Introduction to the special issue on Data Science for COVID-19

Abstract

An introduction to this Special Issue on Data Science for COVID-19 is included in this paper. It contains a general overview about methods and applications of nonparametric inference and other flexible data science methods for the COVID-19 pandemic. Specifically, some methods existing before the COVID-19 outbreak are surveyed, followed by an account of survival analysis methods for COVID-related times. Then, several nonparametric tools for the estimation of certain COVID rates are revised, along with the forecasting of most relevant series counts, and some other related problems. Within this setup, the papers published in this special issue are briefly commented in this introductory article.

Keywords:

• COVID-19
• generalised additive models
• local polynomials
• spatiotemporal models
• survival analysis

1. Introduction

Over the past decades, nonparametric statistical inference has been used to model important features in epidemiology. It has been also used for data analysis related to the evolution and side effects of pandemics. In the case of the current COVID-19 pandemic, a huge amount of data-analytical procedures have been proposed or used over the past three years.

Section 2 deals with some nonparametric statistical methods for epidemiology proposed and used before COVID-19. A short overview of nonparametric procedures proposed for COVID-19 is also included in the rest of the paper. Section 3 deals with survival analysis methods proposed or used for the study of the distribution of relevant times related to COVID-19: incubation period, time until hospitalisation, length of stay in hospital, as well as COVID-19 infection of a patient and survival time for patients. Nonparametric estimation of recovery rates and in-hospital mortality rates are also considered in this section. Nonparametric estimation of COVID-19 incidence, infection rate, transmission rate and mortality rate are considered in Section 4. Forecasting methods for COVID-19 cases, hospitalisations and deaths are presented in Section 5, while Section 6 deals with some relevant classification problems related to COVID-19. Section 7 deals with several nonparametric statistical procedures proposed for other setups related to the COVID-19 pandemic. Finally, Section 8 describes in more detail the contributions of this Special Issue.

2. Some pre-COVID-19 nonparametric statistical methods in epidemiology

Survival analysis methods, including censored and truncated data, have been widely applied in Biomedicine, and specifically in Epidemiology. Rao, Talwaker, and Kundu (1991) proposed confidence intervals for the relative risk ratio under random censorship. Hoover et al. (1993) proposed using events from dropouts to construct a nonparametric survival function estimate for interval-censored and left-truncated data and they applied these methods to the incubation time of AIDS. Gauderman and Thomas (1994) proposed some methods for the analysis of diseases with variable age at onset. They presented a modification of the Cox proportional hazards model to include both measured (environmental) covariates and latent (genetic) variables, as well as their interactions. Three right censoring strategies were evaluated in Van Benthem et al. (1997) to model AIDS incubation time. In the same setup Geskus (2001) presents some methods for estimating the AIDS incubation time distribution when the date of seroconversion is censored. A method for assessing dose-response effects from case-control and cohort studies with interval-censored exposure information is presented in Cook, Brumback, Wigg, and Ryan (2001). Cure models are used in Stoltenbere, Nordeng, Ystrom, and Samuelsen (2020) in the context of perinatal epidemiology. They study the relation between a gestational exposure to paracetamol and attention-deficit hyperactivity disorder, a condition that can only be ascertained after several years. Recurrent events in observational studies are considered in Hernández-Herrera, Morina, and Navarro (2022). Since it is common to include subjects who became at risk before follow-up, this leads to left censoring, and simply ignoring these prior episodes can lead to biased and inefficient estimates. The authors proposed statistical methods in this context, that are included in the R package miRecSurv.

Statistical methods original from other fields have been also used for Epidemiology. For instance, Liu, Yue, Lai, Huang, and Zhang (2019) discussed the EWMA control chart based on rank methods for a multivariate process, and showed that the proposed method is efficient in detecting shifts for multivariate processes, by applying it to Japanese influenza data.

