On the derivation of the renewal equation from an age-dependent branching process: an epidemic modelling perspective

Renewal processes are a popular approach used in modelling infectious disease outbreaks. In a renewal process, previous infections give rise to future infections. However, while this formulation seems sensible, its application to infectious disease can be difficult to justify from first principles. It has been shown from the seminal work of Bellman and Harris that the renewal equation arises as the expectation of an age-dependent branching process. In this paper we provide a detailed derivation of the original Bellman Harris process. We introduce generalisations, that allow for time-varying reproduction numbers and the accounting of exogenous events, such as importations. We show how inference on the renewal equation is easy to accomplish within a Bayesian hierarchical framework. Using off the shelf MCMC packages, we fit to South Korea COVID-19 case data to estimate reproduction numbers and importations. Our derivation provides the mathematical fundamentals and assumptions underpinning the use of the renewal equation for modelling outbreaks

Estimating the number of infections and the impact of non-pharmaceutical interventions on COVID-19 in European countries: technical description update

Following the emergence of a novel coronavirus (SARS-CoV-2) and its spread outside of China, Europe has experienced large epidemics. In response, many European countries have implemented unprecedented non-pharmaceutical interventions including case isolation, the closure of schools and universities, banning of mass gatherings and/or public events, and most recently, wide-scale social distancing including local and national lockdowns. In this technical update, we extend a semi-mechanistic Bayesian hierarchical model that infers the impact of these interventions and estimates the number of infections over time. Our methods assume that changes in the reproductive number - a measure of transmission - are an immediate response to these interventions being implemented rather than broader gradual changes in behaviour. Our model estimates these changes by calculating backwards from temporal data on observed to estimate the number of infections and rate of transmission that occurred several weeks prior, allowing for a probabilistic time lag between infection and death. In this update we extend our original model [Flaxman, Mishra, Gandy et al 2020, Report #13, Imperial College London] to include (a) population saturation effects, (b) prior uncertainty on the infection fatality ratio, (c) a more balanced prior on intervention effects and (d) partial pooling of the lockdown intervention covariate. We also (e) included another 3 countries (Greece, the Netherlands and Portugal). The model code is available at https://github.com/ImperialCollegeLondon/covid19model/ We are now reporting the results of our updated model online at https://mrc-ide.github.io/covid19estimates/ We estimated parameters jointly for all M=14 countries in a single hierarchical model. Inference is performed in the probabilistic programming language Stan using an adaptive Hamiltonian Monte Carlo (HMC) sampler

Report 13: Estimating the number of infections and the impact of non-pharmaceutical interventions on COVID-19 in 11 European countries

Following the emergence of a novel coronavirus (SARS-CoV-2) and its spread outside of China, Europe is now experiencing large epidemics. In response, many European countries have implemented unprecedented non-pharmaceutical interventions including case isolation, the closure of schools and universities, banning of mass gatherings and/or public events, and most recently, widescale social distancing including local and national lockdowns. In this report, we use a semi-mechanistic Bayesian hierarchical model to attempt to infer the impact of these interventions across 11 European countries. Our methods assume that changes in the reproductive number – a measure of transmission - are an immediate response to these interventions being implemented rather than broader gradual changes in behaviour. Our model estimates these changes by calculating backwards from the deaths observed over time to estimate transmission that occurred several weeks prior, allowing for the time lag between infection and death. One of the key assumptions of the model is that each intervention has the same effect on the reproduction number across countries and over time. This allows us to leverage a greater amount of data across Europe to estimate these effects. It also means that our results are driven strongly by the data from countries with more advanced epidemics, and earlier interventions, such as Italy and Spain. We find that the slowing growth in daily reported deaths in Italy is consistent with a significant impact of interventions implemented several weeks earlier. In Italy, we estimate that the effective reproduction number, Rt, dropped to close to 1 around the time of lockdown (11th March), although with a high level of uncertainty. Overall, we estimate that countries have managed to reduce their reproduction number. Our estimates have wide credible intervals and contain 1 for countries that have implemented all interventions considered in our analysis. This means that the reproduction number may be above or below this value. With current interventions remaining in place to at least the end of March, we estimate that interventions across all 11 countries will have averted 59,000 deaths up to 31 March [95% credible interval 21,000-120,000]. Many more deaths will be averted through ensuring that interventions remain in place until transmission drops to low levels. We estimate that, across all 11 countries between 7 and 43 million individuals have been infected with SARS-CoV-2 up to 28th March, representing between 1.88% and 11.43% of the population. The proportion of the population infected to date – the attack rate - is estimated to be highest in Spain followed by Italy and lowest in Germany and Norway, reflecting the relative stages of the epidemics. Given the lag of 2-3 weeks between when transmission changes occur and when their impact can be observed in trends in mortality, for most of the countries considered here it remains too early to be certain that recent interventions have been effective. If interventions in countries at earlier stages of their epidemic, such as Germany or the UK, are more or less effective than they were in the countries with advanced epidemics, on which our estimates are largely based, or if interventions have improved or worsened over time, then our estimates of the reproduction number and deaths averted would change accordingly. It is therefore critical that the current interventions remain in place and trends in cases and deaths are closely monitored in the coming days and weeks to provide reassurance that transmission of SARS-Cov-2 is slowing

Abolishment of the audit requirement– the choice of auditing for small firms and the impacts on other parties

Uppsatsens syfte är att ta reda på vilka faktorer som påverkar valet att inte ha revisionsgranskning för mindre företag som går under den frivilliga revisionen. Vi vill även undersöka vilka effekter de mindre företagen och andra aktörer som berörts av revisionspliktens avskaffande märkt av. Vi har samlat in data genom en kvalitativ ansats som underlag för intervjuerna med bakgrund av teorin. Teorikapitlet i uppsatsen är sorterad i två delar, först en teoridel med två vetenskapliga teorier och sedan en referensram. Utifrån metoden och teorin har vi gjort kvalitativa intervjuer som redovisas i empirin. Vi har intervjuat Skatteverket, två mindre aktiebolag, revisor, auktoriserad redovisningskonsult och en bank. Vår analys och slutsats tyder på att mindre aktiebolag inte har behov av revision och därför kan inte kostnaden för den motivera

The effect of composition and temperature on the ultraviolet absorption of glass

Thesis (M.S.)--University of Rochester. College of Engineering and Applied Science. Institute of Optics, 1962. This thesis was digitized by the Institute of Optics in 2014 and was determined to have lapsed into the public domain. If you are the author and have questions about the digitization of your work, please contact Kari Brick, Graduate Program Coordinator for the Institute of Optics, at [email protected]. Other contact information for the Institute is available at http://www.optics.rochester.eduA study of the ultraviolet absorption spectra of glasses of 19 different compositions has been carried out. A systematic investigation has been made into the mechanism of ultraviolet absorption in glass and the influence of the various network formers and modifiers on ultraviolet transmission. The results of a series of ultraviolet transmission measurements on the sodium borate glass system are presented. The significance of these measurements in the support of the Warren-Stevels explanation of the boron anomaly is discussed. The results of a study of the effect of temperature on the ultraviolet spectra of a number of glasses are presented. A comparison is made between the results obtained in this study and the data presented by other investigators