  Phylogenetic tree reconciliation is employed for the examination of
coevolution between host and symbiont species. An important concern is the
requirement for dependable cost values when selecting event-based parsimonious
reconciliation. Although certain approaches deduce event probabilities unique
to each pair of host and symbiont trees, which can subsequently be converted
into cost values, a significant limitation lies in their inability to model the
invasion of diverse host species by the same symbiont species (termed as a
spread event), which is believed to occur in symbiotic relationships. Invasions
lead to the observation of multiple associations between symbionts and their
hosts (indicating that a symbiont is no longer exclusive to a single host),
which are incompatible with the existing methods of coevolution. We present
AmoCoala, an enhanced version of the tool Coala, that provides a more realistic
estimation of cophylogeny event probabilities for a given pair of host and
symbiont trees, even in the presence of spread events. We expand the classical
4-event coevolutionary model to include 2 additional spread events (vertical
and horizontal spreads) that lead to multiple associations. By incorporating
spread events, our reconciliation model enables a more accurate consideration
of multiple associations. This improvement enhances the precision of estimated
cost sets, paving the way to a more reliable reconciliation of host and
symbiont trees. Our results showcase that AmoCoala produces biologically
plausible reconciliation scenarios, further emphasizing its effectiveness. The
software is accessible at https://github.com/sinaimeri/AmoCoala
