Say you have 10 infected people enter a susceptible population. The reproductive number, R0 tells you how many people on average those infected individuals will go on to infect, before they recover. If they manage to infect no additional people (R=0), the virus dies straight away. If each individual manages to infect 1 other person on average (R=1) then that initial population of 10 infected will infect another 10, who will infect another 10 and so on. In that case, the virus is growing linearly in the population (10x per time period), but the number of currently infected individuals will remain at 10.
Now if each individual manages to infect
less than 1 individual on average (R<1), then those 10 infected individuals will not be able to infect another 10. They may only infect 9 before they recover. Well, those 9 individuals may only be to infect another 8, and so on. In this case, the number of currently infected individuals would be continuously reducing and eventually the virus will die completely. So when we say we want to achieve herd immunity, what we really mean is want infected individuals only to be able to infect less than 1 person on average, so that all outbreaks of the virus eventually die out.
Vaccines can achieve this, by reducing the number of susceptible individuals an infected person comes in contact with. Say R0=5, because an infected individual comes in contact with 10 susceptible individuals on average and infects half of them. Now suppose that half the population is immune to transmitting the virus. In this case, each infected individual will only come across 5 susceptible individuals, so we have effectively reduced R to 2.5. Now what we really want is a formula that will tell us what proportion of the population needs to be immune, in order for R<1. That formula is
1 - 1/R0
Where R0 is the reproductive number in a 100% susceptible population. (As we have seen the reproductive number in practice –sometimes called Rt – reduces as people become immune). There is quite an accessible explanation of how the formula is derived here:
Simple math shows how widespread vaccination can disrupt the exponential spread of disease and prevent epidemics.
www.quantamagazine.org
So, to answer your question, the 85% is derived from having an estimated R0 of 20/3: 1 - 1/20/3 = 0.85
Every inhabitant, or every
adult inhabitant? I saw that Gibraltar was ready to start vaccinating children, but haven't seen any confirmation that it has gone ahead.
But suppose that every susceptible member of the population has been vaccinated, including children, and the incidence of the virus in the population is still increasing. What does that show? Well, that there are not enough individuals immune to transmission, to hit the herd immunity threshold. If that threshold is really 85%, then perhaps the effectiveness of the vaccines is less than anticipated. But the key takeaway is that every individual who has been given immunity through vaccination is one less individual that needs to be infected to reach that 85% threshold. So it's completely wrong to think that just because they may not be sufficient to achieve herd immunity in a population, vaccines do nothing to limit spread.
The other thing to think about is that R0 is not a fixed quantity but varies depending on the interconnectedness of the population. And Gibraltar has the 5th greatest population density in the world. So the 20/3 figure that was estimated for R0 probably doesn't apply to Gibraltar at all. It could be 10 or more, and in that case, you would need at least 90% of the population immune. So in that case even if you vaccinated everyone, if the efficiency of the vaccine dropped below 90%, the virus would start spreading again.
Or that you have some level of immunity through vaccination, which we know can be variable.
