The way it works with stray or feral animals is you trap them, have them neutered or spayed so they can no longer breed and then you keep feeding them for as long as they live.  At least that's how it work in the animal organizations that I support for feral and stray animals.  This of course isn't legal to enact for humans but the thing we can do for humans is educate them, provide them with birth control and safe and affordable abortions when the first two options don't work.

Sometimes the education actually allows them to be come part of the big economic picture and pay their own way, sometimes it doesn't, but there is a 5-year cap on federal welfare benefits, except for some areas in Alaska and other poverty-stricken regions.  The best we can do is try to educate, provide birth control and abortion options and then hope for the best.