The kernel method for curve estimation has been used in epidemiology over the past few years. Adaptive and fixed bandwidth-based kernel density estimates are used in Lemke, Mattauch, Heidinger, Pebesma, and Hense (2015) for spatial cancer epidemiology. A kernel weighting approach has been used by Wang, Graubard, Katki, and Li (2020) to improve the external validity of epidemiologic cohort analyses. Kernel density estimation and approximate Bayesian computation have been also used in Irvine and Hollingsworth (2018) to produce flexible epidemiological model fitting.

The Bayesian paradigm was also present in several other papers using nonparametric techniques for epidemiology. The paper (Xu, Kypraios, and O'Neill 2016) develops a Bayesian nonparametric approach using Gaussian Processes to estimate the infection process in an epidemic. A specific problem of estimating a disease risk from highly correlated environmental exposure covariates and a censored survival outcome is considered in Belloni, Laurent, Guihenneuc, and Ancelet (2020). Bayesian profile regression mixture models were extended by assuming an instantaneous excess hazard ratio disease sub-model. A specific adaptive Metropolis-Within-Gibbs algorithm-including label switching moves was implemented to infer the model. This nonparametric Bayesian approach was applied to the estimation of the risk of death by lung cancer in a cohort of French uranium miners. Nonparametric Bayesian meta-regression, incorporating functional meta-predictors and spatial dependency was proposed by Yu et al. (2021) in the context of environmental epidemiology. The method was applied to a temperature-mortality association study and to a suicide seasonality study. A Bayesian penalised B-spline estimation approach was proposed in Meng and Tao (2017) to estimate the parameters and initial values for ordinary differential equations used in epidemiology. The approach was applied to the transmission dynamics of hepatitis C virus in mainland China.

Splines and penalised splines have been also considered in many more papers. For instance, Bergen, Sheppard, Kaufman, and Szpiro (2016) considered measurement errors in air pollution epidemiology studies. They developed analytic bias correction methods when using penalised regression splines to predict exposure. The method is applied to analyse the association of systolic blood pressure with levels of two pollutants, PM2.5 and NO2. In the presence of unmeasured spatial confounding, Bobb et al. (2022) studied the bias increase. These authors proposed an exposure-penalised spline approach that selects the degree of spatial smoothing to explain spatial variability in the exposure. Confounder control is also considered in Howe et al. (2011), where splines are also used for trend analysis. Smoothing splines are proposed in semiparametric ANOVA models in Wahba, Wang, Gu, Klein, and Klein (1995). The methods are applied to the data from the Wisconsin Epidemiologic Study of Diabetic Retinopathy to model the risk of progression of diabetic retinopathy as a function of glycosylated haemoglobin, duration of diabetes and body mass index.

Nonparametric and semiparametric forecasting is of primary interest in Epidemiology. Nonparametric estimation of time-dependent transmission rates using SEIR models are proposed in Smirnova, de Camp, and Chowell (2019). The method is illustrated using case incidence data for various epidemics including the 1918 influenza pandemic in San Francisco and the 2014–2015 Ebola epidemic in West Africa. Forecasting methods originally proposed for epidemics have also been used to predict other social indicators. For instance, Cortes, Sánchez-Sánchez, Santonja, and Villanueva (2013) considered nonparametric probabilistic forecasting of academic performance in Spanish high school using ordinary differential equations coming from epidemiological modelling. A bootstrap approach is employed for forecasting.

Some other relevant problems in Epidemiology have been addressed using nonparametric methods (Savilov and Astafyev 1987, among others). Nonparametric exact tests based on the number of records for early detection of emerging events in epidemiology have been proposed in Khraibani, Jacob, Ducrot, Charras-Garrido, and Sala (2015). A nonparametric approach for estimating cut-offs in continuous risk indicators was applied to type 2 diabetes in Klotsche, Ferger, Pieper, Rehm, and Wittchen (2009). Nonparametric backcasting methods have been used in Punyacharoensin and Viwatwongkasem (2009) to estimate the trends along three decades of the HIV/AIDS epidemic in Thailand. The infection rate and its derivatives of the 2003 Severe Acute Respiratory Syndrome (SARS) epidemic in Beijing, China, were estimated nonparametrically in Chen, Huggins, Yip, and Lam (2008) considering multiplicative counting processes.

