Doctor despaired after western epidemic (10)
I believe the answer is:
'epidemic' is the definition.
'doctor despaired after western' is the wordplay.
'doctor' indicates anagramming the letters.
'after' is a charade indicator (letters next to each other).
'western' becomes 'w'.
'despaired' is an anagram of 'idespread'.
'idespread' after 'w' is 'WIDESPREAD'.
(Other definitions for widespread that I've seen before include "Sweeping" , "Generally prevalent" , "common" , "available to lots of people" , "found in many places" .)