Why are different estimates of the effective reproductive number so different? A case study on COVID-19 in Germany.

The effective reproductive number Rt has taken a central role in the scientific, political, and public discussion during the COVID-19 pandemic, with numerous real-time estimates of this quantity routinely published. Disagreement between estimates can be substantial and may lead to confusion among decision-makers and the general public. In this work, we compare different estimates of the national-level effective reproductive number of COVID-19 in Germany in 2020 and 2021. We consider the agreement between estimates from the same method but published at different time points (within-method agreement) as well as retrospective agreement across eight different approaches (between-method agreement). Concerning the former, estimates from some methods are very stable over time and hardly subject to revisions, while others display considerable fluctuations. To evaluate between-method agreement, we reproduce the estimates generated by different groups using a variety of statistical approaches, standardizing analytical choices to assess how they contribute to the observed disagreement. These analytical choices include the data source, data pre-processing, assumed generation time distribution, statistical tuning parameters, and various delay distributions. We find that in practice, these auxiliary choices in the estimation of Rt may affect results at least as strongly as the selection of the statistical approach. They should thus be communicated transparently along with the estimates.

Scoring epidemiological forecasts on transformed scales.

Forecast evaluation is essential for the development of predictive epidemic models and can inform their use for public health decision-making. Common scores to evaluate epidemiological forecasts are the Continuous Ranked Probability Score (CRPS) and the Weighted Interval Score (WIS), which can be seen as measures of the absolute distance between the forecast distribution and the observation. However, applying these scores directly to predicted and observed incidence counts may not be the most appropriate due to the exponential nature of epidemic processes and the varying magnitudes of observed values across space and time. In this paper, we argue that transforming counts before applying scores such as the CRPS or WIS can effectively mitigate these difficulties and yield epidemiologically meaningful and easily interpretable results. Using the CRPS on log-transformed values as an example, we list three attractive properties: Firstly, it can be interpreted as a probabilistic version of a relative error. Secondly, it reflects how well models predicted the time-varying epidemic growth rate. And lastly, using arguments on variance-stabilizing transformations, it can be shown that under the assumption of a quadratic mean-variance relationship, the logarithmic transformation leads to expected CRPS values which are independent of the order of magnitude of the predicted quantity. Applying a transformation of log(x + 1) to data and forecasts from the European COVID-19 Forecast Hub, we find that it changes model rankings regardless of stratification by forecast date, location or target types. Situations in which models missed the beginning of upward swings are more strongly emphasised while failing to predict a downturn following a peak is less severely penalised when scoring transformed forecasts as opposed to untransformed ones. We conclude that appropriate transformations, of which the natural logarithm is only one particularly attractive option, should be considered when assessing the performance of different models in the context of infectious disease incidence.

Collaborative nowcasting of COVID-19 hospitalization incidences in Germany.

Real-time surveillance is a crucial element in the response to infectious disease outbreaks. However, the interpretation of incidence data is often hampered by delays occurring at various stages of data gathering and reporting. As a result, recent values are biased downward, which obscures current trends. Statistical nowcasting techniques can be employed to correct these biases, allowing for accurate characterization of recent developments and thus enhancing situational awareness. In this paper, we present a preregistered real-time assessment of eight nowcasting approaches, applied by independent research teams to German 7-day hospitalization incidences during the COVID-19 pandemic. This indicator played an important role in the management of the outbreak in Germany and was linked to levels of non-pharmaceutical interventions via certain thresholds. Due to its definition, in which hospitalization counts are aggregated by the date of case report rather than admission, German hospitalization incidences are particularly affected by delays and can take several weeks or months to fully stabilize. For this study, all methods were applied from 22 November 2021 to 29 April 2022, with probabilistic nowcasts produced each day for the current and 28 preceding days. Nowcasts at the national, state, and age-group levels were collected in the form of quantiles in a public repository and displayed in a dashboard. Moreover, a mean and a median ensemble nowcast were generated. We find that overall, the compared methods were able to remove a large part of the biases introduced by delays. Most participating teams underestimated the importance of very long delays, though, resulting in nowcasts with a slight downward bias. The accompanying prediction intervals were also too narrow for almost all methods. Averaged over all nowcast horizons, the best performance was achieved by a model using case incidences as a covariate and taking into account longer delays than the other approaches. For the most recent days, which are often considered the most relevant in practice, a mean ensemble of the submitted nowcasts performed best. We conclude by providing some lessons learned on the definition of nowcasting targets and practical challenges.

