campbellking, let me explain!
I owned and a Pest Control business and worked as a Pest Control Officer for 13 years. I have so often heard of cats coming in through the cat flap with a live mouse, invariably the cat flap will lead into the kitchen where there are many hiding places, under the many appliances etc. So live mouse is now in your kitchen, now along the myriad of water pipe that have holes leading to most areas in the house via the wall cavities it will will travel until it finds an area to nest, often under floors or in the wall cavities. Now consider the fact that most female mice spend most of their lives pregnant, the have a litter of young that will be interbreeding within just a few weeks and hey presto your cat has just introduced a major mouse infestation! And due to the mouse feeding habits they are lot more difficult to get shot of than rats!