Maybe the rats are too light to trip the trap, since the trap is meant for rabbits. Or maybe you need a different brand or type of trap (assuming there is more than one kind). Is the trap in good shape? I had once lent a cat trap to someone and they put stuff on top of it in their garage so it dented in a little, and it never worked right after that, even after I tried to bang it back into shape with a hammer. What I used to do when I trapped cats is put the food all the way against the back of the trap, and to make sure they didn't step over the bar that trips it I put a piece of newspaper that covered the whole bottom of the trap so they didn't know what they were stepping on (also made it more comfortable for their paws since the bottom wasn't solid).
If you love animals called pets, why do you eat animals called dinner?