Comparing trained and untrained probabilistic ensemble forecasts of COVID-19 cases and deaths in the United States.

The U.S. COVID-19 Forecast Hub aggregates forecasts of the short-term burden of COVID-19 in the United States from many contributing teams. We study methods for building an ensemble that combines forecasts from these teams. These experiments have informed the ensemble methods used by the Hub. To be most useful to policymakers, ensemble forecasts must have stable performance in the presence of two key characteristics of the component forecasts: (1) occasional misalignment with the reported data, and (2) instability in the relative performance of component forecasters over time. Our results indicate that in the presence of these challenges, an untrained and robust approach to ensembling using an equally weighted median of all component forecasts is a good choice to support public health decision-makers. In settings where some contributing forecasters have a stable record of good performance, trained ensembles that give those forecasters higher weight can also be helpful.

Human judgement forecasting of COVID-19 in the UK.

Background: In the past, two studies found ensembles of human judgement forecasts of COVID-19 to show predictive performance comparable to ensembles of computational models, at least when predicting case incidences. We present a follow-up to a study conducted in Germany and Poland and investigate a novel joint approach to combine human judgement and epidemiological modelling. Methods: From May 24th to August 16th 2021, we elicited weekly one to four week ahead forecasts of cases and deaths from COVID-19 in the UK from a crowd of human forecasters. A median ensemble of all forecasts was submitted to the European Forecast Hub. Participants could use two distinct interfaces: in one, forecasters submitted a predictive distribution directly, in the other forecasters instead submitted a forecast of the effective reproduction number R t. This was then used to forecast cases and deaths using simulation methods from the EpiNow2 R package. Forecasts were scored using the weighted interval score on the original forecasts, as well as after applying the natural logarithm to both forecasts and observations. Results: The ensemble of human forecasters overall performed comparably to the official European Forecast Hub ensemble on both cases and deaths, although results were sensitive to changes in details of the evaluation. R t forecasts performed comparably to direct forecasts on cases, but worse on deaths. Self-identified "experts" tended to be better calibrated than "non-experts" for cases, but not for deaths. Conclusions: Human judgement forecasts and computational models can produce forecasts of similar quality for infectious disease such as COVID-19. The results of forecast evaluations can change depending on what metrics are chosen and judgement on what does or doesn't constitute a "good" forecast is dependent on the forecast consumer. Combinations of human and computational forecasts hold potential but present real-world challenges that need to be solved.

National and subnational short-term forecasting of COVID-19 in Germany and Poland during early 2021.

BACKGROUND: During the COVID-19 pandemic there has been a strong interest in forecasts of the short-term development of epidemiological indicators to inform decision makers. In this study we evaluate probabilistic real-time predictions of confirmed cases and deaths from COVID-19 in Germany and Poland for the period from January through April 2021. METHODS: We evaluate probabilistic real-time predictions of confirmed cases and deaths from COVID-19 in Germany and Poland. These were issued by 15 different forecasting models, run by independent research teams. Moreover, we study the performance of combined ensemble forecasts. Evaluation of probabilistic forecasts is based on proper scoring rules, along with interval coverage proportions to assess calibration. The presented work is part of a pre-registered evaluation study. RESULTS: We find that many, though not all, models outperform a simple baseline model up to four weeks ahead for the considered targets. Ensemble methods show very good relative performance. The addressed time period is characterized by rather stable non-pharmaceutical interventions in both countries, making short-term predictions more straightforward than in previous periods. However, major trend changes in reported cases, like the rebound in cases due to the rise of the B.1.1.7 (Alpha) variant in March 2021, prove challenging to predict. CONCLUSIONS: Multi-model approaches can help to improve the performance of epidemiological forecasts. However, while death numbers can be predicted with some success based on current case and hospitalization data, predictability of case numbers remains low beyond quite short time horizons. Additional data sources including sequencing and mobility data, which were not extensively used in the present study, may help to improve performance.

We compare forecasts of weekly case and death numbers for COVID-19 in Germany and Poland based on 15 different modelling approaches. These cover the period from January to April 2021 and address numbers of cases and deaths one and two weeks into the future, along with the respective uncertainties. We find that combining different forecasts into one forecast can enable better predictions. However, case numbers over longer periods were challenging to predict. Additional data sources, such as information about different versions of the SARS-CoV-2 virus present in the population, might improve forecasts in the future.

