Gaussian process priors with Markov properties for effective reproduction number inference

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 Matérn 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.