  Disease propagation between countries strongly depends on their effective
distance, a measure derived from the world air transportation network (WAN). It
reduces the complex spreading patterns of a pandemic to a wave-like propagation
from the outbreak country, i.e. a linear relationship to the arrival time of
the unmitigated spread of a disease. However, in the early stage of an
outbreak, what matters to countries' decision makers is knowledge about the
relative risk of arrival of active cases, i.e. how likely it is that an active
case that boarded at the outbreak location will arrive in their country. As
accurate mechanistic models to estimate such risks are still lacking, we
propose here the ``import risk'' model that defines an import probability by
means of the effective-distance framework. The model assumes that airline
passengers are distributed along the shortest path tree that starts at the
outbreak's origin. In combination with a random walk, we account for all
possible paths, thus inferring predominant connecting flights. Our model
outperforms other mobility models, such as the radiation and gravity model with
varying distance types, and it improves further if additional geographic
information is included. The import risk model's precision increases for
countries that are more connected within the WAN, and recovers a geographic
distance-dependence that suggests a pull- rather than a push- dynamic of the
distribution process.
