"But -- correct me if I'm wrong -- didn't the Doctor say that each of "the flesh" had been infected with all the diseases -- like about a thousand diseases each? "

Well, yeah, Which is daft, as you note. So I was imagining the somewhat more plausible scenario that, say, each "floor" of the "intensive care" place was used for studying a different disease, or something like that. Which would be a far more useful way for the cats to be studying them.