Correction: Evaluating epidemic forecasts in an interval format.

[This corrects the article DOI: 10.1371/journal.pcbi.1008618.].

Comparing human and model-based forecasts of COVID-19 in Germany and Poland.

Forecasts based on epidemiological modelling have played an important role in shaping public policy throughout the COVID-19 pandemic. This modelling combines knowledge about infectious disease dynamics with the subjective opinion of the researcher who develops and refines the model and often also adjusts model outputs. Developing a forecast model is difficult, resource- and time-consuming. It is therefore worth asking what modelling is able to add beyond the subjective opinion of the researcher alone. To investigate this, we analysed different real-time forecasts of cases of and deaths from COVID-19 in Germany and Poland over a 1-4 week horizon submitted to the German and Polish Forecast Hub. We compared crowd forecasts elicited from researchers and volunteers, against a) forecasts from two semi-mechanistic models based on common epidemiological assumptions and b) the ensemble of all other models submitted to the Forecast Hub. We found crowd forecasts, despite being overconfident, to outperform all other methods across all forecast horizons when forecasting cases (weighted interval score relative to the Hub ensemble 2 weeks ahead: 0.89). Forecasts based on computational models performed comparably better when predicting deaths (rel. WIS 1.26), suggesting that epidemiological modelling and human judgement can complement each other in important ways.

The United States COVID-19 Forecast Hub dataset.

Academic researchers, government agencies, industry groups, and individuals have produced forecasts at an unprecedented scale during the COVID-19 pandemic. To leverage these forecasts, the United States Centers for Disease Control and Prevention (CDC) partnered with an academic research lab at the University of Massachusetts Amherst to create the US COVID-19 Forecast Hub. Launched in April 2020, the Forecast Hub is a dataset with point and probabilistic forecasts of incident cases, incident hospitalizations, incident deaths, and cumulative deaths due to COVID-19 at county, state, and national, levels in the United States. Included forecasts represent a variety of modeling approaches, data sources, and assumptions regarding the spread of COVID-19. The goal of this dataset is to establish a standardized and comparable set of short-term forecasts from modeling teams. These data can be used to develop ensemble models, communicate forecasts to the public, create visualizations, compare models, and inform policies regarding COVID-19 mitigation. These open-source data are available via download from GitHub, through an online API, and through R packages.

Analysis of pesticide and persistent organic pollutant residues in German bats.

Bats are strictly protected throughout Europe. They are a highly diverse order of mammals in terms of body size, body weight, migratory behaviour, trophic niche specialisation and habitat use. The latter ranges from urban areas and arable land to forest. Due to their low reproductive rate, environmental stressors can have a major impact on bat populations. Pesticides in particular are discussed as an important driver of bat population declines. In this work, we analysed nearly 400 animals of five different species (Eptesicus serotinus, Myotis myotis, Nyctalus noctula, Pipistrellus pipistrellus, and Plecotus auritus) from all over Germany for residues of 209 pesticides and persistent organic pollutants. Residue analysis was conducted with a previously developed method using a miniaturized quick, easy, cheap, effective, rugged and safe (QuEChERS) sample preparation and gas chromatography-tandem mass spectrometry for separation and detection. These analytical data were statistically correlated with the known data on the animals (e.g. age, sex, place and time of finding). Of 209 pesticides and pollutants investigated, 28 compounds were detected, the most frequent being organochlorine insecticides and polychlorinated biphenyls, which have been banned for decades by the Stockholm Convention on Persistent Organic Pollutants. Detection of more recent pesticides that were legally used for the last decade included azole antifungals and the insecticide fipronil. The bats contained between four and 25 different residues. Statistical data analyses showed that the distribution throughout Germany is largely comparable, and single exceptions were observed in specialized ecological niches. In conclusion, this work provides the largest dataset of pesticide and persistent organic pollutant residues in European bats to date.

Evaluation of individual and ensemble probabilistic forecasts of COVID-19 mortality in the United States.

