First, `1/1-R_0`. This is exactly what Wikipedia says: https://en.wikipedia.org/wiki/Herd_immunity#Mechanics
And it makes intuitive sense that once you infect <= 1.00 people, growth is no longer exponential. So I don't think that point is wrong at all.
> The idea that vaccination and epidemics need the same percent of immunity is completely 100% wrong. An active epidemic is going to infect way more people before dying out than the simple, naive herd immunity calculation will give you. In fact, while an r0 of 2 indicates that you get herd immunity with just 50% of the population, a typical active epidemic with the same r0 would infect 80+% before dying out.
This doesn't make any sense to me. Vaccination is just an artificial way to build immunity. So I don't understand your active epidemic point at all. Or put another way, as the proportion of % recovered goes up, we start approaching the eventual limit where `1/1-R_0`->1. That is to say, in your example we start with R_0 = 2, then as people recover that value slowly starts decreasing until it hits 1 at the herd immunity threshold. It's not clear to me what the discrepancy you're detecting is here.