I have potatoes popping up like crazy but they aren't purple. We have potatoes in beds that haven't had potatoes planted in them for years and years. Maybe purple potatoes are especially prolific but IME they're all pretty stubborn, if you miss harvesting a potato it's just going to become a
seed potato for the next generation, whether you want it to or not.
I would guess, if you don't want it to spread, stop distributing dirt between your beds, even when you
think everything is dead. Also if you see your intentionally-planted potatoes blooming, pinch off the flowers. Potatoes do flower and go to seed and I'm not sure what the success rate is, but if you don't want them spreading, then there's no reason to let them go to seed.