On Stable Parameter Estimation and Forecasting in Epidemiology by the Levenberg-Marquardt Algorithm with Broyden's Rank-one Updates for the Jacobian Operator.

Abstract

Rigorously calibrating dynamic models with time-series data can pose roadblocks. Oftentimes, the problem is ill-posed and one has to rely on appropriate regularization techniques to ensure stable parameter estimation from which forward projections with quantified uncertainty could be generated. If the inversion procedure is cast as nonlinear least squares constrained by a system of nonlinear differential equations, then the system has to be solved numerically at every step of the iterative process and the corresponding parameter-to-data map cannot be used to evaluate the Fréchet derivative analytically. To address challenges related to both instability and Jacobian approximation, we propose a novel regularized Levenberg-Marquardt algorithm with iterative rank-one updates for computation of the derivative operator. In order to test the efficiency of this scheme, we conduct numerical experiments using a mathematical model of infectious disease transmission and real incidence data of historic measles outbreaks in the UK.