Actually antibiotic resistance is a bit more complex. It is when a pathogen is present and exposed to an antibiotic but not completely killed that the pathogen has an opportunity to adapt to the antibiotic. A much larger contributor to antibiotic resistance is non-compliance. Someone who is prescribed an antibiotic who takes it for a short period, starts feeling better, and stops rather than completing the dosage and if not completely 100% clear, does not follow up with the physician for more or a different antibiotic if needed. What happens is that there are individuals (or groups of individuals) in any population that have a naturally higher resistance to some drug or chemical like an antibiotic. Those with the least resistance are removed from the population first when exposed. If the individual's immune system does not overwhelm the rest and they can produce future generations, the new population will have a much higher percentage of individuals (or groups) that have a resistance to the drug.
Given the consequences of Lyme disease along with the inconclusiveness of current testing, prophylactic use of antibiotics is not unreasonable.
It is Doxycycline and another drug that has been effective if administered early is Minocycline (a cousin).
Thanks,
Jack