Oldschoolers like Baraka Obam and Scabdraggr claim it's contagious through fluid transfer, not casual contact/ airborne fiber transmission. I think they are probably right.
IDK about the alcohol baths, I'd recommend switching to Epsom Salts and mixing in some borax sometimes. (2 cups Epsom Salts , and/or a sprinkling of borax detergent)
Stay away from the permithrian, I think it makes the issue worse.