School of Mathematics and Statistics - Theses
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ItemQuantitative Epidemiology: A Bayesian Perspective( 2019)Influenza inflicts a substantial burden on society but accurate and timely forecasts of seasonal epidemics can help mitigate this burden by informing interventions to reduce transmission. Recently, both statistical (correlative) and mechanistic (causal) models have been used to forecast epidemics. However, since mechanistic models are based on the causal process underlying the epidemic they are poised to be more useful in the design of intervention strategies. This study investigate approaches to improve epidemic forecasting using mechanistic models. In particular, it reports on efforts to improve a forecasting system targeting seasonal influenza epidemics in major cities across Australia. To improve the forecasting system we first needed a way to benchmark its performance. We investigate model selection in the context of forecasting, deriving a novel method which extends the notion of Bayes factors to a predictive setting. Applying this methodology we found that accounting for seasonal variation in absolute humidity improves forecasts of seasonal influenza in Melbourne, Australia. This result holds even when accounting for the uncertainty in predicting seasonal variation in absolute humidity. Our initial attempts to forecast influenza transmission with mechanistic models were hampered by high levels of uncertainty in forecasts produced early in the season. While substantial uncertainty seems inextricable from long-term prediction, it seemed plausible that historical data could assist in reducing this uncertainty. We define a class of prior distributions which simplify the process of incorporating existing knowledge into an analysis, and in doing so offer a refined interpretation of the prior distribution. As an example we used historical time series of influenza epidemics to reduce initial uncertainty in forecasts for Sydney, Australia. We explore potential pitfalls that may be encountered when using this class of prior distribution. Deviating from the theme of forecasting, we consider the use of branching processes to model early transmission in an epidemic. An inhomogeneous branching process is derived which allows the study of transmission dynamics early in an epidemic. A generation dependent offspring distribution allows for the branching process to have sub-exponential growth on average. The multi-scale nature of a branching process allows us to utilise both time series of incidence and infection networks. This methodology is applied to data collected during the 2014–2016 Ebola epidemic in West-Africa leading to the inference that transmission grew sub-exponentially in Guinea, Liberia and Sierra Leone. Throughout this thesis, we demonstrate the utility of mechanistic models in epidemiology and how a Bayesian approach to statistical inference is complementary to this.
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ItemPromoting change in teacher practice through supported differentiation of instruction in mathematics( 2019)Differentiated instruction has been shown to be effective in improving student learning outcomes; however, the resulting work load can be difficult for teachers to manage. A teaching package known as the NRP (Number Resource Package) was created to support teachers to differentiate their instruction, and used effectively in two classrooms. The package allows teachers to identify their students’ current understanding using a diagnostic test and a Guttman Chart, and then provides appropriate material for the area in which students need further consolidation. It assists teachers to identify, and provide instruction for, several different knowledge levels within the one classroom. Use of the NRP in the two experimental classes was compared with five classes that did not use the NRP and continued to follow their school’s mathematics curriculum. This study involved a quasi-experimental approach, using qualitative and quantitative data. Involved were an experimental group (two teachers) and a control group (five teachers) and a total of 147 year 7 students. The research took place in a large school in western Melbourne, Australia. The qualitative data consisted of three surveys and provided information on the effectiveness of the components in the NRP. The quantitative data consisted of a pre- and a post-test completed by students in both the experimental and control groups. These tests were completed at the beginning and the end of a nine-week teaching cycle and the learning gains were determined for each student (i.e. the difference between the pre- and post-test). There was a statistically significant difference between the experimental group and the control group when these learning gains were analysed. The results demonstrated that students in the experimental group who were taught using the NRP showed greater improvement on the post-test when compared to students in the control group. It was noted that those students who performed ‘below’ the expected level and those students who performed ‘above’ the expected level showed the most improvement in the experimental group, when compared with the control group.
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ItemCores of vertex-transitive graphs( 2013)The core of a graph $\Gamma$ is the smallest graph $\Gamma^\ast$ for which there exist graph homomorphisms $\Gamma\rightarrow\Gamma^\ast$ and $\Gamma^\ast\rightarrow\Gamma$. Thus cores are fundamental to our understanding of general graph homomorphisms. It is known that for a vertex-transitive graph $\Gamma$, $\Gamma^\ast$ is vertex-transitive, and that $\left|V(\Gamma^\ast)\right|$ divides $\left|V(\Gamma)\right|$. The purpose of this thesis is to determine the cores of various families of vertex-transitive and symmetric graphs. We focus primarily on finding the cores of imprimitive symmetric graphs of order $pq$, where $p
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ItemMathematical modelling of layering in nature( 2012)This thesis examines convection-based models to simulate the formation of structured populations. Specifically, the thesis focuses on systems that exhibit characteristic layered patterns. Such structures are important throughout the natural world and range from the molecular layers formed on substrates to the shifting rock layers within the earth's crust. Our primary motivation here, however, stems from the formation of cortical layers within the mouse brain. The creation of layers within the cortex of the mammalian brain has been of interest to researchers in developmental biology. Experiments on mice have found a counter-intuitive layering sequence. The layers are formed in an inside-out arrangement, with the deepest layers formed first and the outermost last. It is believed that the glycoprotein reelin is linked to the mechanisms behind layering. In the reeler mutant, where this glycoprotein is absent, it is observed that the layer arrangement is reversed. This has led to a number of mechanisms being proposed for how reelin may affect the neurons that form the cortical layers. No conclusive theory of this phenomenon has been established so far. Mathematical modelling provides an elegant method to evaluate the feasibility of different layering mechanisms. In this context, the thesis provides models for the purpose of understanding layer formation in nature. The replication of both age-uniform and inverted structures is a key goal. We examine the effect that modelling perspective, population models versus the individual agent models, has on our results. Results from each perspective are utilised to strengthen and augment the work. Furthermore, the major issue of volume constraints is also addressed. Modelling is used to investigate the connections between the motion of invading agents and their conversion to settled states. Base models will be initially formulated from a quite general foundation. Published experimental results are utilised as inspiration for later extensions to these base models. This research culminates with a proposed theoretical framework for cortical layer formation.
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ItemDuality groups( 2011)This thesis focuses mainly on the subclass of complex reflection groups called duality groups. We have worked towards a new classification of the duality groups. As well as this, an analysis of affine diagrams has been done to set the framework for the possibility of defining a new class of groups, affine duality groups defined analogously to affine Weyl groups. We have studied one family of affine duality groups.