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Infinite-activity completely random measures (CRMs) have become important building blocks of complex Bayesian nonparametric models. They have been successfully used in various applications such as clustering, density estimation, latent feature models, survival analysis or network science. Popular infinite-activity CRMs include the (generalised) gamma process and the (stable) beta process. However, except in some specific cases, exact simulation or scalable inference with these models is challenging and finite-dimensional approximations are often considered. In this work, we propose a general and unified framework to derive both series representations and finite-dimensional approximations of CRMs. Our framework can be seen as a generalisation of constructions based on size-biased sampling of Poisson point process (Probab. Theory Related Fields 92 (1992) 21–39). It includes as special cases several known series representations and finite approximations as well as novel ones. In particular, we show that one can get novel series representations for the generalised gamma process and the stable beta process. We show how these constructions can be used to derive novel algorithms for posterior inference, including a generalisation of the slice sampler for normalised CRMs mixture models introduced by (J. Comput. Graph. Statist. 20 (2011) 241–259). We also provide some analysis of the truncation error.

The SARS-CoV-2 Gamma variant spread rapidly across Brazil, causing substantial infection and death waves. We use individual-level patient records following hospitalisation with suspected or confirmed COVID-19 to document the extensive shocks in hospital fatality rates that followed Gamma's spread across 14 state capitals, and in which more than half of hospitalised patients died over sustained time periods. We show that extensive fluctuations in COVID-19 in-hospital fatality rates also existed prior to Gamma's detection, and were largely transient after Gamma's detection, subsiding with hospital demand. Using a Bayesian fatality rate model, we find that the geographic and temporal fluctuations in Brazil's COVID-19 in-hospital fatality rates are primarily associated with geographic inequities and shortages in healthcare capacity. We project that approximately half of Brazil's COVID-19 deaths in hospitals could have been avoided without pre-pandemic geographic inequities and without pandemic healthcare pressure. Our results suggest that investments in healthcare resources, healthcare optimization, and pandemic preparedness are critical to minimize population wide mortality and morbidity caused by highly transmissible and deadly pathogens such as SARS-CoV-2, especially in low- and middle-income countries. NOTE: The following manuscript has appeared as 'Report 46 - Factors driving extensive spatial and temporal fluctuations in COVID-19 fatality rates in Brazilian hospitals' at https://spiral.imperial.ac.uk:8443/handle/10044/1/91875 . ONE SENTENCE SUMMARY: COVID-19 in-hospital fatality rates fluctuate dramatically in Brazil, and these fluctuations are primarily associated with geographic inequities and shortages in healthcare capacity.

Bayesian Statistics provide us with a powerful approach to model real-world phenomena and quantify the uncertainty therein by treating unknown factors as random. The Bayesian paradigm prescribes quantifying the prior knowledge about some state of the world, and after having obtained new information updating that knowledge in order to update the prior beliefs and propose posterior knowledge. A central approach to Bayesian Statistics is modelling, i.e. to represent a data generating process using statistical models equipped with some parameters which are to be estimated via Bayesian inference. Bayesian Nonparametric modelling comes with great flexibility as it provides infnitely many parameters. Bayesian Nonparametric models are data adaptive because when given a finite set of data, the size of the finite set of parameters to be used, adapts to the complexity of the observed data. Bayesian Nonparametric models have been used in many applications of machine learning such as density estimation, clustering, latent feature models, survival analysis and function approximation. One of the contributions of this thesis to Bayesian Nonparametrics is the formulation of priors on random graphs to model networks. Our motivation for networks sources from the fact that networks are found in numerous areas of modern society reflecting the patterns of connection across a wide range of physical and social phenomena. Our research focus is on modelling two types of networks; undirected static networks and directed networks with temporal connections. In both cases our objective is to overcome the limitations of the current literature and propose network methodology that captures real-world network properties. Precisely, the property of graph sparsity is particularly important. However, traditional Bayesian network models that assume the desirable property of exchangeability are necessarily not sparse. The first random graph to allow sparsity as well as exchangeability was recently introduced by Caron and Fox [2017] whose framework we follow. Our objective is to propose models with communities in their latent structure, and we therefore propose a generalisation of existing undirected network models with community structure to the sparse regime. Regarding temporal interaction network data, our goal is to capture reciprocation in interactions and therefore define a family of network models that generalises existing classes of reciprocating models to the sparse graph case. The key building blocks of our models as well as for various other popular Bayesian Nonparametric constructions, are infnite-activity completely random measures. The use of these random measures offers flexibility, but is also accompanied by computational complexity. The third objective of this thesis is a methodological and theoretical contribution to Bayesian Nonparametrics consisting of a novel framework to form approximations of infinite-activity completely random measures. By providing finite approximations to these infinite data structures we offer practicality and efficiency. Our constructions can be used to develop efficient posterior inference algorithms and shed light on current Bayesian Nonparametric issues of computational complexity.

