Machine learning classification based on k-Nearest Neighbors for PolSAR data.

In this work, we focus on obtaining insights of the performances of some well-known machine learning image classification techniques (k-NN, Support Vector Machine, randomized decision tree and one based on stochastic distances) for PolSAR (Polarimetric Synthetic Aperture Radar) imagery. We test the classifiers methods on a set of actual PolSAR data and provide some conclusions. The aim of this work is to show that suitable adapted standard machine learning methods offer excellent performances vs. computational complexity trade-off for PolSAR image classification. In this work, we evaluate well-known machine learning techniques for PolSAR (Polarimetric Synthetic Aperture Radar) image classification, including K-Nearest Neighbors (KNN), Support Vector Machine (SVM), randomized decision tree, and a method based on the Kullback-Leibler stochastic distance. Our experiments with real PolSAR data show that standard machine learning methods, when adapted appropriately, offer a favourable trade-off between performance and computational complexity. The KNN and SVM perform poorly on these data, likely due to their failure to account for the inherent speckle presence and properties of the studied reliefs. Overall, our findings highlight the potential of the Kullback-Leibler stochastic distance method for PolSAR image classification.

On the Use of Machine Learning Techniques and Non-Invasive Indicators for Classifying and Predicting Cardiac Disorders.

This research aims to enhance the classification and prediction of ischemic heart diseases using machine learning techniques, with a focus on resource efficiency and clinical applicability. Specifically, we introduce novel non-invasive indicators known as Campello de Souza features, which require only a tensiometer and a clock for data collection. These features were evaluated using a comprehensive dataset of heart disease cases from a machine learning data repository. Our findings highlight the ability of machine learning algorithms to not only streamline diagnostic procedures but also reduce diagnostic errors and the dependency on extensive clinical testing. Three key features-mean arterial pressure, pulsatile blood pressure index, and resistance-compliance indicator-were found to significantly improve the accuracy of machine learning algorithms in binary heart disease classification. Logistic regression achieved the highest average accuracy among the examined classifiers when utilizing these features. While such novel indicators contribute substantially to the classification process, they should be integrated into a broader diagnostic framework that includes comprehensive patient evaluations and medical expertise. Therefore, the present study offers valuable insights for leveraging data science techniques in the diagnosis and management of cardiovascular diseases.

Similarity-Based Predictive Models: Sensitivity Analysis and a Biological Application with Multi-Attributes.

Predictive models based on empirical similarity are instrumental in biology and data science, where the premise is to measure the likeness of one observation with others in the same dataset. Biological datasets often encompass data that can be categorized. When using empirical similarity-based predictive models, two strategies for handling categorical covariates exist. The first strategy retains categorical covariates in their original form, applying distance measures and allocating weights to each covariate. In contrast, the second strategy creates binary variables, representing each variable level independently, and computes similarity measures solely through the Euclidean distance. This study performs a sensitivity analysis of these two strategies using computational simulations, and applies the results to a biological context. We use a linear regression model as a reference point, and consider two methods for estimating the model parameters, alongside exponential and fractional inverse similarity functions. The sensitivity is evaluated by determining the coefficient of variation of the parameter estimators across the three models as a measure of relative variability. Our results suggest that the first strategy excels over the second one in effectively dealing with categorical variables, and offers greater parsimony due to the use of fewer parameters.

Distributional regression modeling via generalized additive models for location, scale, and shape: An overview through a data set from learning analytics.

The advent of technological developments is allowing to gather large amounts of data in several research fields. Learning analytics (LA)/educational data mining has access to big observational unstructured data captured from educational settings and relies mostly on unsupervised machine learning (ML) algorithms to make sense of such type of data. Generalized additive models for location, scale, and shape (GAMLSS) are a supervised statistical learning framework that allows modeling all the parameters of the distribution of the response variable with respect to the explanatory variables. This article overviews the power and flexibility of GAMLSS in relation to some ML techniques. Also, GAMLSS' capability to be tailored toward causality via causal regularization is briefly commented. This overview is illustrated via a data set from the field of LA. This article is categorized under:Application Areas > Education and LearningAlgorithmic Development > StatisticsTechnologies > Machine Learning.

ModInterv COVID-19: An online platform to monitor the evolution of epidemic curves.

