  Traditionally, studies of coevolving systems have considered cases where a
parasite may inhabit only a single host. The case where a parasite may infect
many hosts, widespread parasitism, has until recently gained little traction.
This is due in part to the computational complexity involved in reconstructing
the coevolutionary histories where parasites may infect only a single host,
which is NP-Hard. Allowing parasites to inhabit more than one host has been
seen to only further compound this computationally intractable problem.
Recently however, well-established algorithms for estimating the problem
instance where a parasite may infect only a single host have been extended to
handle widespread parasites. Although this has offered significant progress, it
has been noted that these algorithms poorly handle parasites that inhabit
phylogenetically distant hosts.
  In this work we extend these previous algorithms to handle cases where
parasites inhabit phylogenetically distant hosts using an additional
evolutionary event which we call spread. Our new framework is shown to infer
significantly more congruent coevolutionary histories compared to existing
methods over both synthetic and biological data sets. We then apply the newly
proposed algorithm, which we call WiSPA (WideSpread Parasitism Analyser), to
the well studied coevolutionary system of Primates and Enterobius (pinworms),
where existing methods have been unable to reconcile the widespread parasitism
present without permitting additional divergence events. Using WiSPA and the
new biological event, spread, we provide the first statistically significant
coevolutionary hypothesis for this system.
