This paper from 2011 proves scientifically that if you test positive by PCR for borrelia, then the infection is still there. Dead borrelia cannot give a positive PCR result.
https://attachment.fbsbx.com/file_download.php?id=517568781694021&eid=ASvS_ZMisLusU2tB29_-4DndfZUSJAaTSEvi16VmCr15F5rLgX1vSKnC8BkdsE0bksM&inline=1&ext=1394468607&hash=ASsUtUskiYfztKAd