Lyme is a peculiar bacteria that can create a whole range of different symptoms in different people. If the infection is very recent, chances are good that a month on antibiotics will completely eliminate the bacteria without any further symptoms.
Different doctors have very different beliefs about the illness and optimal treatment, however. It's a tricky illness, and if you don't feel confident your doctor is treating it aggressively enough, get a second or third opinion, sooner and not later.
Watch the documentary "Under Our Skin" (available streaming on Netflix) for a great deal of information on the illness and the aftermath of lax or inadequate treatment. I make this recommendation cautiously, though, because it will probably scare your socks off. We have a strong interest in this illness because we're pretty sure my husband contracted it several years ago. He had to fight hard to get his doctor to even prescribe a 2-week course of treatment. But he did, and we apparently caught it in time.
I wish the best for your family.