We present the software ModInterv as an informatics tool to monitor, in an automated and user-friendly manner, the evolution and trend of COVID-19 epidemic curves, both for cases and deaths. The ModInterv software uses parametric generalized growth models, together with LOWESS regression analysis, to fit epidemic curves with multiple waves of infections for countries around the world as well as for states and cities in Brazil and the USA. The software automatically accesses publicly available COVID-19 databases maintained by the Johns Hopkins University (for countries as well as states and cities in the USA) and the Federal University of Viçosa (for states and cities in Brazil). The richness of the implemented models lies in the possibility of quantitatively and reliably detecting the distinct acceleration regimes of the disease. We describe the backend structure of software as well as its practical use. The software helps the user not only to understand the current stage of the epidemic in a chosen location but also to make short term predictions as to how the curves may evolve. The app is freely available on the internet (http://fisica.ufpr.br/modinterv), thus making a sophisticated mathematical analysis of epidemic data readily accessible to any interested user.

Weibull Regression and Machine Learning Survival Models: Methodology, Comparison, and Application to Biomedical Data Related to Cardiac Surgery.

In this article, we propose a comparative study between two models that can be used by researchers for the analysis of survival data: (i) the Weibull regression model and (ii) the random survival forest (RSF) model. The models are compared considering the error rate, the performance of the model through the Harrell C-index, and the identification of the relevant variables for survival prediction. A statistical analysis of a data set from the Heart Institute of the University of São Paulo, Brazil, has been carried out. In the study, the length of stay of patients undergoing cardiac surgery, within the operating room, was used as the response variable. The obtained results show that the RSF model has less error rate for the training and testing data sets, at 23.55% and 20.31%, respectively, than the Weibull model, which has an error rate of 23.82%. Regarding the Harrell C-index, we obtain the values 0.76, 0.79, and 0.76, for the RSF and Weibull models, respectively. After the selection procedure, the Weibull model contains variables associated with the type of protocol and type of patient being statistically significant at 5%. The RSF model chooses age, type of patient, and type of protocol as relevant variables for prediction. We employ the randomForestSRC package of the R software to perform our data analysis and computational experiments. The proposal that we present has many applications in biology and medicine, which are discussed in the conclusions of this work.

Multiple waves of COVID-19: a pathway model approach.

A generalized pathway model, with time-dependent parameters, is applied to describe the mortality curves of the COVID-19 disease for several countries that exhibit multiple waves of infections. The pathway approach adopted here is formulated explicitly in time, in the sense that the model's growth rate for the number of deaths or infections is written as an explicit function of time, rather than in terms of the cumulative quantity itself. This allows for a direct fit of the model to daily data (new deaths or new cases) without the need of any integration. The model is applied to COVID-19 mortality curves for ten selected countries and found to be in very good agreement with the data for all cases considered. From the fitted theoretical curves for a given location, relevant epidemiological information can be extracted, such as the starting and peak dates for each successive wave. It is argued that obtaining reliable estimates for such characteristic points is important for studying the effectiveness of interventions and the possible negative impact of their relaxation, as it allows for a direct comparison of the time of adoption/relaxation of control measures with the peaks and troughs of the epidemic curve.

Ingredients for Responsible Machine Learning: A Commented Review of The Hitchhiker's Guide to Responsible Machine Learning.

In The hitchhiker's guide to responsible machine learning, Biecek, Kozak, and Zawada (here BKZ) provide an illustrated and engaging step-by-step guide on how to perform a machine learning (ML) analysis such that the algorithms, the software, and the entire process is interpretable and transparent for both the data scientist and the end user. This review summarises BKZ's book and elaborates on three elements key to ML analyses: inductive inference, causality, and interpretability.

ModInterv: An automated online software for modeling epidemics.

The COVID-19 pandemic has proven the importance of mathematical tools to understand the evolution of epidemic outbreaks and provide reliable information to the general public and health authorities. In this perspective, we have developed ModInterv, an online software that applies growth models to monitor the evolution of the COVID-19 epidemic in locations chosen by the user among countries worldwide or states and cities in the USA or Brazil. This paper describes the software capabilities and its use both in recent research works and by technical committees assisting government authorities. Possible applications to other epidemics are also briefly discussed.

A New Wavelet-Based Privatization Mechanism for Probability Distributions.

