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The formula used on the program was (1-1/R0)/Efficacy so if the vaccines are 100% effective then herd immunity for R0=3 remains at 67%, but with an effectiveness of 90% then herd immunity rises to 74%.
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This is the mathematically correct formula to calculate the needed vaccination fraction. But we should always be clear that it only describes reality if many simplifications are assumed. First it assumes that the population, the R0, and the immunity levels are uniform in location and time. But that is not really the case, rather you could have local outbreaks where R0 changes (due to distancing behavior, for example during certain religious services; new variants, for example it is believed that several of the newer ones have higher R0; higher or lower susceptibility of the population, for example from what we see so far children are not only less likely to get severe cases but less like to get infected at all)
I like to view the situation more like fire danger in forests in western states: the danger may well be extreme, but if there is no trigger somewhere, there is no fire. But even if the danger is not extreme, some careless campfire somewhere can start a big fire. And then you need to bring to bear firebreaks (social distancing for trees), local water (masks for trees) and whatever else is in the arsenal of firefighters, or ideally, rain as water from above (the virus accidentally mutating to something less dangerous, which is likely what happened with the 1918 flu).
The latest estimates of Ro that I have seen are 2.1 to 2.4 (based on Korean data)
https://europepmc.org/article/MED/33612125 and 3.3 (based on Danish data)
https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0247021 If it is 3.0, that does not necessarily mean that you need to vaccinate 74% of the population, as some will have immunity from a prior infection.
Indeed the 74% would be needed to keep the "average" fire danger low, and it can be either due to previous infection, or due to vaccine. But to avoid dangers in more dangerous pockets, you need more than that, and you need a rapid response system (fire department) to move in and deal with local outbreaks. Also, the Danish data are likely higher because there they are currently seeing the UK variant, which is more infectious. Overall, the more infectious variants propagate better, so what we are seeing is an overall upwards creep of R0.
It has been estimated that perhaps 33% of the population of the US has been infected already.
https://www.npr.org/sections/health...the-pandemic-is-10-times-worse-than-you-think
If infection and recovery conveys immunity comparable to the vaccine (and only 57 people around the world are known to have been infected twice
https://www.marketwatch.com/story/o...medical-experts-are-on-high-alert-11613743994 ), that would mean you only need to reach another 40% or so with vaccines before herd immunity starts to develop.
Although, to be sure, there will be some duplication in that people who may already have been infected may be part of the group getting the vaccine.
Indeed. If say 1/3 of the population is already protected and you want to achieve 3/4 immunity by randomly vaccinating a fraction x of the population, then you need to have 1/3 + 2/3 x = 3/4. If my morning coffee has kicked in sufficiently already, this means x=5/8 or about 62%. Still a hill to climb.
And then the problem is that if this coronavirus behaves like the four others that cause common cold, then immunity doesn't last very long. So get ready for annual Covid-19 shots.