Your rights aren't clearly defined. As you state, you have a right to move into a rat-free environment. After that, whether preventing and eliminating rat problems depends on culpability which depends on facts which are rarely clearly defined. On balance, I'd expect that a tenant is responsible for a rodent problem unless caused by the negligence of the landlord. Facts here seem conflicting in that regard. So you can hire an attorney to fight about liability or you and your landlord could cooperate in figuring out how to best split the repairs and the costs since it's likely neither of you are interested in rats or litigation.
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