Here is a long reply with a lot of info. We have had major problems with various large rodent species at our home. Most of our problems involve packrats and some nonnative ground squirrels but I suspect our solution could be adapted to work with rats. We have been very successful in controlling populations.
We use mostly live traps and have found tomahawk brand work best among livetraps because they are not shiny (dull finish) and they rattle around less than have-a-hart traps. Tomahawks are used more by professionals. (I'm a wildlife biologist by training, so that is how I knew about this brand). We set 10 or so around our property. Two key things are (1) keep traps out 24-7-365. We bait them most of the time but the important thing is don't wait until you have a problem, out-of-control population. Keep traps out all the time. You can dispatch animals you capture with any preferred method (drowning, which we do in our pond, pellet gun, etc.). And (2) Find a location that works. Not all locations work. Keep moving the traps every 2-3 days until you find a location that catches your problem species, and then keep traps at these locations all the time. A lot of locations will not work and do not get discouraged. Keep moving traps until you find a good spot. For us, we find that areas under large rocks and boulders are best and we set the trap with the entrance facing a burrow under a rock, but not blocking the entrance. It took us a couple of weeks of trial-and-error to figure this out. We caught 0 squirrels/rats prior to figuring out these ideal locations, even though they were running around EVERYWHERE. Then scatter bait at the entrance and going into the trap (leave a trail of bait so they can nibble it as they walk in).
We have tried other traps. We had success with a squirrel-inator trap that we picked up on amazon, as well as a tube trap. I posted links below. Note that the tube trap works really well with our problem species, but it is lethal so we use it sparingly. With the live traps, if we catch a non-target animal, we can release it.
Tomahawk trap: https://www.livetrap.com/index.php?dispatch=categories.view&category_id=182
Squirrelinator trap: Rugged Ranch Squirrelinator Squirrel Trap
Tube body grip trap: https://www.forestry-suppliers.com/product_pages/products.php?mi=44491&itemnum=35803&redir=Y
If you have a small property and your neighbors have rats, you will have more problems. We have a couple of neighbors who let us set traps in key locations on their property and it helps stem-the-tide (so to speak) and prevent a lot of rats/squirrels from even making it to our property boundary.
The first year, we caught hundreds of squirrels. Now, we catch and kill less than 100 per year in a bad year, probably closer to 80 in most years.
When they are reproducing in spring and summer, we catch a lot more and check traps about every 1-3 days. During slow times of year (winter) we only check a couple times monthly.
Anti-coagulant rodenticide poisons will kill non-target species so we never use them. I lost a few barn cats and a dog at a previous property where the owners had used poison years prior, because rodents will cache the poison and future generations can then eat it. Predatory wildlife that eat a rodent that ate anti-coagulant rodenticides die a miserable death. But I do understand if you have a terrible rat problem, you may want to use poison. I would try our method first, if you can.
I realize you have different species that are causing the problem but like I said, I bet a similar approach could be adapted to rats but with using different bait.
It sounds awful, what you are going through. I hope this info helps. Best of luck!!
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and I’ve set out poison which has killed some not all though
. Not interested in rabbits in the slightest 