In this paper, we propose a new privatization mechanism based on a naive theory of a perturbation on a probability using wavelets, such as a noise perturbs the signal of a digital image sensor. Wavelets are employed to extract information from a wide range of types of data, including audio signals and images often related to sensors, as unstructured data. Specifically, the cumulative wavelet integral function is defined to build the perturbation on a probability with the help of this function. We show that an arbitrary distribution function additively perturbed is still a distribution function, which can be seen as a privatized distribution, with the privatization mechanism being a wavelet function. Thus, we offer a mathematical method for choosing a suitable probability distribution for data by starting from some guessed initial distribution. Examples of the proposed method are discussed. Computational experiments were carried out using a database-sensor and two related algorithms. Several knowledge areas can benefit from the new approach proposed in this investigation. The areas of artificial intelligence, machine learning, and deep learning constantly need techniques for data fitting, whose areas are closely related to sensors. Therefore, we believe that the proposed privatization mechanism is an important contribution to increasing the spectrum of existing techniques.

Adaptive kernel fuzzy clustering for missing data.

Many machine learning procedures, including clustering analysis are often affected by missing values. This work aims to propose and evaluate a Kernel Fuzzy C-means clustering algorithm considering the kernelization of the metric with local adaptive distances (VKFCM-K-LP) under three types of strategies to deal with missing data. The first strategy, called Whole Data Strategy (WDS), performs clustering only on the complete part of the dataset, i.e. it discards all instances with missing data. The second approach uses the Partial Distance Strategy (PDS), in which partial distances are computed among all available resources and then re-scaled by the reciprocal of the proportion of observed values. The third technique, called Optimal Completion Strategy (OCS), computes missing values iteratively as auxiliary variables in the optimization of a suitable objective function. The clustering results were evaluated according to different metrics. The best performance of the clustering algorithm was achieved under the PDS and OCS strategies. Under the OCS approach, new datasets were derive and the missing values were estimated dynamically in the optimization process. The results of clustering under the OCS strategy also presented a superior performance when compared to the resulting clusters obtained by applying the VKFCM-K-LP algorithm on a version where missing values are previously imputed by the mean or the median of the observed values.

Prevalence ratio estimation via logistic regression: a tool in R.

The interpretation of odds ratios (OR) as prevalence ratios (PR) in cross-sectional studies have been criticized since this equivalence is not true unless under specific circumstances. The logistic regression model is a very well known statistical tool for analysis of binary outcomes and frequently used to obtain adjusted OR. Here, we introduce the prLogistic for the R statistical computing environment which can be obtained from The Comprehensive R Archive Network, https://cran.r-project.org/package=prLogistic. The package prLogistic was built to assist the estimation of PR via logistic regression models adjusted by delta method and bootstrap for analysis of independent and correlated binary data. Two applications are presented to illustrate its use for analysis of independent observations and data from clustered studies.

Power law behaviour in the saturation regime of fatality curves of the COVID-19 pandemic.

We apply a versatile growth model, whose growth rate is given by a generalised beta distribution, to describe the complex behaviour of the fatality curves of the COVID-19 disease for several countries in Europe and North America. We show that the COVID-19 epidemic curves not only may present a subexponential early growth but can also exhibit a similar subexponential (power-law) behaviour in the saturation regime. We argue that the power-law exponent of the latter regime, which measures how quickly the curve approaches the plateau, is directly related to control measures, in the sense that the less strict the control, the smaller the exponent and hence the slower the diseases progresses to its end. The power-law saturation uncovered here is an important result, because it signals to policymakers and health authorities that it is important to keep control measures for as long as possible, so as to avoid a slow, power-law ending of the disease. The slower the approach to the plateau, the longer the virus lingers on in the population, and the greater not only the final death toll but also the risk of a resurgence of infections.

Situation of the Covid-19 Epidemic in Brazil in August 2020: Most Northern and Northeastern States in a Saturation Phase, while Southern States in Acceleration or Slight Deceleration / Situação da Epidemia de Covid-19 no Brasil em Agosto de 2020: Maioria do Estados do Norte-Nordeste em Saturação e Estados do Sul em Aceleração ou Leve Desaceleração

In this Technical Note we analyze the cumulative curves of deaths attributed to Covid-19 in the 26 Brazilian states and the Federal District until August 21, 2020. Mathematical growth models implemented by the ModInterv Covid-19 application, which can be accessed via internet browser or via a mobile app available at the Google Play Store, were used to investigate at which stage the epidemic is in each of these entities of the Federation. The analysis revealed that almost all states in the Northern and Northeastern regions are in the saturation phase, when the epidemic is relatively under control, while in all Southern states and in most states in the Midwest the epidemic is still accelerating or shows only a slight deceleration. The Southeastern region presents a great diversity of epidemic stages, with each state at a different stage, ranging from acceleration to saturation.

