title: Numerical discretization-based estimation methods for ordinary differential equation models via penalized spline smoothing with applications in biomedical research.
abstract: Differential equations are extensively used for modeling dynamics of physical processes in many scientific fields such as engineering, physics, and biomedical sciences. Parameter estimation of differential equation models is a challenging problem because of high computational cost and high-dimensional parameter space. In this article, we propose a novel class of methods for estimating parameters in ordinary differential equation (ODE) models, which is motivated by HIV dynamics modeling. The new methods exploit the form of numerical discretization algorithms for an ODE solver to formulate estimating equations. First, a penalized-spline approach is employed to estimate the state variables and the estimated state variables are then plugged in a discretization formula of an ODE solver to obtain the ODE parameter estimates via a regression approach. We consider three different order of discretization methods, Euler's method, trapezoidal rule, and Runge-Kutta method. A higher-order numerical algorithm reduces numerical error in the approximation of the derivative, which produces a more accurate estimate, but its computational cost is higher. To balance the computational cost and estimation accuracy, we demonstrate, via simulation studies, that the trapezoidal discretization-based estimate is the best and is recommended for practical use. The asymptotic properties for the proposed numerical discretization-based estimators are established. Comparisons between the proposed methods and existing methods show a clear benefit of the proposed methods in regards to the trade-off between computational cost and estimation accuracy. We apply the proposed methods t an HIV study to further illustrate the usefulness of the proposed approaches.
authors.affiliation: Department of Biostatistics and Computational Biology, University of Rochester, 601 Elmwood Avenue, Box 630, Rochester, New York 14642, USA. hwu@bst.rochester.edu
authors.forename: Hulin
authors.initials: H
authors.lastname: Wu
authors.forename: Hongqi
authors.initials: H
authors.lastname: Xue
authors.forename: Arun
authors.initials: A
authors.lastname: Kumar
mesh_terms: D000465:Algorithms; D035843:Biomedical Research / Q000706:statistics & numerical data; D001699:Biometry / Q000379:methods*; D003198:Computer Simulation; D015658:HIV Infections / Q000188:drug therapy / Q000821:virology; D015497:HIV-1; D006801:Humans; D008954:Models, Biological; D015233:Models, Statistical; D017711:Nonlinear Dynamics; D012044:Regression Analysis; D019562:Viral Load