Fast likelihood calculations for emerging epidemics

Abstract Statistical inference for epidemic outbreaks is often complicated by only partial observation of the epidemic process. Recently in Ball and Neal (Adv Appl Probab 55:895-926, 2023) the distribution of the number of infectives (individuals alive) given only the times of removals (death) in a Markovian SIR epidemic (time-inhomogeneous birth–death process) was derived. We show that this allows us to derive an explicit expression for the likelihood of the observed inter-removal times of the epidemic without recourse to data augmentation techniques. Moreover, the time-inhomogeneous birth–death process provides a good approximation for the SIR epidemic model for which we are able to obtain both, the exact likelihood of the inter-arrival death times, and a fast to compute Gaussian-based approximation of the likelihood. The explicit expressions for the likelihood enable us to reveal bi-modality in the likelihood of the ongoing Markovian SIR epidemic model and to devise scaleable MCMC algorithms which are applied to the emergence of the Covid-19 epidemic in Europe (March–May 2020).