Rats

I’m not sure if the decking is the main problem, our neighbours feed the birds and rats come to feed from their garden.
We had field mice nesting under our decking until we removed it, we also had rats in our compost bins.
I think your only solution is to buy the small green boxes with rat poison placed in them, but it could get costly as the rats will probably keep coming, especially during the Winter months.
I think the days of the local authority ‘Rat man’ coming out, free of charge, are long gone. Unless your neighbour is hoarding rubbish, I think it’s something that you will have to deal with yourself.
Rats are a common problem nowadays I’m afraid.
We live in the countryside and have a constant battle with rats, mice, rabbits and a young deer that sleeps at the bottom of the garden and sets the dog off!

Hmm. A property I own was having problems with rats attracted by a neighbour's rubbish.

I expected the neighbour to resolve the issue, which they did, but IF they hadn't I was going to environmental health. I had photographs and videos of the problem as evidence.