A Bayesian adaptive shift-mixture approach for estimating incubation period and exposure time in small outbreaks

Rapid estimation of infection time and the incubation-period distribution is essential for evidence-based quarantine and contact tracing. However, outbreak data typically mix primary and higher-generation cases with unobserved exposure times. We propose the infection-induced adaptive shifted-knots Dirichlet process mixture (IASK-DPM) model, a nonparametric Bayesian framework applying a Dirichlet process prior on the shift parameter in a three-parameter lognormal kernel. Incorporating infectious dynamics, the base measure is derived from a kernel-density estimate of observed onset times, concentrating mixture components where the epidemic curve is dense while preserving support in the left tail, where earlier exposure may plausibly lie. The proposed framework is intended primarily for small-outbreak or early-phase epidemic settings in which infections are concentrated around a limited number of dominant transmission episodes or narrow exposure windows. Posterior inference uses Hamiltonian Monte Carlo and is implemented in R/Stan. In nine Monte Carlo scenarios representing COVID-19, measles, and O-157 outbreaks, IASK-DPM reduced mean squared error in shift parameter estimates (i.e. exposure time) by 35% to 70% and in the 95th incubation percentile by 30% to 68%, relative to fixed finite-mixture models. These gains persisted even when the competing Dirichlet-process model used a conventional base measure. Applied to the 2015 Korean MERS outbreak, the method recovered four exposure dates (4.7-19.5 days after May 10, 2015), coinciding with field-investigated superspreading events, and estimated a posterior median incubation of 5.8 days (95th percentile 13.7 days), aligning with exhaustive contact tracing. By allowing the data, rather than the analyst, to determine latent exposure generations, IASK-DPM enables accurate inference on key epidemic parameters with minimal tuning.