  We numerically study disease dynamics that lead to the disease switching from
one host species to another, resulting in diseases gaining the ability to
infect, e.g., humans. Unlike previous studies that focused on branching
processes starting with the first infected humans, we begin by considering a
disease pathogen that initially cannot infect humans. We model the entire
process, starting from an infection in the animal population, including
mutations that eventually enable the disease to cause an epidemic outbreak in
the human population. We use an SIR model on a network consisting of 132 dog
and 1320 human nodes, with a single parameter representing the gene of the
pathogen.
  We use numerical large-deviation techniques, specifically the $1/t$
Wang-Landau algorithm, to calculate the potentially very small probability of
the host switching event. With this approach we are able to resolve
probabilities as small as $10^{-120}$. Additionally the $1/t$ Wang-Landau
algorithm allows us to obtain the complete probability density function $P(C)$
of the cumulative fraction $C$ of infected humans, which is an indicator for
the severity of the disease in the human population.
  We also calculate correlations of $C$ with selected quantities $q$ that
characterize the outbreak. Due to the application of the rare-event algorithm,
this is possible for the entire range of $C$ values.
