Illness proceeded very rapidly, we hear (3)
I believe the answer is:
flu
'illness' is the definition.
(flu is a kind of health problem)
'proceeded very rapidly we hear' is the wordplay.
'proceeded very rapidly' becomes 'flew'.
'we hear' indicates a 'sounds like' (homophone) clue.
'flew' sounds like 'FLU'.
(Other definitions for flu that I've seen before include "Minor ailment" , "Viral infection (abbr)" , "A bad winter bug" , "Influenza, in short" , "Viral illness" .)