I recall a briefing where I learned that if you are sick, your medical certificate is invalid for that period of time. That is, your medical cert is only good when you're in the same shaper you were when you were at the flight doc. Heaven forbid something happens, even something not your fault like a birdstrike, and if the FAA gets involved you might have bigger problems than you might have expected.
I wonder how it would work if you got your medical renewed *while* you were sick...!
Get well soon,
LC