  Identification of parameters in ordinary differential equations (ODEs) is an
important and challenging task when modeling dynamic systems in biomedical
research and other scientific areas, especially with the presence of
time-varying parameters. This article proposes a fast and accurate method,
TVMAGI (Time-Varying MAnifold-constrained Gaussian process Inference), to
estimate both time-constant and time-varying parameters in the ODE using noisy
and sparse observation data. TVMAGI imposes a Gaussian process model over the
time series of system components as well as time-varying parameters, and
restricts the derivative process to satisfy ODE conditions. Consequently,
TVMAGI completely bypasses numerical integration and achieves substantial
savings in computation time. By incorporating the ODE structures through
manifold constraints, TVMAGI enjoys a principled statistical construct under
the Bayesian paradigm, which further enables it to handle systems with missing
data or unobserved components. The Gaussian process prior also alleviates the
identifiability issue often associated with the time-varying parameters in ODE.
Unlike existing approaches, TVMAGI can be applied to general nonlinear systems
without specific structural assumptions. Three simulation examples, including
an infectious disease compartmental model, are provided to illustrate the
robustness and efficiency of our method compared with numerical integration and
Bayesian filtering methods.
