A major open question, affecting the decisions of policy makers, is the estimation of the true number of Covid-19 infections. Most of them are undetected, because of a large number of asymptomatic cases. We provide an efficient, easy to compute and robust lower bound estimator for the number of undetected cases. Methods: A modified version of the Chao estimator is proposed, based on the cumulative time-series distributions of cases and deaths. Heterogeneity has been addressed by assuming a geometrical distribution underlying the data generation process. An (approximated) analytical variance of the estimator has been derived to compute reliable confidence intervals at 95% level. Results: A motivating application to the Austrian situation is provided and compared with an independent and representative study on prevalence of Covid-19 infection. Our estimates match well with the results from the independent prevalence study, but the capture-recapture estimate has less uncertainty involved as it is based on a larger sample size. Results from other European countries are mentioned in the discussion. The estimated ratio of the total estimated cases to the observed cases is around the value of 2.3 for all the analyzed countries. Conclusions: The proposed method answers to a fundamental open question: "How many undetected cases are going around?". CR methods provide a straightforward solution to shed light on undetected cases, incorporating heterogeneity that may arise in the probability of being detected.

A major open question, affecting the policy makers decisions, is the estimation of the true size of COVID-19 infections. Most of them are undetected, because of a large number of asymptomatic cases. We provide an efficient, easy to compute and robust lower bound estimator for the number of undetected cases. A "modified" version of the Chao estimator is proposed, based on the cumulative time-series distribution of cases and deaths.

Estimating the size of an elusive target population is of prominent interest in many areas in the life and social sciences. Our aim is to provide an efficient and workable method to estimate the unknown population size, given the frequency distribution of counts of repeated identifications of units of the population of interest. This counting variable is necessarily zero-truncated, since units that have never been identified are not in the sample. We consider several applications: clinical medicine, where interest is in estimating patients with adenomatous polyps which have been overlooked by the diagnostic procedure; drug user studies, where interest is in estimating the number of hidden drug users which are not identified; veterinary surveillance of scrapie in the UK, where interest is in estimating the hidden amount of scrapie; and entomology and microbial ecology, where interest is in estimating the number of unobserved species of organisms. In all these examples, simple models such as the homogenous Poisson are not appropriate since they do not account for present and latent heterogeneity. The Poisson-Gamma (negative binomial) model provides a flexible alternative and often leads to well-fitting models. It has a long history and was recently used in the development of the Chao-Bunge estimator. Here we use a different property of the Poisson-Gamma model: if we consider ratios of neighboring Poisson-Gamma probabilities, then these are linearly related to the counts of repeated identifications.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS436 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

BackgroundWhile the number of detected COVID-19 infections are widely available, an understanding of the extent of undetected cases is urgently needed for an effective tackling of the pandemic. The aim of this work is to estimate the true number of COVID-19 (detected and undetected) infections in several European countries. The question being asked is: How many cases have actually occurred?MethodsWe propose an upper bound estimator under cumulative data distributions, in an open population, based on a day-wise estimator that allows for heterogeneity. The estimator is data-driven and can be easily computed from the distributions of daily cases and deaths. Uncertainty surrounding the estimates is obtained using bootstrap methods.ResultsWe focus on the ratio of the total estimated cases to the observed cases at April 17th. Differences arise at the country level, and we get estimates ranging from the 3.93 times of Norway to the 7.94 times of France. Accurate estimates are obtained, as bootstrap-based intervals are rather narrow.ConclusionsMany parametric or semi-parametric models have been developed to estimate the population size from aggregated counts leading to an approximation of the missed population and/or to the estimate of the threshold under which the number of missed people cannot fall (i.e. a lower bound). Here, we provide a methodological contribution introducing an upper bound estimator and provide reliable estimates on the dark number, i.e. how many undetected cases are going around for several European countries, where the epidemic spreads differently.

While the number of detected Monkeypox infections are widely available, an understanding of the extent of undetected cases is urgently needed for an effective tackling of its spread. The aim of this study is to estimate the true number of Monkeypox (detected and undetected) infections in most affected countries. The question being asked is: How many cases have actually occurred? We propose a lower bound estimator for the true number of Monkeypox cases. The estimator is data‐driven and can be easily computed from the cumulative distributions of weekly cases. We focused on the ratio of the total estimated cases to the observed cases on July 31, 2022: The proportion of undetected cases was relevant in all countries, with countries whose estimated true number of infections could be more than three times the observed one. We provided a practical contribution to the understanding of the current Monkeypox wave and reliable estimates on how many undetected cases are going around in several countries, where the epidemic spreads differently.