I have the same problem next door, it's now unoccupied because the owners have recently moved into a nursing home, they have a fig and an apple tree, which partly drapes over my fence.
I've seen 2 rats, who tease my cat by running along the top of my fence into the apple tree, then jump onto next doors rainwater tank and into their roof.
2 weeks ago, one lost balance and fell into my side of the yard. It lasted about 10 seconds. I'll tell you what, for a 14 year old cat, Jackie still moves bloody fast
So whilst Jackie was playing with this almost dead rat, the other rat was watching from the apple tree, with a mini apple in it's mouth. I warned him that this would be him if he hangs around. I let Jackie play with her catch for a while, lord knows it's the most exciting thing that's happened to her for 2 years, poor thing. I stopped her once she bit it's head off though. In the bin it went.
Funnily enough, i haven't seen the other rat since. I have no doubt he's still around though.
My problem is, is that i'm fairly comfortable that they won't get into my place or roof, however I don't want to go chucking ratsack around incase another one falls into the yard and Jackie eats it when I'm not home.
I'd probably need to find a petsafe solution If I found they're still around the place.
Now I just have bees instead (in the same neighbours tree)