The health department generally won't take care of a mouse problem. It will be up to you to control the rats on YOUR property, but you won't be able to do it if there is a problem next door. It will be a never-ending battle. And, for rats, you will need to use baits. Traps will not catch all of the rats, you'll only get a couple of young rats, then you won't catch any more.
Baits are the only way to go.
Forget the moth balls and do your cleaning with whatever you normally use. Soap and water are just fine, you don't need anything extraordinary. Probably the easiest for homeowners to use are bait packs. The rodents chew through the packs to get to the bait inside. You can purchase these locally, probably, if not, they are available from us.