There are people who have ferrets and terriers and such and who will come out for a few hours and literally kill almost all the rats for you. The dogs are bred for that, and they are GOOD at it. See if you can google anyone local who does this; it's a one-evening-fix. Then, follow up with poison for any new rats that move in.
We have a pair of collie-type dogs; one is probably australian shepherd/lab/border/something, the other is a scotch collie. They are death on four feet to the rats. The very few times we had a rat come in, the dogs went out at night, found it, and, well, no more rat. They've taken out three, and then we put out poison, and no more rats.
Our ONLY concern is that the chickens will absolutely eat mice and small young rats, so we have to check for rodent bodies just-in-case. Haven't found one

But the dogs go and check and will absolutely destroy any that show up, and they aren't even terriers. A collie-type dog (border, aussie, heeler, kelpie, corgi, scotch, rough, mcnab) should also be a very good vermin dog, they're bred to it. And we can attest to their effectiveness, too!