I don't think this should be a core component of the game.
If needed a quest \ mission which isn't part of the main story line. But a random event that can happen. Then with a branching quest line to find he cure for the said disease. Also with consequences of when and type of cure you choose.
For example you get Osteomyelitis [Bone infection] probably due to the fact you for some reason keep shooting your self in the leg. It becomes infected, till you get it treated your movement speed is reduced, as well as your weight carrying capacity.
Now if you can get it treated within say a month. Which requires you to visit various towns, to find a orthopedic surgeon or something of the sort. You talk to various people who refer you to various places.
some one offers this kind of treatment, others offer you something else.
Some one may even offer you experimental surgery that may even give a bonus when you get treated.
So your basically looking around for the best option.
But during your looking you may not have enough time to reach the previous doctor who could have treated you, resulting in you loosing your leg. Which means you need a prosthetic which gives you a permentant handicap. Now this permentant handicap can be fixed if you find a better prosthetic [not by a doctor] but by random encounter \ knowledge skill \ etc. Or if you get cybernetic surgery done or of the sort, which requires a high cost to get.
But i wouldn't want it every 5 minutes i get the cold, plague, or erectile dysfunciton.
Gotta love that Sick Demented Dark Humor.