Gaussian process priors with Markov properties for effective reproduction number inference

TL;DR

This paper introduces Gaussian Markov process priors, especially the Integrated Brownian Motion, for more accurate and computationally efficient inference of the effective reproduction number in infectious disease modeling.

Contribution

The paper proposes novel Gaussian Markov process priors, including IBM, for $R_t$ inference, improving over traditional Gaussian random walk priors in epidemiological analysis.

Findings

IBM matches or exceeds other priors in simulated data

IBM produces epidemiologically reasonable results on SARS-CoV-2 data

Proposed priors are computationally efficient and easy to implement

Abstract

Many quantities characterizing infectious disease outbreaks - like the effective reproduction number ($R_t$), defined as the average number of secondary infections a newly infected individual will cause over the course of their infection - need to be modeled as time-varying parameters. It is common practice to use Gaussian random walks as priors for estimating such functions in Bayesian analyses of pathogen surveillance data. In this setting, however, the random walk prior may be too permissive, as it fails to capture prior scientific knowledge about the estimand and results in high posterior variance. We propose several Gaussian Markov process priors for $R_t$ inference, including the Integrated Brownian Motion (IBM), which can be represented as a Markov process when augmented with its corresponding Brownian Motion component, and is therefore computationally efficient and simple to implement and tune. We use simulated outbreak data to compare the performance of these proposed priors with the Gaussian random walk prior and another state-of-the-art Gaussian process prior based on an approximation to a Mat\'ern covariance function. We find that IBM can match or exceed the performance of other priors, and we show that it produces epidemiologically reasonable and precise results when applied to county-level SARS-CoV-2 data.