An approximate diffusion process for environmental stochasticity in infectious disease transmission modelling

Modelling the transmission dynamics of an infectious disease is a complex task. Not only it is difficult to accurately model the inherent non-stationarity and heterogeneity of transmission, but it is nearly impossible to describe, mechanistically, changes in extrinsic environmental factors including public behaviour and seasonal fluctuations. An elegant approach to capturing environmental stochasticity is to model the force of infection as a stochastic process. However, inference in this context requires solving a computationally expensive “missing data” problem, using data-augmentation techniques. We propose to model the time-varying transmission-potential as an approximate diffusion process using a path-wise series expansion of Brownian motion. This approximation replaces the “missing data” imputation step with the inference of the expansion coefficients: a simpler and computationally cheaper task. We illustrate the merit of this approach through three examples: modelling influenza using a canonical SIR model, capturing seasonality using a SIRS model, and the modelling of COVID-19 pandemic using a multi-type SEIR model.Author summary: Over the course of an epidemic, the ability to transmit infection is a key parameter in the epidemic models that seek to quantify and forecast epidemic burden. This parameter will evolve considerably over time due to the emergence of new variants, changing governmental advice and access to testing and possible immunisation. Ignoring these factors can lead to models that produce overly confident estimates and biased predictions. In this paper we propose a computationally efficient method to quantifying this uncertainty in such a way that makes no assumptions about the timing and the magnitude of the impacts of these external stimuli influencing transmission. Our approach relies on the application of an approximate diffusion process to characterise changes in transmission over time. This results in a sizeable computational advantage over previous methods that use a true diffusion process, while retaining a similar quality of estimation. Our methodology would be very valuable to modellers attempting to monitor epidemics in real time.