I'm having the same issue with rats getting into my run. Based on the droppings found in our shed, they have been around awhile, but I just never saw them until we got the chickens. I have my food in the run, not the coop, and I bring it in every night, but the little buggers still keep getting into the run (And if my girls are spilling food, it isn't much because I look for it and don't find any to clean up). I put traps out in the run at night (chickens locked up so can't get near them) for a few weeks (get multiple traps. I read somewhere that 12 traps for a minor infestation is not excessive) and caught 6 rats. Haven't caught any in the last few weeks because now all that are left are the smart rats that know how to avoid the traps. I rent and am not going to trench and bury wire, but I did put 12 inch pavers up against the fence all the way around my run. The rats are smarter than me, though, and they somehow are finding a way to burrow in the miniscule space between the pavers and the run fence and still tunnel under the fence. I went out and bought more hardware cloth and this weekend I will put down a curtain of hardware cloth from the bottom of the fence extending out one foot from the run and hold the edge down with the aforementioned 12 inch pavers. If this doesn't keep them out of the run, I don't know what I'm going to do. At least I got rid of enough of them that I don't hear them rustling all around in the leaves and behind the fence when I go out to lock the girls up at night (right now it's already dark when I get home from work). That really creeped me out - made me think the evil little things were watching me, just waiting for a chance to attack! LOL!
Also, don't know about where you live, but I randomly met someone from our county's vector control office and asked him about this - he told me to call for an appointment and they will come out and inspect the property and give me a report with suggestions for things I can improve/do to discourage the rats from living in my yard. And in my county, this service is apparently free (Yay tax dollars!). Haven't done it yet, but I plan to set up an appointment after the holidays.