  Ordinary differential equations (ODEs) are a mathematical model used in many
application areas such as climatology, bioinformatics, and chemical engineering
with its intuitive appeal to modeling. Despite ODE's wide usage in modeling,
the frequent absence of their analytic solutions makes it challenging to
estimate ODE parameters from the data, especially when the model has lots of
variables and parameters. This paper proposes a Bayesian ODE parameter
estimating algorithm which is fast and accurate even for models with many
parameters. The proposed method approximates an ODE model with a state-space
model based on equations of a numeric solver. It allows fast estimation by
avoiding computations of a complete numerical solution in the likelihood. The
posterior is obtained by a variational Bayes method, more specifically, the
approximate Riemannian conjugate gradient method (Honkela et al. 2010), which
avoids samplings based on Markov chain Monte Carlo (MCMC). In simulation
studies, we compared the speed and performance of the proposed method with
existing methods. The proposed method showed the best performance in the
reproduction of the true ODE curve with strong stability as well as the fastest
computation, especially in a large model with more than 30 parameters. As a
real-world data application, a SIR model with time-varying parameters was
fitted to the COVID-19 data. Taking advantage of the proposed algorithm, more
than 50 parameters were adequately estimated for each country.
