Your lease should be reviewed to determine who is responsible for maintainence of the affected area although residential landlords remain responsible for the "habitability" of the premises. I would bring the matter to the landlord's attention, ask for immediate assistance, if none is forthcoming, you need to protect yourself by getting out of there or having someone come and fix it and then worrying about who is going to pay for it. (I would think that the City has animal safety people that would handle this for no charge any way.)
Good luck!
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