If it is not in the lease, pest control may not be required of the landlord UNLESS local housing or health codes require.
If the pest problem in the apartment is severe (and most BB infestations are) the landlord may be required to address the problem because the property's condition violates local health ordinances--that is your point of entry to getting the LL to act, and to pay. Recommend you immediately contact your county housing code enforcement and report the matter.
LL has the responsibility to treat for infestations, treat the whole building. LL must pay for that treatment and cannot be unreasonable in delaying treatment. IF LL can prove who introduced the BBs to the property, LL MAY go after that party for the costs--but it is VERY hard to prove who did what in a multiunit dwelling. If it were me, I would refuse to pay, demand the LL treat the infestation, and force the LL to prove it was me--they likely cannot. See http://www.dca.state.ga.us/housing/housingdevelopment/programs/LandlordTenantHandbook.asp and http://consumer.georgia.gov/consumer-topics/landlord-tenant-issues-and-handbook
READ THIS BEFORE CALLING OR EMAILING ME: I am licensed to practice before the state and federal courts in Virginia. We have not established an attorney-client relationship unless we have a signed representation agreement and you have paid me. I do five my 100% effort to get you on the right track with your issue. Sometimes that means legal educational information, sometimes that means counseling and non-legal guidance. You should speak with an attorney to whom you have provided all the facts, before you take steps that may impact your legal rights. I am not obligated to answer subsequent emails or phone calls unless you have hired me. I wish you the best of luck with your situation.
If the issue of pest control is not addressed in the lease, then there is no hard-and-fast rule about who has liability. Generally, the landlord is responsible for structural repairs and the tenant is responsible for any damages that they cause. I would suspect that the issue would become something for the judge to decide, but may be controlled over other factors (how long the tenant lived there before the bed bugs arrived, whether there was a bedbug problem in an adjoining apartment, etc.)
I hope this information helps answer your question(s).
~ Kem Eyo
The above answer is a general explanation of legal rights and procedures. It does not constitute legal advice. Nor does it establish an attorney-client relationship between the individual posting the question and the attorney providing the answer.