Little animal swallows first of pills for infectious disease (8)
I believe the answer is:
smallpox
'for infectious disease' is the definition.
'little animal swallows first' is the wordplay.
'little' becomes 'small' ('small' can be a synonym of 'little').
'animal swallows first' becomes 'pox' (I can't justify this - if you can you should believe this answer much more).
'small'+'pox'='SMALLPOX'
'of pills' is the link.
I am not very happy about this link. Some or all of it may be part of another bit of the clue.
Can you help me to learn more?
(Other definitions for smallpox that I've seen before include "Viral disease, now almost eradicated by vaccination" , "Acute viral disease, technical name variola" , "Highly contagious viral disease with fever and skin pustules that leave permanent scars" .)