I got the plague once in my second campaign. It burned out in about 4 turns. THe interesting thing is that I tried to save some of the garrison by moving them out and putting them in a neighboring fort. It didn't work. In fact the soldiers that stayed in the infected city did not lose any men in the next turn, whereas those that I moved out lost men for 3 more turns.

Moral of the story: do NOT move people in or out of infected towns, unless you are moving out a character.

BTW, if my capital got infected, I would just switch capital. Much easier than moving characters.