Mice don't eat birds so it will be a bit harder unless the mice are lapping up bird poop.
They eat out of bird feeders, we are all doomed.
But again, if the cats guard the bird feeders, the mice and birds can't get together. But if the cats get sick and die, no one will be guarding the bird feeders. But if the cats guard the bird feeders the mice and the birds can't get to the bird feeders to infect the cats.
I have ten cats and need to get rid of a few of them. If you want to save the world by posting guards on bird feeders, this is your big chance. Where should I send your cat to?