitude des simulations. Les résultats des simulations montrent que le modèle est en mesure d'évaluer correctement les effets des conditions limites qui s'exercent sur la concurrence bactérienne. Enfin, ce modèle a été étendu pour inclure le phénomène de colonisation microbienne. Le nouveau modèle est capable de prendre en compte l'invasion de nouvelles espèces en se basant sur un ensemble d'équations non linéaires aux dérivées partielles hyperboliques pour ce qui concerne le processus de croissance. De plus, le processus d'invasion biologique d'espèces nouvelles dans le biofilm a été modélisé par un système d'équations non linéaires aux dérivées partielles paraboliques. Ce modèle d'invasion a été appliqué avec succès pour simuler l'invasion des bactéries hétérotrophes dans les biofilms autotrophes / This dissertation relates to the applications of a one-dimensional mathematical model for multispecies biofilm formation and growth. The model consists of a system of nonlinear hyperbolic partial differential equations, describing the growth of microbial species in biofilms, and a system of semilinear parabolic partial differential equations, which governs substrate diffusion from the surrounding aqueous phase into the biofilm. Overall, this leads to a free boundary value problem, essentially hyperbolic. In a first study, the analysis and simulations of the initial phase of biofilm growth have been addressed. The resulting mathematical problem has been discussed by using the method of characteristics and the fixed-point theorem has been used to obtain existence, uniqueness and properties of solutions. A second aspect of the thesis deals with the analysis and prediction of population dynamics in multispecies biofilms for wastewater treatment. The model has been applied to simulate the bacterial competition and to evaluate the influence of substrate diffusion on microbial stratification for a nitrifying multispecies biofilm including Anammox bacteria and a sulfate-reducing biofilm. In both cases, the method of characteristics has been used for numerical purposes and the mass conservation equation plays a crucial role in checking the accuracy of simulations. The simulation results reveal that the model is able to evaluate properly the effects that boundary conditions exert on bacterial competition. Finally, the biofilm model has been extended to include the colonization phenomenon. The new model is able to take into account the invasion of new species diffusing from bulk liquid to biofilm, still based on a set of nonlinear hyperbolic partial differential equations for what concerns growth process. Indeed, the biological invasion process of new species into the biofilm has been modeled by a system of nonlinear parabolic partial differential equations. The invasion model has been successfully applied to simulate the invasion of heterotrophic bacteria in a constituted autotrophic biofilm and viceversa