The things reproduce very quickly. It is a matter of math. Once you have reduced them to two or three reproducing couples, you can keep up with them and at some point you will completely eliminate them. Just keep going with the traps. The mice may look for a warm place only but on top of that make sure there is no source of food. Usually, if you have any grain, seeds etc., they will come to live with you!
Cats stay on top of the population. There would never be such growth with a cat around. Most cats I had would chew the mice open for the most tasty bits and then leave the rest. Thats why you want to make sure your cat is dewormed on a regular basis if it stays with you in the house.
Good Luck, George