Ok, under AZ law, (AZ 33.1324 and1361) the landlord must:
Comply with the requirements of applicable building codes
materially affecting health and safety
· Make all repairs and do whatever is necessary to put and keep the premises in a fit and habitable condition
· Keep all common areas of the premises in a clean and safe condition
· Maintain in good and safe working order and condition all electrical
, sanitary, heating, ventilating, air-conditioning and other facilities and appliances, including elevators
, supplied or required to be supplied by him
· Provide and maintain appropriate receptacles and conveniences for the removal of ashes, garbage, rubbish and other waste incidental to the occupancy of the dwelling unit and arrange for their removal
· Supply running water and reasonable amounts of hot water at all times, reasonable heat and reasonable air-conditioning or cooling where such units are installed and offered, when required by seasonal weather conditions, except where the building that includes the dwelling unit is not required by law to be equipped for that purpose or the dwelling unit is so constructed that heat, air-conditioning, cooling or hot water is generated by an installation within the exclusive control of the tenant and supplied by a direct public utility connection.
So if the landlord is allowing an insect infestation, then this would violate his legal duty to provide a fit and habitable dwelling.
If there is a noncompliance by the landlord with section 33-1324 materially affecting health and safety, the tenant may deliver a written notice to the landlord specifying the acts and omissions constituting the breach and that the rental agreement
will terminate upon a date not less than five days after receipt of the notice if the breach is not remedied in five days. The tenant may also sue for damages
and an injunction.
So your recourse here is to give them a written 5 day notice to cure the problems or you will terminate the lease and pursue legal action against them for any damages for breach of contract.