This rat sounds like a roof rat. They enter the house on the electical and cable lines. They love the insulation on electrical wires, so you had better get rid of it pronto. They reproduce like crazy too.
I have been told that they dislike fiberglass insulation, so they will run along uncovered rafters. That is where you need to set the traps.
Stuff all the holes with steel wool; they have a hard time with that.
Also, pick all ripe citrus and other fruits from the trees. When they get hungry, make them move on to the neighbor's place where they didn't strip the trees. They really like dates, so maybe you could try them on the traps.
Rufus