Disease picked up, passed quickly (4)

I believe the answer is:

flew

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'passed quickly' is the definition.
'flew' can be an answer for 'passed' (flying is a kind of passing). I'm not certain of the 'quickly' bit.

'disease picked up' is the wordplay.
'disease' becomes 'flu' (flu is a kind of health problem).
'picked up' indicates a 'sounds like' (homophone) clue.
'flu' sounds like 'FLEW'.

Can you help me to learn more?

(Other definitions for flew that I've seen before include "Rushed < from local flat" , "bolted" , "Travelled by air" , "Dashed" , "Travelled by plane" .)

I've seen this clue in The Guardian.

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