Nesta Nota Técnica nós analisamos as curvas acumuladas de mortes atribuídas à Covid-19 nos 26 estados e Distrito Federal até o dia 21 de agosto de 2020. Foram utilizados modelos matemáticos de crescimento implementados pelo aplicativo ModInterv Covid-19, que pode ser acessado via internet (http://fisica.ufpr.br/modinterv) ou através de aplicativo para celular disponível na Play Store (https://play.google.com/store/apps/details?id=com.tanxe.covid_19), para investigar em qual fase da epidemia cada um dessas unidades da federação se encontra. A análise revelou que quase todos os estados das Regiões Norte e Nordeste encontram-se em uma fase de saturação, quando a epidemia está relativamente sob controle, ao passo que em todos os estados do Sul e a maioria dos estados do Centro-Oeste a epidemia ainda está em aceleração ou apresenta apenas uma leve desaceleração. A Região Sudeste apresenta uma grande diversidade de estágios da epidemia, com cada estado em um estágio diferente, indo de acelerado à saturação.

Modelling fatality curves of COVID-19 and the effectiveness of intervention strategies.

The main objective of the present article is twofold: first, to model the fatality curves of the COVID-19 disease, as represented by the cumulative number of deaths as a function of time; and second, to use the corresponding mathematical model to study the effectiveness of possible intervention strategies. We applied the Richards growth model (RGM) to the COVID-19 fatality curves from several countries, where we used the data from the Johns Hopkins University database up to May 8, 2020. Countries selected for analysis with the RGM were China, France, Germany, Iran, Italy, South Korea, and Spain. The RGM was shown to describe very well the fatality curves of China, which is in a late stage of the COVID-19 outbreak, as well as of the other above countries, which supposedly are in the middle or towards the end of the outbreak at the time of this writing. We also analysed the case of Brazil, which is in an initial sub-exponential growth regime, and so we used the generalised growth model which is more appropriate for such cases. An analytic formula for the efficiency of intervention strategies within the context of the RGM is derived. Our findings show that there is only a narrow window of opportunity, after the onset of the epidemic, during which effective countermeasures can be taken. We applied our intervention model to the COVID-19 fatality curve of Italy of the outbreak to illustrate the effect of several possible interventions.

Recife and Belém are currently the only capitals that are already in the saturation phase of Covid-19 in Brazil / Recife e Belém são atualmente as únicas capitais que já estão na fase de saturação da Covid-19 no Brasil

In this technical note we analyze the accumulated fatality curves attributed to Covid-19 in the 27 Brazilian state capitals until July 19, 2020. We employed three mathematical growth models to assess at which stage of the epidemic each of these cities is at. These models were implemented in the Modinterv Covid-19 application, developed by the Federal Universities of Paraná, Pernambuco and Sergipe, which can be accessed through the <http://fisica.ufpr.br/modinterv> page. The analysis reveals that only Recife and Belém appear to have reached the saturation phase of the epidemic, when the accumulated fatality curve begins to approach the plateau. Among the other capitals, eight are still in the initial phase of rapid growth and seventeen are in the intermediate phase, when the epidemic curve has already passed through the inflection point but is still relatively far from the plateau.

Nessa nota técnica analisamos as curvas acumuladas de mortes atribuídas à Covid-19 nas 27 capitais brasileiras até o dia 19 de julho de 2020. Empregamos três modelos matemáticos de crescimento para avaliar em que fase da epidemia encontra-se cada uma dessas cidades. Esses modelos  foram implementados  no aplicativo Modinterv Covid-19, desenvolvido pelas Universidades Federais do Paraná, Pernambuco e Sergipe, o qual pode ser acessado através da página http://fisica.ufpr.br/modinterv. A análise revela que apenas Recife e Belém aparentam ter atingido a fase de saturação da epidemia, quando a curva acumulada de morte começa a se aproximar do platô. Entre as demais capitais, oito ainda estão na fase inicial de crescimento rápido e dezessete estão na fase intermediária, quando a curva epidêmica já passou pelo ponto de inflexão mas ainda está relativamente distante do platô.

