I lived in a 250 year old New England farmhouse for eight years that was riddle with mice, squirrels and who knows what. I don't think we had any rats but can't be certain. I fought the hoards the entire time. One weekend I snared about 50 mice and didn't even make a dent in the population.
At any rate, control your dog and do what ever you can to annihilate the disease ridden vermin. The amount of filth and nastiness that a family of rats can leave is not worth worrying about how humane you need to be. If you have to club the little bastards, shoot them, poison them, take a hatchet to them, burn them out, smoke them out but get them out of your house.
A cat is a good thing to have around sometimes.
No offense Haz.....