I honestly think your concerns would be better addressed by one of the vets on the Ask a Vet forum here at Med Help. The only thing I know about Lyme, which is admittedly little, is that it can take months to eridicate in both humans and animals. It's a frustratingly difficult disease to treat. Best of luck to you both. :-)
I am always concerned about tick-borne diseases, because there are ticks that transmit Lyme Disease etc in my area. I found this website, I don't know if you will find your answer here or not. I hope it is helpful.
http://www.plainfieldvet.com/petcare_anaplasmosis.asp
For the Lyme positive test, you could ask that this be followed up with the Lyme Quant C6 test, which helps to monitor infection level (dormant to active) so treatment can be given before the onset of noticeable symptoms. As far as I know, a broad-spectrum antibiotic is normally used. It is also possible re-infection by fresh tick-bites could occur, especially if you live in a winter frost-free area. And that may account for the new 'positive' results.