# On Herd Immunity

**Definition**

When would we be able to say that a region or country has achieved “herd immunity” to a contagious disease, for example the UK to Covid?

In my previous post I was guilty of using the term “herd immunity” sloppily, as implied by Martin Sewell’s comment. As Martin suggests, one might claim herd immunity has been achieved if R<1 and if we use that strict definition, then my conclusions should be that: “1 The UK is closer to *sustained* herd immunity than is generally supposed”, etc.

However, in my defence, I was using the term “herd immunity” in the vague way it has been used in every article I’ve read on the subject. And more concerned with exposing the apparent arithmetical difficulties in determining a population’s herd immunity threshold (HIT).

Viral transmission obviously varies considerably from month to month, week to week and even from day to day, depending on multitudinous factors including whether kids are in school; the weather; what public events, such as sports matches, are taking place; the day of the week; what infection controls have been put in place by government; how afraid people are of infection etc. etc, as well as vaccination rates; immunity from recent infection; and declining immunity from previous infection and/or vaccination. It would be daft to claim, for example, that a population had herd immunity at the weekend, but not during the week.

So I looked for a coherent definition of “herd immunity”. Michael le Page of New Scientist claims (and I’ve pointed out to NS that le Page’s article is undated) “there is no formal definition”, but Wikipedia, citing the Encyclopedia Britannica, suggests that “[h]erd immunity is a form of indirect protection from infectious [sic: should be “contagious”, surely] disease that can occur with some diseases when a sufficient percentage of a population has become immune to an infection, whether through vaccination or previous infections”.

Everyone uses the same arithmetic for determining the herd immunity threshold, i.e. that proportion (1 – 1/R_{0}) of a population must be immune, but no-one seems concerned that this may vary over time. Not only are there changes in patterns of contacts between people, the properties of the virus itself, such as its persistence in the environment, are likely to vary significantly with the weather. It’s as if they’re concerned more with a simple mathematical model than the real world.

I’m not going to go back and change my previous post, but from now on will use the term “sustained herd immunity”.

The question we need to answer, I suggest, is how close is the UK to sustained herd immunity, given the risks of increases in contacts due to the start of the educational year, in particular, and of behavioural and biological changes as we move into autumn and winter?

**Arithmetical points**

Michael le Page suggests that:

“With the more transmissible B.1.1.7 [Alpha] variant first identified in the UK, each person may infect four or five others, suggesting

at least80 per cent of people need to be immune to get R below 1 and reach the herd immunity threshold.”

Maybe that’s a slip: it’d be 75% if R_{0} for Alpha is 4; 80% of the population if R_{0} is 5.

But, as I pointed out in my previous post, *if immunity is achieved through infection and not just vaccination, the herd immunity threshold (HIT) will always be less than (1 – 1/R _{0}) of the population, because the individuals with most contacts, who spread the virus to more other people, are also those who will tend to be infected and develop immunity first. *

One reason I mention this is that it’s been nagging at the back of my mind, from reading around the subject, that the number of people infected in a region or town (in the 1918 flu pandemic as well as unrestrained Covid outbreaks) is always less than would be suggested by the formula.

And that undershoot is *despite* the fact that *the number infected will always overshoot the number necessary to achieve herd immunity, because people still become infected even when R<1. * Wikipedia puts it like this:

“The cumulative proportion of individuals who get infected during the course of a disease outbreak can [actually, will always] exceed the HIT. This is because the HIT does not represent the point at which the disease stops spreading, but rather the point at which each infected person infects fewer than one additional person on average. When the HIT is reached, the number of additional infections does not immediately drop to zero. The excess of the cumulative proportion of infected individuals over the theoretical HIT is known as the overshoot.”

^{}

Perhaps the existence of overshoot is why there isn’t much theoretical concern about sustaining herd immunity. In many epidemics, fluctuations in contagiousness (strictly in R_{0}, an appalling concept which not only varies between populations, but also between groups within any given population), would be masked by overshoot.

**Covid in the UK**

As the previous post showed, case numbers in the UK over the last few weeks have fluctuated, leading to the supposition that we’re flirting with sustained herd immunity.

The risks are, though, that R will increase due to:

- The new educational year. Even younger teenagers have not yet been vaccinated, and vaccination rates in younger adults are lower than in older adults.
- The onset of autumn when the biology of the virus may change (in particular, it may be favoured by lower temperatures and declining vitamin D levels in the population) as may behaviour (more time spent in less well ventilated spaces).
- The continued relaxation of formal restrictions and reduction in infection-prevention habits, such as mask-wearing.

On the other hand, immunity may be continuing to increase significantly due to the continuing vaccination campaign and to infections.

**What is the likely level of immunity in the UK?**

The multibillion pound question!

