If it is Demodex mange then I wouldn't worry too much. Demodex is a mite that lives on a dogs skin and sometimes when a dogs immune system is compromised (usually due to stress) that mite takes over and causes patches of hair loss. Ivermectin is the main treatment and the antibiotic was probably added due to some sort of skin infection associated with it. I will say 9 months sounds really long to be on the medication, unless that's an indication of severity.....hair loss on a large part of the body vs smaller patches, but most Demodex clears up nicely and isn't a problem long term. If it's sarcoptic.....then that's another story.