A computationally efficient framework for realistic epidemic modelling through Gaussian Markov random fields

TL;DR

This work tackles the challenge of environmental stochasticity in epidemic nowcasting by replacing ad hoc stochastic transmission with a Gaussian Markov Random Field over time and population strata. The authors develop a bespoke Bayesian framework and a tailored MCMC sampler that efficiently handles the high-dimensional latent states and parameters, enabling accurate nowcasting and forecasting in SEIR-type models. Through simulation and a UK Covid-19 case study, the approach demonstrates improved predictive calibration and narrower prediction intervals compared with models using independent, piecewise-constant stochastic processes. The framework is computationally scalable and adaptable, offering practical impact for real-time epidemic monitoring and decision-making.

Abstract

We tackle limitations of ordinary differential equation-driven Susceptible-Infections-Removed (SIR) models and their extensions that have recently be employed for epidemic nowcasting and forecasting. In particular, we deal with challenges related to the extension of SIR-type models to account for the so-called \textit{environmental stochasticity}, i.e., external factors, such as seasonal forcing, social cycles and vaccinations that can dramatically affect outbreaks of infectious diseases. Typically, in SIR-type models environmental stochasticity is modelled through stochastic processes. However, this stochastic extension of epidemic models leads to models with large dimension that increases over time. Here we propose a Bayesian approach to build an efficient modelling and inferential framework for epidemic nowcasting and forecasting by using Gaussian Markov random fields to model the evolution of these stochastic processes over time and across population strata. Importantly, we also develop a bespoke and computationally efficient Markov chain Monte Carlo algorithm to estimate the large number of parameters and latent states of the proposed model. We test our approach on simulated data and we apply it to real data from the Covid-19 pandemic in the United Kingdom.

Figures (7)

• Figure 1: Schematic diagram of simple SEIR model
• Figure 2: Posterior means (Black dashed lines) and $95\%$ credible intervals of predictive distributions for the total number of deaths for the data generation mechanism A. The red dots represent the simulated number of daily deaths.
• Figure 3: Posterior means (black dashed lines) and $95\%$ credible intervals of the posterior predictive distribution for the total number of deaths in each of the $14$ out-of-sample days. The red dots represent the true number of daily deaths. Top: simulated scenario A with correlated random walk processes across regions. Bottom: simulated scenario B with independent random walk.
• Figure 4: Mean interval scores across seven regions for scenario A where the simulated random walk processes are independent across the regions. Left: comparison for the models (a) with daily changes in the states of the random walk process $\tilde{\beta}$ and the region-specific random walk process are correlated, (b) with daily changes $\tilde{\beta}$ and the region-specific random walk process are independent and (c) with bi-weekly changes in the states of $\tilde{\beta}$ and independent random walk processes across regions. Right: interval scores only for models (a) and (b).
• Figure 5: Red solid lines: $95\%$ credible intervals and means of the posterior distribution for the number of deaths estimated by using data from 2020-02-16 to 2020-05-27 for the whole England as well as for the regions South West, South East and London. Blue dotted lines: $95\%$ credible intervals and means for the posterior predictive distribution of the number of deaths from 2020-02-17 to 2020-06-16. Black dots: true number of deaths. The gray dashed line indicates the last day of the data used for the estimation of the model.