Use bait, traps and anything you can to get rid of them. Cute they are not Weils disease/ fleas/rabies are a horrible things.
We've bought bait stations and hidden them in "rat run" areas. We've had one rat attack this year which was stemmed very quickly and nothing when the weather was nice. though this might change now rain is back.
Re shed - we had same problem few years back. Removed old shed because too old to keep and it became a rat hotel despite poision & traps. Replaced shed with Summer house on conrete base. No returning pest yet. We also put galvanised wire on the underside of the wooden shed floor and up the sides of shed to stop chewing through wood.
We don't keep food in there. All chicken, rabbit and dog food is kept in the house or locked sealed metal bins. The metal bins all have bait and traps down.
We have also baited and trapped our compost bin and in the roof of our garage. all our concrete sheds have chemicals in - Jeyes fluid, bleech and so on and I've noticed the rats stay well away from these areas. I put bleech down our outside drains once a week too. I also clean our pation once a week too with Jeyes, but that is more to do with cleaning dog wee up.
I've got 129ft of garden to keep rat free which backs on to unmanaged woodland. This means they love to come to our house for a snack when the weather turns inclement. :twisted: