  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 two examples:
modelling influenza using a canonical SIR model, and the modelling of COVID-19
pandemic using a multi-type SEIR model.
