Nothing makes a home or dwelling less enjoyable than an invasion by rats, mice, bugs, or other creatures. This is true whether you're living in a city apartment—already feeling the grime and congestion of your cosmopolitan surroundings—or a suburban home. If you own your home, pests can also severely lower its value and make it difficult to sell.
Even more annoying is when you suspect the pests crept in from neighboring apartments or homes. If you believe that your neighbor is the source of these marauders, what should you do?
Your first step should be to ensure that your property is not creating conditions that attract or provide sustenance to the pests. Immediately call an exterminator. Have them assess the situation, lay traps in your apartment or home, and mitigate the creatures to the extent possible.
Make sure you don't have open food containers or crumbs that are attracting them. You wouldn't want to accuse a neighbor of having rodents or bugs if you are the true source!
If you live in a large apartment building, the building might provide extermination or fumigation services free of change. After all, it is in the building's interest to maintain its reputation. In urban real estate, word-of-mouth travels quickly, and news of an infestation would surely hurt rentals and sales. See if your building can approach your neighbor directly about entering the apartment, and at least begin to fumigate the hallways and common spaces.
If you live in a suburban neighborhood, your homeowners' association (HOA) or neighborhood association might provide similar resources, or at least be able to recommend high-quality exterminators who have helped the neighborhood during prior instances of infestation.
Ultimately, you might have little choice but to approach your neighbor directly. Be polite, not accusatory. Mention that you have noticed a large number of pests in recent weeks, and emphasize that you have a common goal in eliminating pests from your mutual home.
See if the neighbor is willing to engage in a discussion about the cause, mention what you're already trying to do in order to reduce the problem, and suggest that each of you contact an exterminator to visit on a common date to deal with the problem holistically.
It's possible that your neighbor hasn't dealt with the situation because cleanup efforts or exterminators are expensive. Perhaps the neighbor lacks the resources to buy rodent traps or hire teams of fumigators.
You might frame the problem as an issue in common, so that the neighbor can save face. Offer to pay a large share, or a fixed amount, if you know how much the services will cost, if the neighbor is willing to open his property to the exterminators.
If your neighbor is unwilling to work on this problem, even with your offer to share costs, you might have no choice but to call animal control.
Animal control might go by a different name depending on your jurisdiction. Many cities have helplines for people to call if unsure of the correct agency to speak with. Such helplines will be able to direct you appropriately. Local or state investigators may need to be called in. It is possible that your neighbor is failing to obey local ordinances on proper disposal of refuse or storage of food, which the proper government agency will address.