Parameter Inference and Model Selection for Differential Equation Models with Applications

Abstract

Dynamic systems are commonly modelled by differential equations (DEs) in epidemiology and biology, among other fields. The parameters in the DEs are often of scientific interest and required for estimation, given a set of noisy observations. The first and oldest general class of methods for the parameter inference problem in DEs is based on numerical solvers. As a preliminary study, we conduct a comparative study of compartmental models for COVID-19 transmission using such numerical solver-based methods. However, this class of methods can be computationally intensive and may only converge to the local optima due to the sensitivity of the numerical solution to the parameters and initial conditions. This thesis begins by presenting this study, which highlights these limitations and motivates the methodological developments that follow.

To address these challenges, Gaussian process-based methods serve as an alternative that bypass the need for numerical solvers. In particular, the recent manifold-constrained Gaussian process inference (MAGI) method demonstrated accurate estimation and fast computational speed. However, the original MAGI method is limited to ordinary differential equations (ODEs), which are inadequate for some dynamic systems, calling for more complex or flexible structures in the specification of the DE model. Motivated by this, this thesis extends the framework of MAGI to facilitate inference for three common but challenging contexts, including (i) delayed differential equations, where system components exhibit time delays in their responses, (ii) mixed-effects ODEs, where experimental data consist of time-course observations on multiple subjects from a population, and (iii) selection of the most appropriate ODE model from a set of candidate models, where there is no true underlying model. The complex structures of these DEs introduce inferential and computational burdens and we address them in this thesis, along with computational and theoretical justifications. We illustrate the efficacy of our methodologies through simulated and real-world applications.