3. Survival analysis methods for the study of the distribution of relevant times

Survival analysis has been a key tool for analysing the distribution of relevant times during the current COVID-19 pandemic. This includes incubation period, time until going to hospital, length-of-stay in hospital or in ICU, duration of COVID-19 of a patient and survival times. Related quantities such as recovery rates or mortality rates can be estimated using information about some of these relevant times.

Parametric methods (e.g. lognormal and Gamma distribution) have been recently proposed for the estimation of the distribution of incubation period of COVID-19 based on the doubly interval-censored data (Yin, Zhu, and Lu 2021). But more flexible methods have been presented as well. They consist of semiparametric regression analysis of doubly censored (Wong, Zhou, and Hu 2022) and a simple semiparametric sieve-estimation method based on Laguerre Polynomials (Kreiss and Van Keilegom 2022). Also Deng, You, Liu, Qin, and Zhou (2021) use theory from renewal process by considering the incubation period as the interarrival time, and the duration between departure from Wuhan and onset of symptoms as the mixture of forward time and interarrival time with censored intervals. As a consequence, a consistent estimator for the distribution of the generation time based on the incubation period and a serial interval is proposed for incubation-infectious diseases. A right truncation mechanism is considered by Linton et al. (2020), who carried out a statistical analysis of publicly available case data.

Different times related to COVID-19 hospital admission have been studied using flexible statistical tools. For instance, time until hospitalisation in Galicia (NW Spain), has been studied by Pedrosa-Laza, López-Cheda, and Cao (2022), who used nonparametric mixture cure models selecting relevant covariates such as sex and age. The distribution of length-of-stay has been considered by López-Cheda, Jácome, Cao, and De Salazar (2021), who also used nonparametric mixture cure models and showed that the proposed model outperformed standard approaches, providing more accurate ICU and hospital ward length-of-stay distribution estimates. Adjusting for sex and age is key for accurately forecasting hospital ward and ICU occupancy, as well as discharge or death outcomes. Concerning home-isolated patients, the duration of SARS-CoV-2 RNA detection in COVID-19 patients was studied by Omar et al. (2020), using interval-censored survival analysis. The method was applied to a 2020 database from Rhineland-Palatinate, Germany.

Several methods are reviewed in Seaman, Presanis, and Jackson (2022) for estimating a time-to-event distribution from right-truncated data in an epidemic. For example, one may be interested in estimating the distribution of time from onset of disease symptoms to death. Since at the beginning of an epidemic the only data available are coming from individuals who have died, right truncation causes the distribution of times to event in the sample to be biased towards shorter times compared to the population distribution. These methods are illustrated by applying them to data from individuals who had died from coronavirus disease by 5 April 2020. Other works have considered explanatory covariates as risk factors for death caused by COVID-19. Cox regression and mixture cure models (Seif, Sharafi, Ghaem, and Kasraei 2022) were used to study factors associated with the survival of Iranian patients with COVID-19. The findings revealed that important factors were age, history of diabetes and hypertension.

A new empirical likelihood method is proposed in Hu, Liang, Dai, and Bao (2022) to analyse the recovery rate of COVID-19 based on a doubly censored dataset. The method provides much less biased estimators than previous approaches when the censoring percentage is large. The analysis of COVID-19 in-hospital mortality has been carried out in Xue, Saeed, Castagna, Jorde, and Agalliu (2022). Competing risk models, mixture cure models and promotion time cure models have been considered and compared for estimating in-hospital mortality. These methods were applied to a cohort of COVID-19 in-hospital patients with diabetes, showing that statin use improved the overall survival of older but not younger patients.

4. Nonparametric estimation of COVID-19 incidence and infection, transmission and mortality rates

Nonparametric smoothing techniques are widely used to estimate COVID-19-related curves. This is the case of the incidence and infection, transmission and mortality rates.

