I had a couple of rats and am glad it wasn't more than that. My exclusion program was aimed at rats too - and I got some rat sized snap traps which worked a treat but are very intimidating to set because they are so huge (like the rats!).
I have lived this. I mean really lived this. Both my chickens and my cats catch mice but I was still over-run!
It took me a year to get to eradication. Now I am in a good place I feel I can share my experience. Others may have different views, I am just sharing that this worked for me and I...