Parameter Estimation for Differential Equation Models Using a Framework of Measurement Error in Regression Models

Differential equation (DE) models are widely used in many scientific fields, including engineering, physics, and biomedical sciences. The so-called forward problem, the problem of simulations and predictions of state variables for given parameter values in the DE models. has been extensively studied by mathematicians, physicists, engineers, and other scientists. However, the inverse problem the problem of parameter estimation based on the measurements of output variables, has not been well explored using modern statistical method, although some least squares-based approaches have been proposed and studied. In this article we propose parameter estimation methods for ordinary differential equation (ODE) models based on the local smoothing approach and a pseudo-least squares (PsLS) principle under a framework of measurement error in regression models. The asymptotic properties of the Proposed PsLS estimator are established. We also compare the PsLS method to the corresponding simulation-extrapolation) (SIMEX) method and evaluate their finite-sample performances via simulation studies. We illustrate the proposed approach using an application example from an HIV dynamic study.