Spline-based Bayesian hierarchical models have been used in Wistuba, Mayr, and Staerk (2022) to estimate the incidence of the COVID-19 pandemic in Germany via death counts. Kernel principal component analysis control charts and K-nearest neighbour control chart were used in Fawzy and Ghalib (2022) to monitor the number of infections of COVID-19. These methods were applied to official data coming from the Public Health Department of the Iraqi Ministry of Health. Generalised additive models were used by de Oliveira, Binner, Mandal, Kelly, and Power (2021) to model cumulative and daily curves for confirmed cases and deaths. The pure Markov-Switching and the family of Markov-Switching GARCH models were used to identify structural breaks in the COVID-19 time series. These methods were applied in the time series of 189 countries, collected from the Centre for Systems Science and Engineering at Johns Hopkins University.

A space-time epidemic modelling framework was proposed in Wang et al. (2022) to study the spatial-temporal pattern in the spread of infectious diseases. A quasi-likelihood approach via the penalised spline approximation and alternatively reweighted least-squares techniques are considered to estimate the model. The proposed method is applied to the COVID-19 pandemic.

Parametric regression methods – including the linear, log-linear, polynomial, generative additive regression, and spline regression – and nonparametric ones – including K-nearest neighbourhood, support vector machines and decision trees – have been used in Atteia, Mengash, and Samee (2021) for estimating models for the total number of COVID-19 cases and total deaths.

A semiparametric Bayesian approach is presented in Schweinberger, Bomiriya, and Babkin (2022) for flexible semiparametric modelling of epidemics. The method was applied to model the spread of the coronavirus MERS in South Korea in 2015.

Official data on the number of people infected with SARS-CoV-2-have been released in most of the countries just on the basis of a non-representative sample of population which tested positive. However a reliable estimation of the number of infected, including asymptomatic people, turns out to be crucial. In order to overcome the current data shortcoming, a bootstrap-driven, estimation procedure for the number of people infected with the SARS-CoV-2 is proposed in Fenga (2021). This method was applied to data from Italy.

Estimation of viral load in wastewater samples is a proxy that has been used worldwide to overcome the problem of asymptomatic patients. In Courbariaux et al. (2022), a flexible smoother adapted to censored data with outliers was used to monitor the viral load of SARS-CoV-2 in wastewaters arriving at wastewater treatment plants (WWTPs) in France.

Generalised additive models, kernel smoothing and LOESS have been used in Vallejo et al. (2022) to monitor the viral load of SARS-CoV-2 in a wastewater plant located in Northwest Spain and to formulate statistical models to predict the number of infected people (symptomatic and asymptomatic) based on the smoothed viral load and some other environmental variables.

Compartmental models that allow the infection rate to assume any continuous-time profile are considered by Bisiacco, Pillonetto, and Cobelli (2022). Closed-form expressions of the infection rate time-course can be derived for such models. As a consequence, a nonparametric estimate of the infection rate is proposed. Using real data collected in Italy, this technique proved to be a useful tool to monitor COVID-19 transmission dynamics.

In Scrucca (2022), a near real-time COVID-19 index is proposed for monitoring the evolution of the pandemic. The index is computed from predictions obtained from a GAM beta regression for modelling the test-positive rate as a function of time. The proposal is illustrated using data on the COVID-19 pandemic in Italy. A minimax approximation criterion and polynomial splines are used in Abbasov et al. (2022) for estimating and predicting dynamics of the spread of the coronavirus infection. The method was applied to model the spread of the coronavirus infection in Russia for a period of more than one year.

Case fatality ratios during COVID-19 have been estimated nonparametrically in Ghosh, Samanta, and Nieto (2021) by means of the Kaplan–Meier estimator. Furthermore, a nonparametric cure model has been estimated using Nadaraya–Watson weights to estimate the cure rate. The methods are illustrated by estimating relative risks and cumulative mortality rates of COVID-19 in Spain, Italy and Israel.

