Assume that some virus is rampant that might kill 1% of the population. A vaccine has been developed that is able to prevent death in 90% of those cases (and has no side effects). All the countries in the world are following some strategy but four stand out:
Ignoristan is very poor and can’t afford the vaccine. Consequently, 1% of the population die from the virus. Vaccine efficacy can not be computed because nobody has been vaccinated.
Smartistan is poor but has access to some magical device that identifies those 1% at risk of dying from the virus. They are vaccinated, and Smartistan ends up with 0.1% of the population dead. Vaccine efficacy is being computed as minus infinity since there are no deaths among the unvaccinated.
Wastistan is rich but authoritarian and vaccinates 100% of the population. As in Smartistan, 0.1% of the population die from the virus. Vaccine efficacy can not be computed in Wastistan because there is no control group.
Foolistan is rich as well but only vaccinates those healthy enough to get to the vaccination centres. Unfortunately, these are exactly the 99% not in danger of dying from the virus. Foolistan ends up with 1% of the population dead but computes vaccine efficacy to be 100% because all deaths are among the unvaccinated.
All the other countries sprinkle the vaccine randomly onto some q% of the population. Result: 1% - 0.9% * q% dead, and vaccine efficacy is determined to be 90%.
Moral: be wary of countries that report high vaccine efficacy.
I swear, when writing this I had no idea about the monkeypox paper, with its four countries of Arnica, Brinia, Cardus, and Dranma.
https://www.nti.org/wp-content/uploads/2021/11/NTI_Paper_BIO-TTX_Final.pdf