Estimating the size of undetected cases of the SARS-CoV-2 outbreak in Europe: An upper bound estimator

Background While the number of detected SARS-CoV-2 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 SARS-CoV-2 (detected and undetected) infections in several European Countries. The question being asked is: How many cases have actually occurred? Methods We 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. Results We 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. Conclusions Many 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.