Improving Parameter Estimation of Epidemic Models: Likelihood Functions and Kalman Filtering

Abstract

Projecting the course of infectious diseases and assessing the likely impact of policies to contain them require reliable estimates of the parameters in dynamic models of disease transmission. However, such estimation is difficult, especially for emerging diseases such as COVID-19, due to incomplete and delayed data, measurement error, and model specification error. Absent reliable estimation, model-based recommendations could be misleading. Here we conduct synthetic data experiments comparing the performance of various estimation methods for calibrating epidemic models, considering both process and measurement noise. We compare the performance of standard least squares against maximum likelihood methods including scaled-variance Gaussian, Poisson, and negative binomial likelihood functions, and assess the effectiveness of (extended) Kalman filtering. We explore the performance of these methods under different assumptions about data availability, from full information on the infection, symptom emergence, and removal rates to the more realistic setting where data are available only for symptom emergence. We find that widely-adopted naive estimation methods (least squares or incomplete Gaussian likelihood, without variance scaling and without Kalman filtering) perform poorly, especially in the estimation of confidence intervals. The negative binomial likelihood function performs well across a range of assumptions, scaling the error variance and Kalman filtering improve performance over the basic Gaussian estimator. We provide modelers with guidance to help choose estimation methods appropriate for their problem setting and data. Results may inform problems beyond the epidemic context, from models of innovation diffusion to other feedback-rich managerial and policy settings with limited data, process noise, and measurement error.

Keywords: parameter estimation, epidemic model, likelihood function, Kalman filtering, negative binomial