A renewal-equation approach to estimating <i> R _t </i> and infectious disease case counts in the presence of reporting delays

During infectious disease outbreaks, delays in case reporting mean that the time series of cases is unreliable, particularly for those cases occurring most recently. This means that real-time estimates of the time-varying reproduction number, [Formula: see text], are often made using a time series of cases only up until a time period sufficiently far in the past that there is some confidence in the case counts. This means that the most recent [Formula: see text] estimates are usually out of date, inducing lags in the response of public health authorities. Here, we introduce an [Formula: see text] estimation method, which makes use of the retrospective updates to case time series which happen as more cases that occurred historically enter the health system; these data encode within them information about the reporting delays, which our method also estimates. These estimates, in turn, allow us to estimate the true count of cases occurring most recently allowing up-to-date estimates of [Formula: see text]. Our method simultaneously estimates the reporting delays, true historical case counts and [Formula: see text] in a single Bayesian framework, allowing the uncertainty in each of these quantities to be accounted for. We apply our method to both simulated and real outbreak data, which shows that the method substantially improves upon naive estimates of [Formula: see text] which do not account for reporting delays. Our method is available in an open-source fully tested R package, <i>incidenceinflation</i>. Our research highlights the value of keeping historical time series of cases since changes to these data can help to characterize nuisance processes, such as reporting delays, which allow these to be accounted for when estimating key epidemic quantities.This article is part of the theme issue 'Uncertainty quantification for healthcare and biological systems (Part 1)'.