Masters Degrees (Mathematical Sciences)
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- ItemPopulation-level considerations for the treatment of lethal diseases in resource-limited settings(Stellenbosch : Stellenbosch University, 2023-03) Sereo, Tumelo Donald; Pulliam, Juliet RC; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.ENGLISH ABSTRACT: Diseases such as Severe Acute Respiratory Syndrome Coronavirus (SARS-CoV) and Ebola Virus disease (EVD), continue to challenge health systems worldwide. At the onset of outbreaks of unknown pathogens, there is often no cure, treatment or vaccine available to limit their impact. As the outbreak unfolds, randomised controlled trials are conducted, usually in patients with severe disease, to investigate candidate treatments. Often, several treatments end up being effective, raising the question of which one is most optimal to deploy. While clinical trials focus on individual-level outcomes, population-level outcomes are often more important for public health decision-making. This study answers two questions: First, when can a hypothetical treatment that increases hospital stay duration and probability of survival be used to improve the population-level mortality outcomes under constrained hospital capacity? Second, when is it preferable to invest in treatments versus beds, in a limited resource setting? We developed a transmission dynamic model, parameterised separately for SARS-CoV2 and Ebola, to address the questions posed. For the first question, we ran the model for baseline (no treatment) and treatment scenarios defined by the probability of surviving and duration of hospital stay. We used cumulative mortality as the metric to compare the population-level outcomes. The model shows that there is a substantial region of parameter space in which it is beneficial to use hypothetical treatments that increase probability of surviving and hospital stay duration. For the second question, we performed a cost-minimization analysis to examine when it is preferable to invest in treatments versus beds, in a limited resource setting. The model identified the number of additional beds that would be needed to obtain approximately the same outcomes compared to what is expected with existing treatments. We found that is it preferable to invest in additional beds rather than the existing treatments when the cost per course of treatment is greater than a threshold that depends on the drug under consideration. We estimated that this threshold is around R5 000 for existing SARS-CoV-2 drugs but higher for available Ebola therapies.
- ItemImage Classification with Graph Neural Networks(Stellenbosch : Stellenbosch University, 2022-04) Neocosmos, Kibidi; Bah, Bubacarr; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.ENGLISH SUMMARY: Convolutional neural networks (CNNs) are a prominent and ubiquitous part of machine learning. They have successfully achieved consistent state-of-the-art performance in areas such as computer vision. However, they require large datasets for such achievements. This is in stark contrast to human-level performance that demands less data for the same task. The question naturally arises as to whether it is possible to develop models that require less data without a significant decrease in performance. I n t his thesis, we address the above question from a different perspective by investigating whether a richer data structure could result in more learning from fewer training examples. We explore the idea by constructing images as graphs – a structure that naturally contains more information about an image than the standard tensor representation. We then use graph neural networks (GNNs) to leverage the graph structure and perform image classification. We found that the graph structure did not enable GNNs to perform well given less data. However, during the process of experimentation, we discovered that the graph topology as well as node features significantly influence performance. Furthermore, some of the proposed GNN models were not able to effectively utilize the graph structure.
- ItemAlgebraic points in tame expansions of fields(Stellenbosch : Stellenbosch University, 2021-12) Harrison-Migochi, Andrew; Boxall, Gareth John; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. Division Mathematics.ENGLISH ABSTRACT: We investigate the behaviour of algebraic points in several expansions of the real, complex and p-adic fields. We build off the work of Eleftheriou, Günaydin and Hieronymi in [17] and [18] to prove a Pila-Wilkie result for a p-adic subanalytic structure with a predicate for either a dense elementary substructure or a dense dcl-independent set. In the process we prove a structure theorem for p-minimal structures with a predicate for a dense independent set. We then prove quantifier reduction results for the complex field with a predicate for the singular moduli and the real field with an exponentially transcendental power function and a predicate for the algebraic numbers using a Schanuel property proved by Bays, Kirby and Wilkie [5]. Finally we adapt a theorem by Ax [2] about exponential fields, key to the proof of the Schanuel property for power functions, to power functions.
