A simple modification to the classical SIR model to estimate the proportion of under-reported infections using case studies in flu and COVID-19.

Background: Under-reporting and, thus, uncertainty around the true incidence of health events is common in all public health reporting systems. While the problem of under-reporting is acknowledged in epidemiology, the guidance and methods available for assessing and correcting the resulting bias are obscure. Objective: We aim to design a simple modification to the Susceptible - Infected - Removed (SIR) model for estimating the fraction or proportion of reported infection cases. Methods: The suggested modification involves rescaling of the classical SIR model producing its mathematically equivalent version with explicit dependence on the reporting parameter (true proportion of cases reported). We justify the rescaling using the phase plane analysis of the SIR model system and show how this rescaling parameter can be estimated from the data along with the other model parameters. Results: We demonstrate how the proposed method is cross-validated using simulated data with known disease cases and then apply it to two empirical reported data sets to estimate the fraction of reported cases in Missoula County, Montana, USA, using: (1) flu data for 2016-2017 and (2) COVID-19 data for fall of 2020. Conclusions: We establish with the simulated and COVID-19 data that when most of the disease cases are presumed reported, the value of the additional reporting parameter in the modified SIR model is close or equal to one, so that the original SIR model is appropriate for data analysis. Conversely, the flu example shows that when the reporting parameter is close to zero, the original SIR model is not accurately estimating the usual rate parameters, and the re-scaled SIR model should be used. This research demonstrates the role of under-reporting of disease data and the importance of accounting for under-reporting when modeling simulated, endemic, and pandemic disease data. Correctly reporting the "true" number of disease cases will have downstream impacts on predictions of disease dynamics. A simple parameter adjustment to the SIR modeling framework can help alleviate bias and uncertainty around crucial epidemiological metrics (e.g.: basic disease reproduction number) and public health decision making.