  This article considers estimation of constant and time-varying coefficients
in nonlinear ordinary differential equation (ODE) models where analytic
closed-form solutions are not available. The numerical solution-based nonlinear
least squares (NLS) estimator is investigated in this study. A numerical
algorithm such as the Runge--Kutta method is used to approximate the ODE
solution. The asymptotic properties are established for the proposed estimators
considering both numerical error and measurement error. The B-spline is used to
approximate the time-varying coefficients, and the corresponding asymptotic
theories in this case are investigated under the framework of the sieve
approach. Our results show that if the maximum step size of the $p$-order
numerical algorithm goes to zero at a rate faster than $n^{-1/(p\wedge4)}$, the
numerical error is negligible compared to the measurement error. This result
provides a theoretical guidance in selection of the step size for numerical
evaluations of ODEs. Moreover, we have shown that the numerical solution-based
NLS estimator and the sieve NLS estimator are strongly consistent. The sieve
estimator of constant parameters is asymptotically normal with the same
asymptotic co-variance as that of the case where the true ODE solution is
exactly known, while the estimator of the time-varying parameter has the
optimal convergence rate under some regularity conditions. The theoretical
results are also developed for the case when the step size of the ODE numerical
solver does not go to zero fast enough or the numerical error is comparable to
the measurement error. We illustrate our approach with both simulation studies
and clinical data on HIV viral dynamics.
