Efficient inference and identifiability analysis for differential equation models with random parameters

Heterogeneity is a dominant factor in the behaviour of many biological processes. Despite this, it is common for mathematical and statistical analyses to ignore biological heterogeneity as a source of variability in experimental data. Therefore, methods for exploring the identifiability of models that explicitly incorporate heterogeneity through variability in model parameters are relatively underdeveloped. We develop a new likelihood-based framework, based on moment matching, for inference and identifiability analysis of differential equation models that capture biological heterogeneity through parameters that vary according to probability distributions. As our novel method is based on an approximate likelihood function, it is highly flexible; we demonstrate identifiability analysis using both a frequentist approach based on profile likelihood, and a Bayesian approach based on Markov-chain Monte Carlo. Through three case studies, we demonstrate our method by providing a didactic guide to inference and identifiability analysis of hyperparameters that relate to the statistical moments of model parameters from independent observed data. Our approach has a computational cost comparable to analysis of models that neglect heterogeneity, a significant improvement over many existing alternatives. We demonstrate how analysis of random parameter models can aid better understanding of the sources of heterogeneity from biological data.Author summary: Heterogeneity is a dominant factor in the behaviour of many biological and biophysical processes, and is often a primary source of the variability evident in experimental data. Despite this, it is relatively rare for mathematical models of biological systems to incorporate variability in model parameters as a source of noise. Therefore, methods for analysing whether model parameters and sources of variability are identifiable from commonly reported experimental data are relatively underdeveloped. As we demonstrate, such identifiability analysis is vital for model selection, experimental design, and gaining biological insights. In this work, we develop a fast, approximate framework for model calibration and identifiability analysis of mathematical models that incorporate biological heterogeneity through random parameters. Our method is highly flexible, and can be employed in both frequentist and Bayesian inference paradigms. Compared to alternative approaches, our approach is computationally efficient, with a computational cost comparable to analysis of standard models that neglect parameter variability.