Identifiability, Observability, Uncertainty and Bayesian System Identification of Epidemiological Models

TL;DR

The study addresses identifiability, observability, and uncertainty in deterministic and Chain-Binomial SIR, SEIR, and SEIAR epidemiological models, with a focus on moderately sized, homogeneous populations. It combines differential-algebraic identifiability analysis via DAISY, structural and local observability analysis using Lie derivatives, and Bayesian inference using Sequential Monte Carlo and Markov Chain Monte Carlo implemented in a custom C++ library alongside PyMC3, to compare parameter distributions across models and dispersion settings. Key findings show that SIR and SEIR are structurally identifiable and observable under mild conditions, while SEIAR generally remains unidentifiable and unobservable; dispersion has small effects in large, homogeneous populations but becomes more important in small populations or near outbreak initiation, where stochasticity dominates and SEIAR may fail to converge to true parameters. The work provides a computationally efficient inference framework and highlights the need for multiple model regimes (deterministic vs stochastic) and targeted data collection to ensure reliable parameter retrieval and robust control implications in epidemiological forecasting.

Abstract

In this project, identifiability, observability and uncertainty properties of the deterministic and Chain Binomial stochastic SIR, SEIR and SEIAR epidemiological models are studied. Techniques for modeling overdispersion are investigated and used to compare simulated trajectories for moderately sized, homogenous populations. With the chosen model parameters overdispersion was found to have small impact, but larger impact on smaller populations and simulations closer to the initial outbreak of an epidemic. Using a software tool for model identifiability and observability (DAISY[Bellu et al. 2007]), the deterministic SIR and SEIR models was found to be structurally identifiable and observable under mild conditions, while SEIAR in general remains structurally unidentifiable and unobservable. Sequential Monte Carlo and Markov Chain Monte Carlo methods were implemented in a custom C++ library and applied to stochastic SIR, SEIR and SEIAR models in order to generate parameter distributions. With the chosen model parameters overdispersion was found to have a small impact on parameter distributions for SIR and SEIR models. For SEIAR, the algorithm did not converge around the true parameters of the deterministic model. The custom C++ library was found to be computationally efficient, and is very likely to be used in future projects.

Figures (17)

• Figure 1: Comparison of overdispersion using PMFs for the Poisson, Negative Binomial, Binomial and Beta Binomial distributions. The dispersion is linearly spaced with additional $\nu = [1, \dots, 8]$ standard deviations compared to the base distributions.
• Figure 2: Comparison of dispersed trajectories for SIR-model ($90$-percentiles from both sides of the stochastic simulation mean). Increasing $\nu$ gradually fades in to gray. 90-percentiles for undispersed trajectory is shown in dotted lines.
• Figure 3: Comparison of dispersed trajectories for SEIR-model ($90$-percentiles from both sides of the stochastic simulation mean). Increasing $\nu$ gradually fades in to gray. 90-percentiles for undispersed trajectory is shown in dotted lines.
• Figure 4: Comparison of dispersed trajectories for SEIAR-model ($90$-percentiles from both sides of the stochastic simulation mean). Increasing $\nu$ gradually fades in to gray. 90-percentiles for undispersed trajectory is shown in dotted lines.
• Figure 5: Comparison of dispersed trajectories for SIR-model ($90$-percentiles from both sides of the stochastic simulation mean). Increasing $\nu$ gradually fades in to gray. 90-percentiles for undispersed trajectory is shown in dotted lines.