Short-term probabilistic forecasts of the trajectory of the COVID-19 pandemic in the United States have served as a visible and important communication channel between the scientific modeling community and both the general public and decision-makers. Forecasting models provide specific, quantitative, and evaluable predictions that inform short-term decisions such as healthcare staffing needs, school closures, and allocation of medical supplies. Starting in April 2020, the US COVID-19 Forecast Hub (https://covid19forecasthub.org/) collected, disseminated, and synthesized tens of millions of specific predictions from more than 90 different academic, industry, and independent research groups. A multimodel ensemble forecast that combined predictions from dozens of groups every week provided the most consistently accurate probabilistic forecasts of incident deaths due to COVID-19 at the state and national level from April 2020 through October 2021. The performance of 27 individual models that submitted complete forecasts of COVID-19 deaths consistently throughout this year showed high variability in forecast skill across time, geospatial units, and forecast horizons. Two-thirds of the models evaluated showed better accuracy than a naïve baseline model. Forecast accuracy degraded as models made predictions further into the future, with probabilistic error at a 20-wk horizon three to five times larger than when predicting at a 1-wk horizon. This project underscores the role that collaboration and active coordination between governmental public-health agencies, academic modeling teams, and industry partners can play in developing modern modeling capabilities to support local, state, and federal response to outbreaks.

Evaluating epidemic forecasts in an interval format.

For practical reasons, many forecasts of case, hospitalization, and death counts in the context of the current Coronavirus Disease 2019 (COVID-19) pandemic are issued in the form of central predictive intervals at various levels. This is also the case for the forecasts collected in the COVID-19 Forecast Hub (https://covid19forecasthub.org/). Forecast evaluation metrics like the logarithmic score, which has been applied in several infectious disease forecasting challenges, are then not available as they require full predictive distributions. This article provides an overview of how established methods for the evaluation of quantile and interval forecasts can be applied to epidemic forecasts in this format. Specifically, we discuss the computation and interpretation of the weighted interval score, which is a proper score that approximates the continuous ranked probability score. It can be interpreted as a generalization of the absolute error to probabilistic forecasts and allows for a decomposition into a measure of sharpness and penalties for over- and underprediction.

A marginal moment matching approach for fitting endemic-epidemic models to underreported disease surveillance counts.

Count data are often subject to underreporting, especially in infectious disease surveillance. We propose an approximate maximum likelihood method to fit count time series models from the endemic-epidemic class to underreported data. The approach is based on marginal moment matching where underreported processes are approximated through completely observed processes from the same class. Moreover, the form of the bias when underreporting is ignored or taken into account via multiplication factors is analyzed. Notably, we show that this leads to a downward bias in model-based estimates of the effective reproductive number. A marginal moment matching approach can also be used to account for reporting intervals which are longer than the mean serial interval of a disease. The good performance of the proposed methodology is demonstrated in simulation studies. An extension to time-varying parameters and reporting probabilities is discussed and applied in a case study on weekly rotavirus gastroenteritis counts in Berlin, Germany.

A spatio-temporal approach to short-term prediction of visceral leishmaniasis diagnoses in India.

BACKGROUND: The elimination programme for visceral leishmaniasis (VL) in India has seen great progress, with total cases decreasing by over 80% since 2010 and many blocks now reporting zero cases from year to year. Prompt diagnosis and treatment is critical to continue progress and avoid epidemics in the increasingly susceptible population. Short-term forecasts could be used to highlight anomalies in incidence and support health service logistics. The model which best fits the data is not necessarily most useful for prediction, yet little empirical work has been done to investigate the balance between fit and predictive performance. METHODOLOGY/PRINCIPAL FINDINGS: We developed statistical models of monthly VL case counts at block level. By evaluating a set of randomly-generated models, we found that fit and one-month-ahead prediction were strongly correlated and that rolling updates to model parameters as data accrued were not crucial for accurate prediction. The final model incorporated auto-regression over four months, spatial correlation between neighbouring blocks, and seasonality. Ninety-four percent of 10-90% prediction intervals from this model captured the observed count during a 24-month test period. Comparison of one-, three- and four-month-ahead predictions from the final model fit demonstrated that a longer time horizon yielded only a small sacrifice in predictive power for the vast majority of blocks. CONCLUSIONS/SIGNIFICANCE: The model developed is informed by routinely-collected surveillance data as it accumulates, and predictions are sufficiently accurate and precise to be useful. Such forecasts could, for example, be used to guide stock requirements for rapid diagnostic tests and drugs. More comprehensive data on factors thought to influence geographic variation in VL burden could be incorporated, and might better explain the heterogeneity between blocks and improve uniformity of predictive performance. Integration of the approach in the management of the VL programme would be an important step to ensuring continued successful control.