OVERVIEW OF THE COVID-19 IN NORTHEAST BRAZIL: ANALYSES AND FORECASTS VIA GROWTH MODELS / PANORAMA DA COVID-19 NO NORDESTE BRASILEIRO: ANÁLISES E PREVISÕES VIA MODELOS DE CRESCIMENTO

The Covid-19 pandemic, caused by the new coronavirus (SARS-CoV-2), is one of the gravest public health crises the world has ever faced. In this context, it is important to have effective models to describe the different stages of the epidemic, in order to offer guidance to the competent authorities regarding the adoption of public policies to contain and control the pandemic. In this work, we present a novel method to analyze epidemic curves based on growth models, using as examples the cumulative curves of deaths attributed to Covid-19 for the states of the Northeastern Region of Brazil. Depending on the case, the q-exponential model, the Richards model or the generalized Richards model were used to make the numerical fits of the respective empirical curves. The models used here describe very well the empirical curves of all the Northeastern Brazilian States, thus allowing a more precise diagnosis of the stage of the epidemic in each of the States.  Among them, only the state of Paraíba is still in the early growth phase, when the epidemic curve does not yet have an inflexion point, being in this case better described by the q-exponential model.  The other states were better described either by the Richards model or by its generalized version. The Richards model, in particular, was able to identify with reasonable reliability the emergence of the inflexion point for states that only recently have reached this stage of the epidemic, such as Piauí, Rio Grande do Norte and Sergipe. This model is also able to predict when the inflection is about to occur, as is the case in Bahia. The generalized Richards model, in turn, has proved more appropriate to describe epidemic curves in states that are in a more developed phase of the epidemic, such as Ceará and Pernambuco, when the epidemic curves already show a more consolidated trend of saturation toward the plateau.

A pandemia da Covid-19, causada pelo novo coronavírus (SARS-CoV-2), é uma das maiores crises de saúde pública que o mundo já enfrentou. Nesse contexto, é importante ter modelos eficazes para descrever os diferentes estágios da evolução da epidemia, a fim de orientar as autoridades competentes na adoção de políticas públicas para o enfrentamento e controle da pandemia. No presente trabalho, nós propomos um novo método de análise de curvas epidêmicas com base na seleção criteriosa de modelos de crescimento, tomando como exemplo as curvas acumuladas de óbitos atribuídos à Covid-19 para os estados da região Nordeste do Brasil. A depender do caso, foram utilizados o modelo q-exponencial, o modelo de Richards ou o modelo generalizado de Richards para fazer o ajuste numérico das respectivas curvas empíricas. Verificou-se que os modelos utilizados descrevem muito bem as curvas empíricas de todos os estados do Nordeste, permitindo assim diagnosticar mais precisamente o estágio da epidemia em cada um dos estados. Dentre eles, apenas o estado da Paraíba ainda encontra-se na fase inicial de crescimento, quando a curva epidêmica ainda não apresenta um ponto de inflexão, sendo nesse caso melhor descrita pelo modelo q-exponencial. Os demais estados foram mais bem descritos ou pelo modelo de Richards ou por sua versão generalizada. O modelo de Richards, em particular, foi capaz de identificar com razoável confiabilidade o surgimento do ponto de inflexão para os estados que só recentemente alcançaram esse estágio da epidemia, como foi o caso do Piauí, Rio Grande do Norte e Sergipe. Esse modelo também é capaz de prever quando a inflexão está prestes a acontecer, como é o caso da Bahia. O modelo generalizado de Richards, por sua vez, mostrou-se mais apropriado para descrever curvas epidêmicas de estados que estão em uma fase mais desenvolvida da epidemia, como Ceará e Pernambuco, quando as curvas epidêmicas já apresentam uma tendência mais consolidada de saturação em direção ao platô.

