Rats and mice can carry many many diseases, many of them deadly. Bred ones usually are checked out and are safe, but you need to be mindful of the ones out in the wild and the sewers. Weil's disease, salmonella, Hantavirus, leptospirosis, rat-bite fever, the plague, etc. Breathing in the dust they leave is enough to get some of these diseases.  I'd not be too bothered by a tame leashed rat, but when this winter came it turned out mice could get into this apartment by walking along certain pipes. So I stomped on those I could find and used PU foam to seal everything. No telling what they might carry.