Variational Bayes method for ODE parameter estimation with application to time-varying SIR model for Covid-19 epidemic

Ordinary differential equation (ODE) is a mathematical model for dynamical systems. For its intuitive appeal to modelling, the ODE is used in many application areas such as climatology, bioinformatics, disease modelling and chemical engineering. Despite ODE's wide usage in modelling, there are difficulties in estimating ODE parameters from the data due to frequent absence of their analytic solutions. The ODE model typically requires enormous computing time and shows poor performance in estimation especially when the model has a lot 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 whole numerical solution in the likelihood. The posterior is obtained by a variational Bayes method, more specifically, the \textit{approximate Riemannian conjugate gradient method} \cite**{honkela2010approximate}, which avoids samplings based on Markov chain Monte Carlo (MCMC). In simulation studies we compared the speed and performance of 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 our proposed algorithm, 30 parameters were adequately fitted for each country.