In Pillonetto, Bisiacco, Palu, and Cobelli (2021), a new class of nonparametric compartmental models is developed to describe how the impact of the lockdown varies in time. Hospitalised data are mapped into an infinite-dimensional space, hence obtaining a function which also takes into account how social distancing measures and people's growing awareness of infection's risk evolves as time progresses. This also allows reconstructing a continuous-time profile of the SARS-CoV-2 reproduction number. When applied to data collected in Lombardy, the model illustrates how people behaviour changed during the restrictions and its importance to contain the epidemic.

The trend pattern of COVID-19 has been studied in Kumar, Agiwal, and Yau (2022) using a spline-based time series model. The estimation of the model parameters is obtained under the Bayesian setup for the best-fitted model. In Pijpers (2021), a nonparametric method for determining epidemiological reproduction numbers is proposed and applied to the timeline of hospitalisation admissions for COVID-19 in the Netherlands up to 20 May 2020. A state-space model is considered in Zhou and Ji (2020) for the transmission dynamics of COVID-19. Semiparametric Bayesian inference is used to fit the model. The approach is applied to COVID-19 data from six states of the United States. An analysis of COVID-19 with spline regression at the province level during the first-level response to a major public health emergency out of Hubei, China, was carried out in Liang and Shen (2021). The 22 Chinese provinces considered from 19 January to 12 March 2020 were grouped into 3 regions, namely eastern, central and western provinces, and the trends between adjacent knots were compared among the 3 regions. Spline regression was applied to the data.

In Shi et al. (2021a, 2021b) an extended weight kernel density estimation model is used to forecast COVID-19 onset risk and to identify spatiotemporal variations of lockdown effects in China. Other simple nonparametric methods, including time-delay correlation analysis, have been considered in Diebner and Timmesfeld (2021) to explore COVID-19 daily records of diagnosed cases and fatalities. The estimates for Germany, France, Italy, United States of America, United Kingdom, Spain, Switzerland, and Brazil are obtained and compared.

5. Forecasting COVID-19 cases, hospitalisations and deaths

Forecasting the most relevant series for a pandemic like COVID-19 is an important task. Public health decision making can benefit from a reliable forecast. This implies that forecasting techniques are one of the priorities of health authorities.

A semiparametric mixed model for short-term projection of daily COVID-19 incidence in Canada is proposed in Mullah and Yan (2022). The proposed model is shown to describe the historical trend very well with an excellent ability to predict the short-term trajectory. In O'Dea and Drake (2022), a semiparametric, Gaussian infection state-space compartmental model with time-dependent parameters has been formulated. The model is useful for forecasting several COVID-19 relevant series. Its performance is evaluated one to four weeks ahead of forecasts of COVID-19 cases, hospital admissions and deaths in the state of California.

An exploratory approach based on the complex network-defined splines is considered in Demertzis, Tsiotas, and Magafas (2020). The method is applied to the evolution of COVID-19 in Greece. A VAR epidemiological model is proposed in Shang, Galow, and Galow (2021) for regional forecasting of COVID-19 caseload. Nonparametric regression is also used. The method is applied across seven major demographically representative New York City metropolitan region counties. Forecasting COVID-19 onset risk is also considered in Shi et al. (2021a) using an extended weight kernel density estimation model.

6. COVID-19 diagnosis, outcomes and other related classification problems

Several classification problems related to COVID-19 are reviewed in this section. It covers from side effects of the pandemic and COVID-19 sentiment to the diagnosis of COVID-19 and prediction of COVID-19 outcomes for a given patient.

Motivation of teleworkers and non-teleworkers in times of COVID-19 in Spain is analysed in Romeo, Yepes-Baldo, and Beltra (2022). Nonparametric statistical analysis and classification and regression trees are used to explore the motivation of employees who teleworked and who did not, during the lockdown. In Rochim, Widyaningrum, and Eridani (2020), COVID-19 sentiment is classified using support vector machines with different kernel functions. The methods are applied to a dataset about comments on Youtube to analyse public sentiment on the increase in cases at the beginning of the COVID-19 pandemic in Indonesia.