The SARS‐CoV‐2 Gamma variant spread rapidly across Brazil, causing substantial infection and death wa ves. We use individual‐level patient records following hospitalisation with suspected or confirmed COVID‐19 to document the extensive shocks in hospital fatality rates that followed Gamma’s spread across 14 state capitals, and in which more than half of hospitalised patients died over sustained time pe riods. We show that extensive fluctuations in COVID‐19 in‐hospital fatality rates also existed prior to Gamma’s detection, and were largely transient after Gamma’s detection, subsiding with hospital d emand. Using a Bayesian fatality rate model, we find that the geo‐graphic and temporal fluctuations in Brazil’s COVID‐19 in‐hospital fatality rates are primarily associated with geo‐graphic inequities and shortages in healthcare c apacity. We project that approximately half of Brazil’s COVID‐19 deaths in hospitals could have been avoided without pre‐pandemic geographic inequities and without pandemic healthcare pressure. Our results suggest that investments in healthcare resources, healthcare optimization, and pandemic preparedness are critical to minimize population wide mortality and morbidity caused by highly trans‐missible and deadly pathogens such as SARS‐CoV‐2, especially in low‐ and middle‐income countries.

Since its emergence in Autumn 2020, the SARS-CoV-2 Variant of Concern (VOC) B.1.1.7 rapidly became the dominant lineage across much of Europe. Simultaneously, several other VOCs were identified globally. Unlike B.1.1.7, some of these VOCs possess mutations thought to confer partial immune escape. Understanding when, whether, and how these additional VOCs pose a threat in settings where B.1.1.7 is currently dominant is vital. This is particularly true for England, which has high coverage from vaccines that are likely more protective against B.1.1.7 than some other VOCs. We examine trends in B.1.1.7’s prevalence in London and other English regions using passive-case detection PCR data, cross-sectional community infection surveys, genomic surveillance, and wastewater monitoring. Our results suggest shifts in the composition of SARS-CoV-2 lineages driving transmission in England between March and April 2021. Local transmission of non-B.1.1.7 VOCs may be increasing; this warrants urgent further investigation.

Updating observations of a signal due to the delays in the measurement process is a common problem in signal processing, with prominent examples in a wide range of fields. An important example of this problem is the nowcasting of COVID-19 mortality: given a stream of reported counts of daily deaths, can we correct for the delays in reporting to paint an accurate picture of the present, with uncertainty? Without this correction, raw data will often mislead by suggesting an improving situation. We present a flexible approach using a latent Gaussian process that is capable of describing the changing auto-correlation structure present in the reporting time-delay surface. This approach also yields robust estimates of uncertainty for the estimated nowcasted numbers of deaths. We test assumptions in model specification such as the choice of kernel or hyper priors, and evaluate model performance on a challenging real dataset from Brazil. Our experiments show that Gaussian process nowcasting performs favourably against both comparable methods, and against a small sample of expert human predictions. Our approach has substantial practical utility in disease modelling -- by applying our approach to COVID-19 mortality data from Brazil, where reporting delays are large, we can make informative predictions on important epidemiological quantities such as the current effective reproduction number., Comment: 26 pages, 31 figures

