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More on 1-1/R formula and herd immunity

The concept of “herd immunity” is still being widely discussed, both from the policy point of view (should we just simply be “flattening the curve” and letting the nature take its course?) and medical and modelling (has Manaus in Brazil reached the “herd immunity” level with 44-66% infected?).

The result that the formula 1-1/R is referring to has been known for a long time, at least since the 1970s. It is quite a powerful result, as it is quite independent of the modelling assumptions. However, it needs to be properly interpreted. As I said in my earlier posts, the term “herd immunity” and the associated 1-1/R formula can be used in different ways.

Firstly, in a generic sense, “herd immunity” simply means the proportion of immune individuals in the population.

Secondly, for an ongoing epidemic, it means a threshold proportion of immune individuals at the point when the incidence starts declining because of the diminishing supply of susceptibles (as opposed to the reduction in the transmission due to lockdown). When the proportion of immune individuals reaches the level of 1-1/R, the number of newly infected individuals stops growing. But, this does not mean the disease will immediately stop. Assuming no loss of immunity (and ignoring that new-born babies will probably be susceptible), we can predict that the disease will then start very slowly decaying.

As can be seen in the graph below, the actual final number of people who will go through disease will be massively higher. For coronavirus, the “herd immunity” level is believed to be around 40-70%, and the corresponding final number of infected individuals will be about 80-99%.

Dependence of the herd immunity level and the final epidemic size on the reproductive number, R. Cited with permission from The Conversation article (TBC).

Thirdly, it is the level of immunity after the outbreak that prevents the emergence or re-emergence of the disease. In this context, 1-1/R determines the vaccination threshold that needs to be reached for protection over the future outbreaks. For example, for measles, this is about 90%, so if the proportion of children vaccinated against measles drops below this level (as it has done in several European countries and the USA), an introduction of measles from outside the country will result in an outbreak.

We also need to understand what R in this formula corresponds to. Normally, we will think of it as the basic reproductive number, i.e. the number of secondary cases resulting from one case, in the absence of any other control measure, like social distancing or masks. In other words, we usually think of it as a potential for the disease to spread under the “business as usual pre-pandemic” scenario. This is where the 40-70% range comes from; a wide range as we are not that sure how fast the virus would have been spreading if we removed all social-distancing measures.

How can we predict/measure “herd immunity”? There are two problems. Firstly, we do not know very well what R is and therefore what 1-1/R is.

Secondly, the formula 1-1/R, while applicable to a wide range of models, is by no means universal. We know, for example, that people and populations differ by how much they are susceptible and how much they contact each other. If the society is highly stratified, with some people living very close to each other and some widely apart, the “herd immunity” level might be smaller (some say 10-20% under some conditions).

There are some speculations that Manaus in Brazil has reached the herd immunity levels at about 44-66% infected, but apparently, they now do see new infections (possibly a result of waning immunity).

Can any country or region achieve herd immunity before the vaccine becomes widely available? As explained above, this depends on what exactly we mean by “herd immunity”. It is not inconceivable that some places in the world where the disease is particularly widely spread – as in Brazil or possibly in Indian slums – but achieving this more widely is going to be difficult and also it is not clear how long it will last.

And then there is a non-lasting immunity. This will make it very difficult to achieve the second objective listed above (eradication) and we will need to be re-vaccinated (or re-infected) multiple times.

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