Gaussian approximations for fast Bayesian inference of partially observed branching processes with applications to epidemiology

We consider the problem of inference for the states and parameters of a continuous-time multitype branching process from partially observed time series data. Exact inference for this class of models, typically using sequential Monte Carlo, can be computationally challenging when the populations that are being modelled grow exponentially or the time series is long. Instead, we derive a Gaussian approximation for the transition function of the process that leads to a Kalman filtering algorithm that runs in a time independent of the population sizes. We also develop a hybrid approach for when populations are smaller and the approximation is less applicable. We investigate the performance of our approximation and algorithms to both a simple and a complex epidemic model, finding good adherence to the true posterior distributions in both cases with large computational speed-ups in most cases. We also apply our method to a COVID-19 dataset with time dependent parameters where exact methods are intractable due to the population sizes involved.