# Herd Immunity

Note: My definition of Herd Immunity is the total % of the population who get infected once the epidemic has completely finished. Most people define it as just the % of the population infected once  the peak is reached. This gives much lower figures – 60% rather than 90% !

When a new disease enters into a community without prior immunity it will spread  if R0 > 1 . R0 is simply the number of people the first person with the disease infects before he/she recovers (or dies). If R0 < 1 then the disease fades quickly away and has no lasting effect,  while if R0 >> 1 the faster the disease spreads. Avian Flu was an example of a lethal disease but fortunately for us had R0 < 1. People got infected by birds but did not easily pass it on to others before they died. As a result there was no pandemic, despite WHO warnings of one at the time.

In the UK there are 65 million people all initially susceptible to any new infectious disease arriving in the country. It is estimated for Covid-19 that  R0 probably was  around 2.4 in Wuhan. However for simplicity let’s consider a general case where some disease has R0 = 4 and that the infectious period is 5 days.

At the start of the epidemic one infected person arrives in UK  and infects 4 others before recovering. These then each infect 4 others so that the infected population will increase as $4^n$ every 5 days.

1,4,16,64,256,2048,8192,……..etc.

It is probably only after ~1 month that anyone really notices that there is a problem, but by then the epidemic is already increasing “exponentially” out of control. However there is a safety catch.  Eventually each new infected person begins to meet some of those also infected or already recovered, so now cannot pass it on to 4 new cases. The reservoir of susceptible people is quickly running out and R is now diminishing fast. It reduces to 1 at the peak of the outbreak and then falls dramatically as the epidemic collapses. The population is then said to have reached “herd immunity”, and the UK is afterwards immune to any new infections arriving from abroad. One interesting fact is that herd immunity is always reached before everyone in the country is infected.  A percentage of the population will always escape any infection, but this percentage depends critically on the initial value of R0. Here are two examples.

Herd immunity for R0 = 1.2.  Only 30% get infected before Herd Immunity is reached.

However if R0 is much larger than 2 it is a different story.

Herd immunity is only reached after 90% of the population have been infected, but at least the epidemic ends much faster!

Governments can reduce “R” through social distancing measures, but this is a tricky process because to return to “normal” life infections would essentially have to drop to zero, perhaps through track & trace. Alternatively we could eventually reach herd immunity with maintaining R=1.2 but that would take 9 months, and even then may not be sustainable while infection rates outside the country remain at R=2.4.  So In both cases international travel might still need to be controlled indefinitely.

The only certain way out of this dilemma remains either a rapid development of an effective vaccine, or a drug treatment which  renders the disease no worse than a cold.

PhD High Energy Physics Worked at CERN, Rutherford Lab, JET, JRC, OSVision
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### 16 Responses to Herd Immunity

1. Jerry says:

Thanks for turning your thought process to this.
Of course Herd Immunity is based upon the fact that people who have been previously infected will not get re-infected for some period of time. I’m not hearing anything conclusive on the chances of a reinfection, let alone for how long that immunity will hold. Fingers crossed.

2. bas says:

With 15000 unchecked arrivals at Heathrow every day international travel needs to be controlled a lot more strictly. It seems that we stay in hoping to stay free of the virus whilst unknown cases wander in and out of the country at will.

• This doesn’t really bother me – any more than a new born baby entering the world. They might be carrying the virus but so long as they pass it to the not vulnerable and the R stays below 1, then we’re cool. If they don’t have the virus then that is also cool.

3. gymnosperm says:

H(i)=1-1/R(o)?

4. I’ve read that 20% of the population are responsible for 80% of the transmission, That’s interesting because it would mean if R0 is 3 and in a sample of 5, one person infects 12, and the other 4 infect only 3 others between them ( so their personal R0 is around 0.75 ).
So let’s fast forward a few months and pretend that R0 is indeed 3, and that 20% of the population have had this thing, and people go back to interacting as they did before. And let’s call those with personal R0 of 12 arseholes, but 80% of the arseholes have already had the virus so cannot spread it again. And let’s start this thing again . . .
By my maths, the new spread rate comes in at 0.96, so we might have a short lived second wave. But territories that had a mild first wave and locked down too early such that tiny numbers of the arseholes got a dose first time are going to be fucked second time around.

• Grope, this may imply that people respond differently, with some naturally being superspreaders and others being less-spreaders. Or it may be a result of variations in viral strains with some being more aggressive (ironically they are probably already spread less rapidly, like SARS-1) and others being milder (they have longer incubation periods). Once a little herd immunity builds up and/or if the community goes into lockdown the milder strains will be favored by natural selection, simply because they cause viral shedding for LONGER (although levels of shedding are lower).

Models such as SIR work quite well for diseases such as polio and myxomatosis but much less well for respiratory diseases.

• Clive Best says:

I think this is an important point. It is probably why Sweden’s outbreak is diminishing without a heavy handed lockdown. There is a paper on this which suggests as a result Herd Immunity can be reached at a low % of infections.

https://www.medrxiv.org/content/10.1101/2020.04.27.20081893v1

• I did see that paper. In fact I emailed the authors – whether they will read my message I have no idea. But I feel they are missing some of the most important points. It’s very clear that nasty viruses slowly become milder – otherwise there would be hundreds of killer strains around like the ones that have emerged recently (SARs Ebola, myxomatosis, Spanish flu etc.) – all the bugs that jumped to humans in (say) the last million years. This paper is focussing on the susceptibility of different age groups, not on changes in the virus.

• Owen Boyle says:

Sweden’s outbreak is not really “diminishing”. It is rumbling along the top of a broad peak. They are clocking up around 500 new infections per day (with a massive variance of about 300 – probably due to slow reporting). Their figure for yesterday (702) is about the same as a month ago (726).

I do not see any evidence of herd immunity in Sweden. Just a controlled infection rate coming from a disciplined population taking sensible hygiene precautions.

• As others have said, infections is a reflection of the number of tests performed. Don’t we need to focus on deaths? Clearly diminishing. Quite surprising. Other nations will be watching.

• Clive Best says:

You do see an effect in the total number of deaths and it fits quite well a model of diminishing R.

5. J Martin says:

@Grope very interesting point about individual R0. And there’s that 80 20 rule again, I have forgotten what it’s called.

6. I’m probably repeating myself, but I think it’s unlikely that the actual numbers will look very like Clive’s graphs. Not that the graphs aren’t useful – they show us where we’re heading right now.

Look at e.g. the graphs of influenza in the 40s – 60s in the UK in this link. Huge epidemics come and go incredibly quickly. This is not explained by mainstream virology:

https://oldwivesandvirologists.blog/the-strange-arrivals-and-departures-of-influenza-epidemics-in-the-uk-1946-1974/

This seems to be driven by the virulence/transmission trade-off model. Viruses mustn’t replicate too fast, or they will fail to transmit themselves, because the illnesses that they cause are too brief. Respiratory viruses seem moderate their replication by being sensitive to temperature – they only replicate in the cooler nose and throat.