Pet rats have red mites!?

Rehomed two male rats who unbeknownst to us have red mites. I'm now wondering if this was one of the reasons the individual wished to rehome them in general. Previous owners kept in very unkempt cage so perhaps this was the culprit. Who knows. All I know is I want to be rid of the mites and feel so bad for these rats. They are the sweetest rats ever! I've since put said ratties in a new "rat mansion" and it is now I've discovered their ailment. One rat in particular has red patches starting now on his back with tiny bits of fur missing. The brown rat, his brother, is not so bad, but you can see the brown/reddish coloring throughout his fur/skin when you brush through it. Besides the obvious, cleaning out their cage, etc., has anyone ever used that ivermectin paste (very small amount) to cure mites in pet rats? What about any topical solutions? Specific wash/shampoos? Our local feed store recommended a tea tree shampoo with Neem oil but I'm not sure. Petsmart suggested a mite/lice topical spray for birds to use. Thus far, I've not done anything as I'm considering checking with my vet tomorrow. Any suggestions? Will the mites return? How did they get them in the first place? Please help.