  Ordinary differential equations (ODEs) are widely used to model dynamical
behavior of systems. It is important to perform identifiability analysis prior
to estimating unknown parameters in ODEs (a.k.a. inverse problem), because if a
system is unidentifiable, the estimation procedure may fail or produce
erroneous and misleading results.
  Although several qualitative identifiability measures have been proposed,
much less effort has been given to developing \emph{quantitative} (continuous)
scores that are robust to uncertainties in the data, especially for those cases
in which the data are presented as a single trajectory beginning with one
initial value.
  In this paper, we first derived a closed-form representation of linear ODE
systems that are not identifiable based on a single trajectory. This
representation helps researchers design practical systems and choose the right
prior structural information in practice. Next, we proposed several
quantitative scores for identifiability analysis in practice. In simulation
studies, the proposed measures outperformed the main competing method
significantly, especially when noise was presented in the data. We also
discussed the asymptotic properties of practical identifiability for
high-dimensional ODE systems and conclude that, without additional prior
information, many random ODE systems are practically unidentifiable when the
dimension approaches infinity.
