  The importation and subsequent establishment of novel pathogenic strains in a
population is subject to a large degree of uncertainty due to the stochastic
nature of the disease dynamics. Mathematical models need to take this
stochasticity in the early phase of an outbreak in order to adequately capture
the uncertainty in disease forecasts. We propose a general branching process
model of disease spread that includes host-level heterogeneity, and that can be
straightforwardly tailored to capture the salient aspects of a particular
disease outbreak. We combine this with a model of case importation that occurs
via an independent marked Poisson process. We use this framework to investigate
the impact of different control strategies, particularly on the time to
establishment of an invading, exogenous strain, using parameters taken from the
literature for COVID-19 as an example. We also demonstrate how to combine our
model with a deterministic approximation, such that longer term projections can
be generated that still incorporate the uncertainty from the early growth phase
of the epidemic. Our approach produces meaningful short- and medium-term
projections of the course of a disease outbreak when model parameters are still
uncertain and when stochasticity still has a large effect on the population
dynamics.
