I'm not going to say much more on this thread, because I don't want to get banned.
However, a few facts.
Most cities require all pets to be on a leash, or otherwise contained. Not allowed to roam. This includes cats. If the law is on your side, you have every right to ask the neighbor to keep Puss inside. If the law in your area doesn't forbid roaming pet cats, ask nicely for your neighbor to keep their cats inside. Explain that their cat is safer inside than out.
Erradicating introduced species doesn't work. For example, House Sparrows and Starlings are both introduced species. Many attempts have been made to erradicate them. There are still butt loads of them around. The house sparrow, especially, has had a terrible impact on native songbirds, due to competition for nesting sites.
When you have a feral colony of cats in an area, that is their turf. They guard it against outsider cats coming in. If you kill those cats off, more cats will move in from other areas. On the other hand, if you neuter the "squatters" and put them back in their area, they will fight to keep invading (and intact) cats from settling into the area.
When a colony of ferals is maintaned well, the ones with diseases are humanely euthanized, or die of natural causes. When they are simply killed, as I said, new cats (and new diseases) will move in.
It's really not that difficult a concept to grasp.
