I doubt it. There are plenty of cities without significant feral cat populations without rat problems.

In my part of the world cats are a rare sight on the streets, and virtually all of them wear a collar so they aren't feral. Meanwhile I can't recall ever having seen a rat, and according to the local statistics almost all rat extermination happens at farms.

Don't want rats in your city? Don't leave trash lying around in places easily-accessible to rats.