Can you play an instant (airlift) to move quarantine specialist prevent infection after the infection card has been drawn?
That is against the rules.

You must resolve each card fully before you can do anything else, so if you draw a card that causes an outbreak, you must first perform all of its steps. After you resolve that, you can play an Event, if you like. And, of course, you could play an instant before drawing and revealing the first Event card.

From the rules, page 7:


  Event cards can be played at any time, except in between
  drawing and resolving a card.


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