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.

More on herd immunity and the “second wave”

The piece on herd immunity and the outbreak in India has been published on the CNN website More than half of Mumbai’s slum residents might have had Covid-19. Here’s why herd immunity could still be a long way off; this follows my article in The Conversation and the blog entry here.

In the meantime, new results have started to appear, adding to the confusion on whether we are nearing the herd immunity or not; see for example here. There are two strands to this research:

Medical strand

I am not an immunologist, but my understanding is that challenged by a virus, the organism responds in different ways, including antibodies, T-cells, B-cells, and the whole cornucopia of other elements. As we are now about 6 months after the outbreak started in serious outside China (Johns Hopkins data set starts on 22nd January 2020), data are now accumulating on the prevalence (how many people have it), variability (how people differ in their response), dose response (have asymptotic cases a lower immunological response), and dynamics (how long do these things stay in the organism). Unfortunately, my understanding is that the picture is mixed. Some experiments show a strong lasting response and some show a decline.

Is SARS-CoV-2 like the common cold viruses which – as far as I understand it – do not induce lasting immunity? After all, you can catch the common cold many times during the winter season and apparently this is not due to any mutations. Or, is COVID-19 like a flu, which infers immunity that lasts for months – if you catch a flu again in the same flu season, this is apparently due to a different strain and certainly each year the strains seem to be different. Or, is it like measles, which infers immunity for life?

Why is it important? If SARS-CoV-2 is like a common cold, we will simply have to learn to live with it for a long time and the whole concept of “herd immunity” is simply not applicable in a long term (in a modelling world, we will be using an SEIRS family of models). On the other hand, if SARS-CoV is like flu (or even better like measles), we can hope that it will at some point “burn out” (SEIR family of models). At the moment, we probably still do not really know (or at least, I do not know).

Modelling strand

The concept of “herd immunity” stems from an observation that for a virus to spread it needs to have a supply of susceptible individuals, the idea of a threshold is a modelling construct. Not surprisingly different models produce different estimates (for an explanation, see this blog entry); the estimates also depend on our understanding of how fast the virus spreads measured in terms of the reproductive rate, R.

But the situation is even more complicated, as the “herd immunity” levels depend on R which in turn is affected by, among other things, social distancing, mask-wearing, hand washing, home working, and other control measures we currently widely implemented. When quoting a number, the researchers usually mean the “herd-immunity-if-we-do-nothing-to-stop-the-virus”, i.e. under conditions of our return completely to the “normal” behaviour (whatever this means) – no lockdown, no masks.

As we believe that the “if-we-do-nothing-to-stop-it” value of R for coronavirus is about 3, the “classical” result (1-1/R) gives 67%, so we would need at least 70% of people to be immune to start seeing the reduction in the new case numbers. The total number of people infected will be much higher. This is also the smallest proportion of the total population who needs to be successfully vaccinated in order to reduce the epidemic and to prevent future outbreaks. If the vaccination success is about 80% (which is optimistic), we would need to inject nearly 90% of the population to get to this level. This is possible but would require very high levels of coercion, a long campaign, and huge investments; difficult to achieve before November elections.

Can this proportion be lower? There is an observation – which could be wishful thinking – that the infection stops spreading around the mark of 20% of people who have had an infection. This includes all people who have had an infection, not just the reported cases; if we believe that there are about 8 times more infections than reported cases, we are looking at a mark of 2-3% of reported cases – for the UK this would be about 2 million reported cases (as of the end of August we are at the 330,000 reported cases).

There are some theoretical results, including a very important and influential paper by my colleague from Strathclyde, Gabriela Gomes, which give some hope to these lower numbers. But, as of the end of August, we simply do not know.

I want to finish with a very sobering quote from The Atlantic:

The very concept of “herd immunity” – which, although it was not the official Sage-determined strategy, became part of a narrative that led British officialdom into a deadly swamp of complacency – implies the existence of one’s very own, very special herd. It is a correlative of genetic distinction, except that it is acquired rather than inherited. The people have absorbed the virus and their collective antibodies have melded into one triumphant, mystical body that is, like the island itself has always been, impregnable.

Why would any scientist fall for this nonsense? You didn’t have to be a specialist in infectious diseases to know that it was based on two entirely untested assumptions: that coronavirus would behave like flu and that it would confer long-term immunity on those who did not die from it. Scientists – let alone highly distinguished and respected ones – are not supposed to jump to such evidence-free conclusions. The only explanation for what happened is that, far from the government being “led by the science”, official scientific thinking was contaminated by its exposure to a political culture in which positive ­assumptions are compulsory, and British difference is taken as read.

https://www.newstatesman.com/2020/07/fatal-delusions-boris-johnson

This is a warning to us, scientists, as much as to politicians. Our ideas might look great on paper – or even on the computer screen – but implementation might have huge consequences.


Speaking about republishing the graphics, the article on the second wave in The Conversation (see also the accompanying piece on this blog) has widely been cited and the graphics used many times. A nice piece in the Polish edition of National Geographic appeared back in June (in Polish).

