This has been a plot of quite a few novels, so I'm struggling to come up with that exact one (if I've even read it). Off the top of my head, Children of Men is similar.
There is a novel written with that premise as the background! I can share the title with you, if you'd like - that particular premise is a very small part of the novel.
michael crichton wrote Prey which is about something similar to this (i don't want to say too much so as to not introduce a spoiler). that wasnt from the 70s or 80s, though.
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