Fast likelihood calculations for emerging epidemics

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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).