We propose a novel class of network models for temporal dyadic interaction data. Our goal is to capture a number of important features often observed in social interactions: sparsity, degree heterogeneity, community structure and reciprocity. We propose a family of models based on self-exciting Hawkes point processes in which events depend on the history of the process. The key component is the conditional intensity function of the Hawkes Process, which captures the fact that interactions may arise as a response to past interactions (reciprocity), or due to shared interests between individuals (community structure). In order to capture the sparsity and degree heterogeneity, the base (non time dependent) part of the intensity function builds on compound random measures following Todeschini et al. (2016). We conduct experiments on a variety of real-world temporal interaction data and show that the proposed model outperforms many competing approaches for link prediction, and leads to interpretable parameters.

We solve analytically the cessation flows of a Newtonian fluid in circular and plane Couette geometries assuming that wall slip occurs provided that the wall shear stress exceeds a critical threshold, the slip yield stress. In steady-state, slip occurs only beyond a critical value of the angular velocity of the rotating inner cylinder in circular Couette flow or of the speed of the moving upper plate in plane Couette flow. Hence, in cessation, the classical no-slip solution holds if the corresponding wall speed is below the critical value. Otherwise, slip occurs only initially along both walls. Beyond a first critical time, slip along the fixed wall ceases, and beyond a second critical time slip ceases also along the initially moving wall. Beyond this second critical time no slip is observed and the decay of the velocity is faster. The velocity decays exponentially in all regimes and the decay is reduced with slip. The effects of slip and the slip yield stress are discussed. © 2016, Springer Science+Business Media Dordrecht. 52 9 2081 2099

The SARS-CoV-2 Gamma variant spread rapidly across Brazil, causing substantial infection and death waves. We use individual-level patient records following hospitalisation with suspected or confirmed COVID-19 to document the extensive shocks in hospital fatality rates that followed Gamma's spread across 14 state capitals, and in which more than half of hospitalised patients died over sustained time periods. We show that extensive fluctuations in COVID-19 in-hospital fatality rates also existed prior to Gamma's detection, and were largely transient after Gamma's detection, subsiding with hospital demand. Using a Bayesian fatality rate model, we find that the geographic and temporal fluctuations in Brazil's COVID-19 in-hospital fatality rates are primarily associated with geographic inequities and shortages in healthcare capacity. We project that approximately half of Brazil's COVID-19 deaths in hospitals could have been avoided without pre-pandemic geographic inequities and without pandemic healthcare pressure. Our results suggest that investments in healthcare resources, healthcare optimization, and pandemic preparedness are critical to minimize population wide mortality and morbidity caused by highly transmissible and deadly pathogens such as SARS-CoV-2, especially in low- and middle-income countries., Note: The following manuscript has appeared as 'Report 46 - Factors driving extensive spatial and temporal fluctuations in COVID-19 fatality rates in Brazilian hospitals' at https://spiral.imperial.ac.uk:8443/handle/10044/1/91875 ., One Sentence Summary: COVID-19 in-hospital fatality rates fluctuate dramatically in Brazil, and these fluctuations are primarily associated with geographic inequities and shortages in healthcare capacity.

Cases of SARS-CoV-2 infection in Manaus, Brazil, resurged in late 2020, despite high levels of previous infection there. Through genome sequencing of viruses sampled in Manaus between November 2020 and January 2021, we identified the emergence and circulation of a novel SARS-CoV-2 variant of concern, lineage P.1, that acquired 17 mutations, including a trio in the spike protein (K417T, E484K and N501Y) associated with increased binding to the human ACE2 receptor. Molecular clock analysis shows that P.1 emergence occurred around early November 2020 and was preceded by a period of faster molecular evolution. Using a two-category dynamical model that integrates genomic and mortality data, we estimate that P.1 may be 1.4-2.2 times more transmissible and 25-61% more likely to evade protective immunity elicited by previous infection with non-P.1 lineages. Enhanced global genomic surveillance of variants of concern, which may exhibit increased transmissibility and/or immune evasion, is critical to accelerate pandemic responsiveness., Competing Interests: Competing interests: Authors declare that they have no competing interests.