He and Carol are convulsed by disease (7)
I believe the answer is:
cholera
'disease' is the definition.
(cholera is a kind of health problem)
'he and carol are convulsed' is the wordplay.
'and' says to put letters next to each other.
'are convulsed' is an anagram indicator (letters convulse or shake).
'he'+'carol'='hecarol'
'hecarol' anagrammed gives 'CHOLERA'.
'by' is the link.
(Other definitions for cholera that I've seen before include "Infectious disease causing severe bowel upset" , "Infectious and often fatal Asiatic disease" , "A severe intestinal disease, severe Asiatic kind" , "Acute intestinal infection from contaminated water or food" , "could kill you" .)