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It was never eradicated. It was renamed. Monkey pox is still extant.

The decrease in infection rates came about by way of sanitation, hygiene, and improvements in knowledge of nutrition.

Seems like it's related but different than smallpox: https://en.wikipedia.org/wiki/Monkeypox

How do you know it's not the combination of all the factors that decrease the infection rates? Why do you single out the vaccination?

Look at the timeline.

Yes. It looks like it's a combination of multiple factors.