A computer-aided detection method for COVID-19 from CT images has been proposed in Saygili (2022). The method is based on a Gaussian mixture model and a kernel support vector machines classifier. Chest CT images have been also used by Turkoglu (2021) to produce a COVID-19 detection system using multiple kernels-extreme learning machines based on deep convolutional neural networks.

In Alotaibi et al. (2022), an optimal kernel extreme learning machine for COVID-19 classification on an epidemiology dataset is performed. Bayesian nonparametric dimensionality reduction was used in Ajirak et al. (2021) for predicting the severity of COVID-19 in pregnant women using categorical data. The method was applied to a dataset of 155 test-positive COVID-19 pregnant women collected at Stony Brook University Hospital. A nonparametric fuzzy hypothesis testing for quantiles applied to clinical characteristics of COVID-19 was developed in Chukhrova and Johannssen (2021). An additional comprehensive case study is performed on COVID-19 in HIV-infected individuals with a focus on human body temperature and related measurement problems. In Gude-Sampedro et al. (2021), a prognostic model based on comorbidities to predict COVID-19 severity is developed and validated. Predicting progression to disease severity, hospitalisation, admission to intensive care unit (ICU) and mortality in patients with COVID-19 infection has been performed for a dataset of COVID-19 patients collected in Galicia (Northwest Spain). A game-theoretic approach was adopted by Davila-Pena, García-Jurado, and Casas-Méndez (2022) to assess the influence of features on classification problems. The method is applied to the same dataset of COVID-19 patients in Galicia.

7. Nonparametric statistical methods for other setups related to COVID-19

Several issues related to COVID-19 are considered in this section. They include under-reporting probability, estimation of highest density regions for COVID-19, and impacts of COVID-19 in meteorology and mobility.

Truncation data analysis has been proposed in Liang, Dai, and Restaino (2022) for the under-reporting probability of the COVID-19 pandemic. Hypothesis testing on the differences in truncation probabilities, that are related to the under-reporting rates, is also considered. These methods are applied to COVID-19 data in several countries, where under-reporting probabilities are expected to be high. In Saavedra-Nieves (2022), a nonparametric estimation of the highest density regions for COVID-19 is proposed. Set estimation methods are used. The behaviour of classical plug-in methods and a recently proposed hybrid algorithm for highest density regions estimation are compared through an extensive simulation study. Both methods are applied to analyse a real data set about COVID-19 cases in the United States.

The meteorological impact of COVID-19 lockdown is analysed in Hua et al. (2021) using generalised additive models. Reductions in nitrogen dioxide (NO2) and fine particulate matter (PM2.5) at 34 sites in the Beijing area are studied. In López-Oriona, D'Urso, Vilar, and Lafuente-Rego (2022), spatial weighted robust clustering of multivariate time series based on quantile dependence is proposed. The method is applied to multivariate time series of mobility indicators concerning the COVID-19 pandemic.

8. Data science for COVID-19

This Special Issue on Data Science for COVID-19 gathers together five important contributions covering different aspects of the study of the COVID-19 pandemic. In this section, a brief overview of the featured papers is provided.

8.1. Semiparametric estimation of features of the incubation and generation times

A situation of interest in epidemics arises when it is known that a person has infected another one (a transmission pair). The exact infection times of both individuals are assumed to be unobserved, and the difference between them (also unobserved) is known as generation time. The distribution of this random variable and some of its features are of great importance in the evolution of epidemics. For instance, the well-known basic reproduction number (the average number of new infections caused by a single individual) can be expressed as a function of the density of this generation time.

Another unobserved random variable of interest is the incubation time, i.e. the difference between the observed time at which one person has symptoms and the infection time. Kreiss and Van Keilegom (2022) exploit the relationship between the observed symptoms time and some related truncation and location variables to propose a semiparametric approach that allows estimating the densities of the incubation and generation times via a sieve of Laguerre polynomials. The new model is then applied to analyse a real dataset of transmission pairs from the early period of the COVID-19 pandemic.

