the world is segregated. No magic sperm transportation devices, no contact.
Do you realize that even in todays society its a difficult thing to do? Not every single sperm is viable, and again, pointless because their isn't a steady supply of it?
Are you ignoring the fact that the majority of scientists, health workers, and other vital workers are now gone? How exactly is this society supposed to function with a bunch of depressed women who have had their entire male family members forcibly moved?
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You think you've Got problems? https://imgur.com/OnEOsdT