Since the infected folks found out relatively quickly that it was the stone causing the problem, they’ve taken to carrying it around with them. This reduces the number of things that join them on their travels (such as large predators) and makes the natives slightly less hostile.
The Nothing Stone is carried by a designated person in the tribe/band/family, as it tends to wear on the mind. This person is normally chosen at birth and carries the stone their entire life (in a shoulder-slung pouch). It makes the stone carriers a bit ‘wonky’, but never in a bad way. They just seem a little more out of touch than everyone else.
Could just name the kids ‘Stone’ and be done with it… hrm