My friends husband used a rat trap (cage type of thing) and got a small scratch from the live rat ,he was quite ill after from infection needed high dose antibiotic. I don't think that kind of trap is good if you don't know what you are doing.We used poison in our garden ,sometimes the rats die near the poison ,just shovel them up and put in bin other times they can go off somewhere else to die and you never find them.
Your neighbour has got to do something about the rats as well or the problem will get worse.