In either of those cases, I would be scheduling a 1-on-1 with whoever is closest to being able to change that policy, in the smallest meeting room I can find, sitting right next to them, and then letting whatever symptoms of my illness take over. Gastric problems causing diarrhea or vomit? Congrats, I'm going to either literally poop in your lap, or vomit in your face. I'm coughing up phlegm? Sorry, here's a towel to wipe off your face. Something's bleeding? You might want to go get yourself tested for STDs and whatnot, gods know what depths I've been plumbing in my spare time. Severe pain such that I can barely sit? Your face looks nice and squishy to use as a stress ball.