Probabilistic forecasting in infectious disease epidemiology: the 13th Armitage lecture.

Routine surveillance of notifiable infectious diseases gives rise to daily or weekly counts of reported cases stratified by region and age group. From a public health perspective, forecasts of infectious disease spread are of central importance. We argue that such forecasts need to properly incorporate the attached uncertainty, so they should be probabilistic in nature. However, forecasts also need to take into account temporal dependencies inherent to communicable diseases, spatial dynamics through human travel and social contact patterns between age groups. We describe a multivariate time series model for weekly surveillance counts on norovirus gastroenteritis from the 12 city districts of Berlin, in six age groups, from week 2011/27 to week 2015/26. The following year (2015/27 to 2016/26) is used to assess the quality of the predictions. Probabilistic forecasts of the total number of cases can be derived through Monte Carlo simulation, but first and second moments are also available analytically. Final size forecasts as well as multivariate forecasts of the total number of cases by age group, by district and by week are compared across different models of varying complexity. This leads to a more general discussion of issues regarding modelling, prediction and evaluation of public health surveillance data. Copyright © 2017 John Wiley & Sons, Ltd.

Using crowdsourced online experiments to study context-dependency of behavior.

We use Mechanical Turk's diverse participant pool to conduct online bargaining games in India and the US. First, we assess internal validity of crowdsourced experimentation through variation of stakes ($0, $1, $4, and $10) in the Ultimatum and Dictator Game. For cross-country equivalence we adjust the stakes following differences in purchasing power. Our marginal totals correspond closely to laboratory findings. Monetary incentives induce more selfish behavior but, in line with most laboratory findings, the particular size of a positive stake appears irrelevant. Second, by transporting a homogeneous decision situation into various living conditions crowdsourced experimentation permits identification of context effects on elicited behavior. We explore context-dependency using session-level variation in participants' geographical location, regional affluence, and local social capital. Across "virtual pools" behavior varies in the range of stake effects. We argue that quasi-experimental variation of the characteristics people bring to the experimental situation is the key potential of crowdsourced online designs.

Long-term air pollution exposure and lung function in 15 year-old adolescents living in an urban and rural area in Germany: The GINIplus and LISAplus cohorts.

INTRODUCTION: The impact of outdoor air pollution exposure on long-term lung development and potential periods of increased lung susceptibility remain unknown. This study assessed associations between early-life and current residential exposure to air pollution and lung function at 15-years of age in two German birth cohorts. METHODS: Fifteen year-old participants living in an urban and rural area in Germany underwent spirometry before and after bronchodilation (N=2266). Annual average (long-term) exposure to nitrogen dioxide (NO(2)), particles with aerodynamic diameters less than 2.5 µg/m(3) (PM2.5) mass and less than 10 µg/m(3) (PM(10)) mass, PM(2.5) absorbance and ozone were estimated to each participant's birth-, 10- and 15-year home address using land-use regression and kriging (ozone only) modelling. Associations between lung function variables and long-term pollutant concentrations were assessed using linear regression models adjusted for host and environmental covariates and recent short-term air pollution exposures. RESULTS: Long-term air pollution concentrations assessed to the birth-, 10- and 15-year home addresses were not associated with lung function variables, before and after bronchodilation, in the complete or study area specific populations. However, several lung function variables were negatively associated with long-term NO2 concentrations among asthmatics. For example, NO(2) estimated to the 15-year home address was associated with the ratio of forced expiratory volume in one second to forced vital capacity (FEV(1)/FVC) and the mean flow rate between 25% and 75% of FVC (-3.5%, 95% confidence interval [-6.0, -1.0] and -297.4 ml/s [-592.6, -2.1] per 5.9 µg/m(3) increase in NO(2), respectively). Nearly all effect estimates for the associations between the short-term PM(2.5) mass, PM(10) mass and ozone concentrations and the lung function variables were negative in the complete population. CONCLUSIONS: Early-life and current long-term air pollution exposures and lung function at the age of 15 years were not associated in the complete study population. Asthmatics may represent a vulnerable group.