8.2. Testing for equal under-reporting in different groups

Under-reporting of infected cases and deaths are likely to occur in the early stages of a pandemic. It might be due, for instance, to people dying from the virus before being properly diagnosed to be infected or, on the contrary, to people that recover from the virus after having only mild symptoms, hence equally failing to be properly diagnosed.

Liang et al. (2022) propose a truncation model to study the difference in under-reporting probabilities in different population groups, based on the nonparametric product limit estimator of their corresponding survival functions. They apply this methodology to test the equality of under-reporting probabilities of different age groups across different countries and conclude that, in some countries, elder people had a significantly higher under-reporting probability than younger people.

8.3. Epidemics modelling with a network-based semiparametric framework

The network of contacts in a population crucially influences the way that infectious diseases spread, and so, a network-based approach to epidemics has been explored by many authors (see, for instance, Britton and O'Neill (2002) or Groendyke, Welch, and Hunter (2011)). Such a population contact network is generated by a random graph model, which is usually based on parametric assumptions. However, the parametric models available in the existing literature usually induce short-tailed distributions of the number of contacts of the population members, whereas there is some empirical evidence that in real life these distributions are long-tailed (Jones and Handcock 2003).

Therefore, Schweinberger et al. (2022) propose a semiparametric Bayesian approach in which the prior over the epidemiological parameter is parametric, but the prior over the network parameter is nonparametric. A simple numerical example shows that this approach can accommodate both short-tailed and also long-tailed distributions of the number of contacts. The new methodology is then successfully applied to data from the partially observed MERS epidemic in South Korea in 2015, where the coronavirus MERS is related to the coronaviruses SARS and COVID-19.

8.4. COVID-19 hotspot identification through highest density regions

Saavedra-Nieves (2022) suggests to study the expansion of COVID-19 in the United States by analysing the distribution of the locations of confirmed cases of coronavirus. In this regard, areas of high concentration of confirmed cases are specifically detected via the regions of high density of such a distribution.

This immediately poses the problem of estimation of the highest density regions. A recent, fully automatic and nonparametric method of estimation based on geometric features (r-convexity) is considered. That methodology was introduced in Rodríguez-Casal and Saavedra-Nieves (2022), where its theoretical properties were studied in depth. In Saavedra-Nieves (2022) we can find a detailed simulation study comparing its finite-sample performance with the plug-in alternatives, and also an application to COVID-19 data, which shows its usefulness at the time of revealing hotspots in the spreading of the pandemic.

8.5. Spatiotemporal modelling of COVID-19 spread

Susceptible-Infectious-Recovered (SIR) models (see Brauer, Van den Driessche, and Wu 2008) provide widely acknowledged tools to describe the dynamics of the spread of an infectious disease. See also Tang et al. (2020) for a recent extensive review. However, they are mostly deterministic and are not intended for handling stochastic variability, or for allowing incorporating auxiliary information in terms of covariates.

Wang et al. (2022) propose to remedy this by considering a nonparametric spatiotemporal model that combines the use of the SIR model as the deterministic skeleton with statistical time series regression models, in a novel spatiotemporal epidemic model (STEM). Consistency and asymptotic normality of the coefficient estimators is shown, and this facilitates making inferences on the proposed models, such as constructing confidence bands for the true model component functions. The new methodology is put into practice with COVID-19 data from the United States.

Acknowledgments

The Guest Editor of this special issue on Data Science for COVID-19 and the Past Editor-in-Chief of the Journal of Nonparametric Statistics would like to thank the members of the Editorial Board of the journal and all the anonymous referees who took part in the evaluation of the manuscripts submitted for possible publication in this special issue.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The research of the first author has been supported by the MICINN grant PID2020-113578RBI00 and by the Xunta de Galicia (Grupos de Referencia Competitiva ED431C-2020-14 and Centro de Investigación del Sistema Universitario de Galicia ED431G 2019/01), all of them through the European Regional Development Fund (ERDF). The research of the second author has been supported by the MICINN grant PID2019-109387GB-I00 and by the Junta de Extremadura grant GR21044.