Aspects of positive definiteness and gaussian processes on planet earth

This thesis studies characterisations and properties of spatial and spatio-temporal Gaussian processes defined over the sphere (or in the spatio-temporal case the product of the sphere and the real line). Such processes are of importance in global weather and climate science, where the geometry is necessarily spherical, but, especially in the dynamic setting, they are less well-studied than their Euclidean counterparts. Beginning with Brownian motion, we first look at characterising Gaussian randomness on the sphere and sphere-cross-line, and how it compares with the Euclidean setting -- we show that the characterisation theorems of Gaussian processes on spaces of types spanning the real line, the sphere and sphere-cross-line can be phrased as consequences of a powerful general theorem of harmonic analysis. We go on to find the answer to a recent question posed about dimension-hopping operators for positive-definite (i.e. covariance) functions on the sphere-cross-line, and consider how we could go about constructing dimension-hopping operators with the semi-group property on the sphere. Later, we address the theory of the path properties of these processes, extending a finite-dimensional result the the infinite-dimensional case and showing that a remarkably elegant approach for processes on Euclidean space carries over to our setting. We finish by finding the analogue of the powerful Ciesielski isomorphism for continuous functions on the two-sphere.Open Acces

Brownian Manifolds, Negative Type and Geo-temporal Covariances

We survey Brownian manifolds -- manifolds that can parametrise Brownian motion -- and those that cannot. We consider covariances of space-time processes, particularly those when space is the sphere -- geo-temporal processes. There are connections with functions of negative type.Comment: 12 pages; corrected typos and acknowledgemen

Indirect effects of the COVID-19 pandemic on malaria intervention coverage, morbidity, and mortality in Africa: a geospatial modelling analysis.

BACKGROUND: Substantial progress has been made in reducing the burden of malaria in Africa since 2000, but those gains could be jeopardised if the COVID-19 pandemic affects the availability of key malaria control interventions. The aim of this study was to evaluate plausible effects on malaria incidence and mortality under different levels of disruption to malaria control. METHODS: Using an established set of spatiotemporal Bayesian geostatistical models, we generated geospatial estimates across malaria-endemic African countries of the clinical case incidence and mortality of malaria, incorporating an updated database of parasite rate surveys, insecticide-treated net (ITN) coverage, and effective treatment rates. We established a baseline estimate for the anticipated malaria burden in Africa in the absence of COVID-19-related disruptions, and repeated the analysis for nine hypothetical scenarios in which effective treatment with an antimalarial drug and distribution of ITNs (both through routine channels and mass campaigns) were reduced to varying extents. FINDINGS: We estimated 215·2 (95% uncertainty interval 143·7-311·6) million cases and 386·4 (307·8-497·8) thousand deaths across malaria-endemic African countries in 2020 in our baseline scenario of undisrupted intervention coverage. With greater reductions in access to effective antimalarial drug treatment, our model predicted increasing numbers of cases and deaths: 224·1 (148·7-326·8) million cases and 487·9 (385·3-634·6) thousand deaths with a 25% reduction in antimalarial drug coverage; 233·1 (153·7-342·5) million cases and 597·4 (468·0-784·4) thousand deaths with a 50% reduction; and 242·3 (158·7-358·8) million cases and 715·2 (556·4-947·9) thousand deaths with a 75% reduction. Halting planned 2020 ITN mass distribution campaigns and reducing routine ITN distributions by 25%-75% also increased malaria burden to a total of 230·5 (151·6-343·3) million cases and 411·7 (322·8-545·5) thousand deaths with a 25% reduction; 232·8 (152·3-345·9) million cases and 415·5 (324·3-549·4) thousand deaths with a 50% reduction; and 234·0 (152·9-348·4) million cases and 417·6 (325·5-553·1) thousand deaths with a 75% reduction. When ITN coverage and antimalarial drug coverage were synchronously reduced, malaria burden increased to 240·5 (156·5-358·2) million cases and 520·9 (404·1-691·9) thousand deaths with a 25% reduction; 251·0 (162·2-377·0) million cases and 640·2 (492·0-856·7) thousand deaths with a 50% reduction; and 261·6 (167·7-396·8) million cases and 768·6 (586·1-1038·7) thousand deaths with a 75% reduction. INTERPRETATION: Under pessimistic scenarios, COVID-19-related disruption to malaria control in Africa could almost double malaria mortality in 2020, and potentially lead to even greater increases in subsequent years. To avoid a reversal of two decades of progress against malaria, averting this public health disaster must remain an integrated priority alongside the response to COVID-19. FUNDING: Bill and Melinda Gates Foundation; Channel 7 Telethon Trust, Western Australia

Spatiotemporal mapping of malaria prevalence in Madagascar using routine surveillance and health survey data.

Malaria transmission in Madagascar is highly heterogeneous, exhibiting spatial, seasonal and long-term trends. Previous efforts to map malaria risk in Madagascar used prevalence data from Malaria Indicator Surveys. These cross-sectional surveys, conducted during the high transmission season most recently in 2013 and 2016, provide nationally representative prevalence data but cover relatively short time frames. Conversely, monthly case data are collected at health facilities but suffer from biases, including incomplete reporting and low rates of treatment seeking. We combined survey and case data to make monthly maps of prevalence between 2013 and 2016. Health facility catchment populations were estimated to produce incidence rates from the case data. Smoothed incidence surfaces, environmental and socioeconomic covariates, and survey data informed a Bayesian prevalence model, in which a flexible incidence-to-prevalence relationship was learned. Modelled spatial trends were consistent over time, with highest prevalence in the coastal regions and low prevalence in the highlands and desert south. Prevalence was lowest in 2014 and peaked in 2015 and seasonality was widely observed, including in some lower transmission regions. These trends highlight the utility of monthly prevalence estimates over the four year period. By combining survey and case data using this two-step modelling approach, we were able to take advantage of the relative strengths of each metric while accounting for potential bias in the case data. Similar modelling approaches combining large datasets of different malaria metrics may be applicable across sub-Saharan Africa