My article on the superspreaders has also been cited in a recent New Scientist article.

Herd immunity and future scenarios

Last week, the media picked up a study from Mumbai, India, where it was reported that 57% of people who had been tested showed exposure to the virus. While it is not completely clear what this means for immunity (the survey looked at antibodies in the blood; see vox.com for a good explanation of what this means) and particularly long-term immunity, it has raised questions about whether we might be nearing “herd immunity” and therefore the magic point when the disease will stop spreading on its own.

Additionally, they reported a relatively low death rate (Infection-Fatality Rate, IFR) – the data seem to suggest that many people are getting infected, but only a few die.

https://indianexpress.com/article/cities/mumbai/in-mumbais-slums-simple-act-of-bathing-and-relieving-oneself-becomes-more-daunting-for-women-6342816/

Before we start jumping to conclusions, there are few words of caution. I have tried to get hold of the details of the report, but a search on medRxiv, the usual place where scientists report their findings, but could not yet find it. We are therefore basing any speculations on newspaper articles. Secondly, the 57% figure was only for Mumbai slums where presumably the transmission was particularly high; for other localities, the number was much lower, perhaps around 16%. This is more in line with places that experienced high coronavirus numbers, like New York (ca 24%).

Thirdly, as mentioned in my The Conversation article a couple of weeks ago, there is still a lot of uncertainty on what exactly these studies tell us about long-term immunity to SARS-CoV-2. Is it for life, like for measles? Or, for a year, like for the flu? Or, weeks, like for the common cold? And, what is the current outlook for the COVID-19 pandemic?

Almost 19 weeks we published the article in The Conversation outlining different scenarios for the pandemic; followed up 10 weeks later with another one on the second wave. How did the outlook change since then?

One of the scenarios that we listed 19 weeks ago was of a large first wave followed by slow eradication:

https://theconversation.com/four-graphs-that-show-how-the-coronavirus-pandemic-could-now-unfold-133979 25th March 2020

While it is a gross oversimplification of what is happening, I still believe this is a correct picture in the long term, although the time scale will be much longer. The “true” graph arguably will be very jagged, with ups and downs of local epidemics, and the peak will last longer than about 30 weeks (a bit like the picture below from the June article), but there will be no “silver bullet” solution that will stop the disease quickly and painlessly.

https://theconversation.com/coronavirus-what-a-second-wave-might-look-like-138980 1st June 2020

Firstly, based on what we currently know, I have strong doubts about the epidemic “burning out” by itself through reaching the “herd immunity” globally – although we might reach “herd immunity” in some confined places. We simply do not know at this stage whether the immunity is lasting and I personally have doubts about it.

Wide availability of the vaccine might change the picture, but again it is not clear to me that it will confer solid and long-lasting immunity. Repeated vaccination (boosters) might be needed to keep the immunity, and it will take a long time to bring the disease worldwide to sufficiently low levels. It has taken 200 years to eradicate smallpox; rinderpest and polio eradication was quicker, but still took many years and a concerted international effort (polio is not yet fully eradicated); measles was never eradicated worldwide, but almost eradicated in Europe and the US and making come back due to decreasing vaccination levels . In my opinion, even with a working vaccine we will have to live with the virus for a long time.

And then, there are likely to be other viruses ready to jump species, unless we become much more cautious.

So, what are the lessons? Firstly, we need to plan long and cautiously. It is clear to me that currently many governments (and people) apply a strategy of “pendulum swings”. We have actually seen this type of strategy emerging in individual people’s behaviour when we challenged them with a simulated epidemic (our 2015 BMC Public Health paper). In this approach, people slam the epidemic when it is threatening, but relax the control measures as soon as it recedes, even if it is not completely gone. As a result, the epidemics come and go in a long-lasting cycle, while the effective reproductive number is close to 1, while the economic losses mount with each cycle of the lockdown. The other problem with this approach is that with each cycle people become more tired of regulations and less likely to obey the rules, as we already see in the crowded beaches and pubs.

I still believe that we should have slammed the virus much stronger and much earlier than what happened in the UK (and other countries). But I now think that this should have now been followed with a much more controlled strategy than reopening everything.

In the long run, we will need to learn to live with the virus. Travel will become much more difficult, with more people having to spend holidays near home; we might need coronavirus vaccination before we travel to places where coronavirus is still present (like polio or typhus).

I think what I am getting at is that we should perhaps look at the virus as an opportunity to change things for better, as a challenge to adapt to the new situation, not desperately hoping to get back to “normality”.

The world at the end of 2020 will be very different to the one a year ago. But these speculations take us too far from the main topic of this blog.


It is only after having written this post I learned that the WHO official has recently used the term “silvery bullet” in the context of the coronavirus pandemic. He said:

“However, there is no silver bullet at the moment and there might never be.”

“It’s completely understandable that people want to get on with their lives, but we will not be going back to the old normal,”

WHO Director-General Tedros Adhanom Ghebreyesus