New Approach to the SIR Inversion Problem: From the 1905-1906 Plague Outbreak in the Isle of Bombay To the 2021-2022 Omicron Surge in New York City

Abstract

AbstractWe describe a novel approach to recovering the underlying parameters of the SIR dynamic epidemic model from observed data on case incidence or deaths. We formulate a discrete-time approximation to the original continuous-time model and search for the parameter vector that minimizes the standard least squares criterion function. We show that the gradient vector and matrix of second-order derivatives of the criterion function with respect to the parameters adhere to their own systems of difference equations and thus can be exactly calculated iteratively. Applying our new approach, we estimate four-parameter SIR models on two datasets: (1) daily reported cases of COVID-19 during the SARS-CoV-2 Omicron/BA.1 surge of December 2021 - March 2022 in New York City; and (2) weekly deaths from a plague outbreak on the Isle of Bombay during December 1905 - July 1906, originally studied by Kermack and McKendrick in their now-classic 1927 paper. The estimated parameters from the COVID-19 data suggest a duration of persistent infectivity beyond that reported in small-scale clinical studies of mostly symptomatic subjects. The estimated parameters from the plague data suggest that the Bombay outbreak was in fact driven by pneumonic rather than bubonic plague.