Novel Estimators for the Number of Susceptible Individuals in SIR Models of Infectious Epidemics

Abstract

Accurate estimation of model parameters early in infectious epidemics may improve planning and resource allocation in mitigating the adverse consequences on affected populations. By applying the Peano-Baker series formula and the Cauchy repeated integral formula, we present the development of three novel estimators which facilitate the estimation of the number of susceptible people as a function of time S(t) for an SIR model of an infectious epidemic. Association of these three estimators by combining them produces new estimators. We present the case for the new estimator, Ŝ<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1,3</inf>, derived from the association of two of the original estimators, Ŝ<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> and Ŝ<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</inf> to estimate S(0). This produces an estimate based on the history of net infection rate, I′(t), from time 0 to t. By assuming parameters for the spread of COVID-19 in Wuhan from the literature, we run numerical simulations starting with an infection rate I′(t) and adding filtered Gaussian noise. Discretization produces inaccuracy in bias and variance, however Ŝ<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> and Ŝ<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</inf> yield accurate figures for S(t), despite noise contamination. Ŝ<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1,3</inf> also yields accurate figures for S(0), with improvements as more observations are accumulated. We plan to investigate other novel estimators and further study their performance with real-world data.