You don't mention if you live in a private home, an apartment building or public housing. You can complaint to the health dept. in Irvington or hire an exterminator and then deduct the amount from your rent (but you should give notice to the landlord. If there are any health effects (like a rat biting a child for instance), then you may have a suit for personal injury. Usually there is a claim for a warranty of habitability, if the apt. becomes uninhabitable for such reasons, the process is go to court and claim damages against your rent.
Easiest first step is call health or building dept. in your city, and also make sure you keep copies of any written complaints you made. Also take pictures if you can.
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I agree with counsel. This is a habitability issue and must be set up properly. Send the LL notice pursuant to the lease and give him time to remedy the situation. If he does nothing or inadequately remedies the situation, withold the rent and escrow it.
1) call the health department about the roaches and rats; 2) before you can deduct from your rent send your letter written notice concerning the condition to your LL. If you do not give your LL written notice you CANNOT deduct. Yes, there is a warrant of habilitabilty, but first you must give written notice.