Giving a financial incentive to shoot them will guarantee a thorough job of killing them. Sportsman like you and I would not hunt them to extinction for the fun of it, but that is what needs to be done.
I would then have a great case of schadenfreude as you are overrun by verminous rodents. Same as spiders, they have their use. I do not kill them around the house, and will try to put them out with killing if they are inside. I prefer them to hordes of flies.