Analysis of Covid-19 epidemic curves via generalized growth models: Case study for the cities of Recife and Teresina / Análise de curvas epidêmicas da Covid-19 via modelos generalizados de crescimento: Estudo de caso para as cidades de Recife e Teresina

Introduction: The Covid-19 pandemic is one of the biggest public health crises the world has ever faced. In this context, it is important to have effective models to describe the different stages of the epidemic's evolution in order to guide the authorities in taking appropriate measures to fight the disease. Objective: To present an analysis of epidemic curves of Covid-19 based on phenomenological growth models, with applications to the curves for the cumulative numbers of confirmed cases of infection by the novel coronavirus (Sars-Cov-2) and deaths attributed to the disease (Covid-19) caused by the virus, for the Brazilian cities of Recife and Teresina. Methods: The Richards generalized model and the generalized growth model were used to make the numerical fits of the respective empirical curves. Results: The models used described very well the empirical curves against which they were tested. In particular, the generalized Richards model was able to identify the appearance of the inflexion point in the cumulative curves, which in turn represents the peak of the respective daily curves. A brief discussion is also presented on the relationship between the fitting parameters obtained from the model and the mitigation measures adopted in each of the municipalities considered. Conclusions: The generalized Richards model proved to be very effective in describing epidemic curves of Covid-19 and estimating important epidemiological parameters, such as the time of the peak of the curve for daily cases and deaths, thus allowing a practical and efficient monitoring of the epidemic evolution.

Introdução: A pandemia da Covid-19 é uma das maiores crises de saúde pública que o mundo já enfrentou. Nesse contexto, é importante ter modelos eficazes para descrever os diferentes estágios da evolução da epidemia, a fim de orientar as autoridades competen- tes na adoção de políticas públicas para o enfrentamento da mesma. Objetivo: Apresentar uma análise de curvas epidêmicas com base em modelos fenomenológicos de crescimento, tomando como exemplo as curvas acumuladas de casos confirmados de infecção pelo novo coronavírus (Sars-Cov-2) e de óbitos atribuídos à doença (Covid-19) causada pelo vírus, para as cidades do Recife e Teresina. Métodos: Foram utilizados o modelo generalizado de Richards e o modelo de crescimento generalizado para fazer o ajuste numérico das respectivas curvas empíricas. Resultados: Verificou-se que os modelos utilizados descrevem muito bem as curvas empíricas em que foram testados. Em particular, o modelo generalizado de Richards é capaz de identificar com razoável confiabilidade o surgimento do ponto de infle- xão nas curvas acumuladas, o qual corresponde ao ponto de máximo das respectivas curvas diárias. Apresenta-se ainda uma breve discussão sobre a relação entre os parâmetros obtidos no ajuste do modelo e as medidas de mitigação adotadas para retardar a propagação da Covid-19 em cada um dos municípios considerados. Conclusões: O modelo generalizado de Richards mostrou-se bastante eficaz para descrever curvas epidêmicas da Covid-19 e es- timar parâmetros epidemiológicos importantes, como o pico das curvas de casos e óbitos diários, permitindo assim realizar de modo prático e eficiente o monitoramento da evolução da epidemia.

Your Face and Moves Seem Happier When I Smile.

In this experiment, we replicated the effect of muscle engagement on perception such that the recognition of another's facial expressions was biased by the observer's facial muscular activity (Blaesi & Wilson, 2010). We extended this replication to show that such a modulatory effect is also observed for the recognition of dynamic bodily expressions. Via a multilab and within-subjects approach, we investigated the emotion recognition of point-light biological walkers, along with that of morphed face stimuli, while subjects were or were not holding a pen in their teeth. Under the "pen-in-the-teeth" condition, participants tended to lower their threshold of perception of happy expressions in facial stimuli compared to the "no-pen" condition, thus replicating the experiment by Blaesi and Wilson (2010). A similar effect was found for the biological motion stimuli such that participants lowered their threshold to perceive happy walkers in the pen-in-the-teeth condition compared to the no-pen condition. This pattern of results was also found in a second experiment in which the no-pen condition was replaced by a situation in which participants held a pen in their lips ("pen-in-lips" condition). These results suggested that facial muscular activity alters the recognition of not only facial expressions but also bodily expressions.

Germany's 'coronavirus anomaly': Statistical evidence that early mass testing leads to low mortality rates

In this note, we present a statistical analysis of the mortality rates of COVID-19 for several selected European countries. We compare the countries' mortality rates with their respective number of tests as a function of the time since the first death. Our analysis shows that countries that either delayed mass testing, such as Italy, or have not fully adopted it, such as France and the UK, have had much higher mortality rates than Germany, which has adopted a policy of wide and early testing. Conversely, countries that have followed Germany's example, such as Portugal, have so far had comparatively low mortality rates.