I've got much less experience with poultry than many here, but I do know a little about trapping rats. Wiping out an infestation CAN be done, just takes time, patience, and attention to detail.
Do all the prevention stuff that others have mentioned above. Rats are there for food, period. Take that away, and that's half the battle.
Forget about any trap that kills the rat. I wouldn't have believed it if I hadn't seen it for myself, but take it from me- rats DO learn to avoid most traps once they see a fellow rat get whacked.
Get yourself a couple of good cage traps, rat/squirrel size (some call them "live traps"). Take a good look around your chicken coop, find out where the rats are spending a lot of time. Look for feces, tracks in the dust, gnawed holes in the floor, etc (a prime spot is where they enter or leave the coop, from the outside). Set your traps in these high-activity areas. For bait just use some chicken feed, or whatever else they're used to eating. Other good baits are sardines, peanut butter, dog food- almost any smelly food. A word of caution- if the trap is outdoors, baited with sardines, stake it down or chain it to something solid! It WILL become a raccoon toy.
Check your traps every day, disposing of any rats, then reset. Great thing about cages is they don't spook the rest of the rats when they're holding one, because the rat in the cage is still alive & unharmed.
I also refuse to use poison. A couple of years ago I worked at a kennel that had a rat infestation. Poison was of course out of the question, because of the dogs. Took me all summer, but I single-handedly wiped out the entire colony of rats with just three cages and the occasional handful of kibble for bait. Try it.
Good luck!