- ItemA deep learning approach to landmark detection in tsetse fly wing images(Stellenbosch : Stellenbosch University, 2021-12) Geldenhuys, Dylan Shane; Hargrove, John; Hazelbag, Marijn; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. Division Mathematics.ENGLISH ABSTRACT: Single-wing images were captured from 14,354 pairs of field-collected tsetse wings of species Glossina pallidipes and G. m. morsitans, and analysed together with relevant biological recordings. To answer research questions regarding these flies, we need to locate 11 anatomical landmark coordinates (x; y) on each wing. The manual location of landmarks is time-consuming, prone to error, and simply infeasible given the number of images. Automatic landmark detection has been proposed to locate these landmark coordinates. We developed a two-tier method using deep learning architectures to classify images and make accurate landmark predictions. The first tier used a classification convolutional neural network to remove most wings that were missing landmarks. The second tier provided landmark coordinates for the remaining wings. For the second tier, we compared direct coordinate regression using a convolutional neural network and segmentation using a fully convolutional network. For the resulting landmark predictions, we evaluate shape bias using Procrustes analysis. We employ a data-centric approach paying particular attention to consistent labelling and data augmentations in training data to improve model performance. The classification model used for the first tier achieved perfect classification on the test set. The regression and segmentation models achieved a mean pixel distance error of 5.34 (95% CI [3,7]) and 3.43 (95% CI [1.9,4.4]) respectively on 1024 1280 images. Segmentation had a higher computational complexity and some large outliers. Both models showed minimal shape bias. Using this two-tier deep learning approach, we accurately filtered damaged tsetse wings with missing landmarks and provided precise landmark coordinates for the remaining wings. We chose to deploy the regression model on the complete un-annotated data since the regression model had a lower computational cost and more stable predictions than the segmentation model.
- ItemTowards a nuanced view of diagnostic test properties: an application to transfusion transmitted risk estimation(Stellenbosch : Stellenbosch University, 2021-03) Bingham, Jeremy; Welte, Alex; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.ENGLISH ABSTRACT: Laboratory screening (rather than pathogen inactivation) is likely to remain, for the foreseeable future, the primary means of ensuring the safety of blood products from transfusion transmissible viruses such as Hepatitis B, Hepatitis C, and Human Immunodeficiency Virus (HIV). Depending on the tests used, there is generically some ‘residual risk’ of transfusion transmitted infection, as no test can guarantee detection of all potentially infectious material. Previouslydescribed risk estimation approaches 1) mostly treat detectability and infectiousness as categories rather than continuously tunable; 2) disregard sources of variability and their correlation; and 3) are not generalizable to arbitrary detection biomarkers – making it difficult to generate estimates of residual risk without extensive programmatic monitoring. We describe a broad framework for modelling test performance which incorporates hitherto neglected sources of variability in parameters governing the infectiousness and detectability of transfusion-transmissible pathogens. We utilise models based on this framework to demonstrate the relationship between test performance and residual risk for various assumptions about the biomarker/infectiousness relationship, and illustrate how the same framework may be used to inform modelling efforts in related fields - such as infection dating and incidence estimation - which rely on realistic representations of test performance. The key findings from our scenario modelling demonstrate: 1) Diminishing returns on increased screening sensitivity not evident in less flexible models; 2) increasing inter-subject variability in detectability and infectiousness leads to increasing residual risk in our general model, but lower risk estimates than in a previously described and widely used semi-mechanistic model. These effects are stronger when the average delay between infectiousness and detectability is short. Planning blood product screening algorithms in light of simulations using our models can generate robust expectations of residual risk over a wide range of test performance and product risk levels. We outline when simpler models may be relied upon, and when additional nuance must be considered.