Efficient Bayesian inference of fully stochastic epidemiological models with applications to COVID-19

Abstract

Epidemiological forecasts are beset by uncertainties about the underlying epidemiological processes, and the surveillance process through which data are acquired. We present a Bayesian inference methodology that quantifies these uncertainties, for epidemics that are modelled by (possibly) non-stationary, continuous-time, Markov population processes. The efficiency of the method derives from a functional central limit theorem approximation of the likelihood, valid for large populations. We demonstrate the methodology by analysing the early stages of the COVID-19 pandemic in the UK, based on age-structured data for the number of deaths. This includes maximum a posteriori estimates, Markov chain Monte Carlo sampling of the posterior, computation of the model evidence, and the determination of parameter sensitivities via the Fisher information matrix. Our methodology is implemented in PyRoss, an open-source platform for analysis of epidemiological compartment models.

Bayesian inference across multiple models suggests a strong increase in lethality of COVID-19 in late 2020 in the UK

“…Using our software package PyRoss, we can calculate the logarithmic likelihood of the observed data for each of the model variants and for any choice of the model parameters and initial conditions [ 6 ]. This computation is based on the inherent stochasticity specified for the model.…”

“…This could be an indication that individuals count as infected for longer in our models than they test positive in a PCR test. Also, the fixed estimate we use for the age-dependent fraction of asymptomatic infections may be incorrect (based on early data [ 21 ], as described in our paper [ 6 ]).…”

“…For a complete analysis of the model evidence, a sampling of the posterior over the parameter space would be necessary, e.g. using Markov Chain Monte Carlo (MCMC) techniques [ 6 ]. Since the evaluation of the likelihood for the given amount of data and complexity of the model is computationally rather expensive, we refrain from the MCMC sampling and instead work with a local Gaussian approximation of the posterior around the MAP parameters, Here, H is the Hessian matrix of the (negative) log-posterior at θ = θ *, calculated using finite differences.…”

“…In order to overcome these challenges while still focusing on (nationally) aggregated data for cases and mortalities, we choose to analyse such data in the context of well-mixed compartmented models, whose complexity is adjustable for this purpose [ 5 , 6 ]. These models take into account not only the stochastic dynamics of infection and progression of the disease, but also the process though which individuals get diagnosed.…”

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“…For a complete analysis of the model evidence, a sampling of the posterior over the parameter space would be necessary, e.g. using Markov Chain Monte Carlo (MCMC) techniques [6]. Since the evaluation of the likelihood for the given amount of data and complexity of the model is computationally rather expensive, we refrain from the MCMC sampling and instead work with a local Gaussian approximation of the posterior around the MAP parameters,…”