Outside of combat, a character who suffers from a disease suffers damage over a period of time.
Resisting the damage inflicted by a disease requires a reflexive Stamina + Resolve roll.
This roll is not contested but it is modified by the severity of the disease. Only one success is necessary to avoid damage each time.
Some diseases are the kind that people don’t heal from without magic. A character’s cancer could go into remission, or he can hold his HIV back with medication, but time alone won’t cure them.
The Storyteller should set a benchmark of how many rolls the character has to succeed at in a row for the disease to go into remission. Medical treatment can offset any penalties to the Stamina + Resolve roll applied by the disease — but might inflict penalties on other rolls, as sometimes the cure is almost as bad as the disease.