I feel for you. People in colder parts of the country don't understand that roof rats are a whole other type of rat and the usual methods used on black/Norway rats aren't particularly effective. The rats in my area, Los Angeles, are also roof rats.
Roof rats are the original species that the white lab rat came from. Even regular rats are smart and adaptable, but these guys take it to another level. They are so difficult to control because they are extremely trap shy and suspicious of any new thing and can learn from watching other rats get killed.
One tip I got from a professional, was to get a LOT of snap traps and put them high on fences, etc where the rats are. Bait, but don't set them for WEEKS. In other words, you are feeding the rats on the traps to get them used to the traps (works best in winter when there is little other food). Then one night set them all. BAM! You will get a bunch of dead rats that night. But that will be it. You won't be able to trap anymore rats for many months, as any survivors will have seen the other rats dead in the traps and won't go near them. I am serious.
Live traps definitely work better for roof rats, because the trapped rat is there still alive for the other rats to see it wasn't harmed, but you still have to wait quite a while for them to get over being spooked by the presence of the trap. You still have to deal with disposing of the live rats (I use an ether bucket).
Using these methods, I still can't get all the rats, but it thins them out. My coop/run is small and totally enclosed in hardware cloth, even the floor of the run under the dirt. Food is not left out at night. I also have fruit trees and a garden, also fully enclosed (although those plant "runs" just have an apron, no floor).
Good luck, it is a hard fight!