The UK government’s dashboard tells us that over 80% of the adult population is fully vaccinated. But that doesn’t give full immunity to infection. There’s been considerable debate recently as to how much immunity the vaccines do provide. And it’s difficult to get an answer more accurate than ~50%. I suggested sampling rather than relying on unreliable statistics, but doubt anyone will take any notice. In any case, there are problems taking account of immunity acquired through infection. The unvaccinated may be immune through infection, making the vaccine seem less effective than it actually is at preventing infection, but on the other hand, the immunity of the vaccinated may be boosted through infection before or after vaccination, making the vaccine seem more effective. It’s not immediately apparent which effect would dominate, though I suspect *it’s most important to take into account the effect of increasing immunity of the unvaccinated through infection*.

I digress slightly to explain why *it’s important to take account of the effect of increasing immunity of the unvaccinated through infection*. The point is that we know that vaccination reduces the risk of infection by a certain amount, let’s say 50%. But we don’t know the *distribution* of this protection. It could, at one extreme, be binary, so that 50% of the population are fully protected by the vaccine and 50% aren’t protected at all. Or it could, at the other extreme, be that everyone’s chance of infection is reduced by 50%. It’s most likely somewhere in between. We know some people are less protected by the vaccine than others, but it seems unlikely the viral load, fluctuations in individuals’ immunity due to stress etc, and pure bad luck are not factors in whether an individual can be infected despite vaccination. The point is that, *if vaccine protection is not entirely binary, the vaccinated population as a whole gains less protection from infection than does the unvaccinated population*. Consider 100 vaccinated and 100 unvaccinated individuals. The 100 vaccinated are initially 50% protected against infection, so 1 will be infected for every 2 unvaccinated. Let’s assume those infected are 100% immune afterwards. If each vaccinated person is 50% protected (the extreme), then after 1 infection, there will be 99 people all 50% protected. There will still be 49.5 persons worth of vulnerability. But in the same time, 2 of the unvaccinated population may be infected, so there would be only 98 persons worth of vulnerability in that group. The vaccinated population is now *less* than 50% protected compared to the unvaccinated population. It may be important to take this into account where substantial numbers of people are becoming infected, such as in the UK. Not doing so might lead to incorrect conclusions, including overestimating the rate of waning of immunity from vaccination.

Anyway, let’s say there is 40% population immunity (half of 80%) in the UK due to vaccination. This will increase as boosters are administered to older age groups, and vaccines are rolled out to younger teenagers over the next few months.

How much immunity is there due to infection? Let’s have a look at some graphs:

What’s interesting here is not so much the proportion of the English population testing positive for antibodies, since we know that doesn’t necessarily imply full immunity. No, what’s really interesting is the number testing positive for antibodies in the younger age groups *before* any vaccination took place. Around 30% of 16 to 24 year-olds had Covid antibodies by December 2020! That is, before the Alpha wave and before the current Delta wave.

What’s more, in older age groups the proportion with antibodies is *less* than the proportion vaccinated, suggesting that immunity acquired through infection is greater than the difference between the antibody positive and vaccinated percentages in the graphs above.

Unfortunately, we don’t really know how long immunity acquired through infection is likely to last. However, it seems reasonable to make a guesstimate that 30% of younger age groups have been infected with Covid sufficiently recently to be immune.

However, here are some more graphs:

Even if those infected with Covid test positive for 10 days, *more than 1% a week* are becoming infected in younger age groups. This is compatible with the government’s PCR test data, since we can assume a significant proportion of cases are unreported.

**Other Evidence**

Earlier Covid variants, in particular Alpha, have declined to very low levels in the UK. This suggests herd immunity against Alpha has been achieved. If R_{0} for Alpha was between 4 and 5, as mooted, immunity against that strain must be of the order of 60-70% (allowing for the effect of preferential infection of individuals with more contacts).

Case numbers have twice declined in recent weeks, as reported in my previous post.

Other countries are reporting falling case numbers without new restrictions, for example Germany and Tunisia.

**Conclusion**

A reasonable guess for the level of population immunity to the Delta Covid variant in the UK at present might be 60-70%. The threshold for sustained herd immunity is *less* than 86%, assuming R_{0} (flawed concept though that is) of 7, perhaps significantly less (my guess, for what it’s worth, would be 75%). With continuing infections, principally in younger age groups, and an ongoing vaccination campaign, including boosters for the over 50s, a guess might be that immunity will increase by around 1% a week going forward (boosters and vaccines for 12-15 year-olds are only just getting underway). This suggests that, *if the seasonal effect is not extreme*, the UK *might* achieve sustained herd immunity to the Delta variant before Christmas in 3 months time.

On the other hand, a lot of this is guesswork…

And let’s hope there’s not a new variant!