Quantifying herd immunity lower bounds

In my previous article on the seasonal flu vaccine’s effectiveness, I wrote:

For the flu, with an R0 of about 1.5, we would need at least a third of the population vaccinated for herd immunity to be effective, assuming a 100% effective vaccine – knowing that flu vaccines’ VE is less than that, we’re looking at closer to 60% for adequate herd immunity. That means you, too. It is vitally important that everybody does their part, even if they themselves don’t expect, or care about, getting the flu.

Since then, I have been approached by a few people, both here and on Facebook, about the maths underlying it. So since I’m officially spending a mandatorily (read: spousally) enforced weekend off work, I thought I’m going to lay out the reasoning behind the maths.

What is herd immunity?

The idea of herd immunity is that an infectious disease needs a given number of available hosts in order to be ‘sustainable’ (able to continue to infect people) within that population. Typically, for illnesses from which people tend to either die or recover as non-infectious cases, this has to do with something called R_0 (pron. arr-nought) or ‘reproduction number’ of the infectious disease in a population.1

R_0 describes, roughly, the number of cases produced by each case. Consider poliomyelitis, which has an R_0 of about 6 (unusually high, given its transmission route).2 That means that every case produces, on average, six new cases. It is obvious from this that unless the R_0 for an infectious disease is above 1, the infection will not be sustainable in the long run – it will burn through the infected and eventually die out. R_0 is determined empirically, and is specific to a naive population, i.e. one where there is no pre-existing immunity to the disease nor is there any significant infection. It is often enough derived from the average contact rate of the population (the number of times the transmission-relevant contact occurs between people, e.g. the number of sexual contacts per unit time for sexually transmitted infections or the number of sufficient vicinity contacts for droplet/airborne transmitted infections) and the inverse of the infectious period, i.e. the multiple of unit time for which a case remains infectious.

The idea of herd immunity relates to R_0 in a particular way. Consider, once again, polio with its R_0 of 6. Now assume that half of the population is vaccinated, and assume for the time being that the vaccine is 100% effective. In practice, that means that 6 successful transmissions will at best yield 3 new cases (the unvaccinated half), whereas there will be 3 guaranteed resisters (the vaccinated half). In effect, vaccinating n% of the population cuts n% off the theoretical R_0 in a naive population – or, in other words, for a population with \frac{1}{n} vaccinated, the adjusted effective R_0 is

R_{0_{post-vaccination}} = R_{0} (1 - \frac{1}{n})

Recall that we have earlier related R_0, and specific values of it, to the survival of an infectious disease – namely that if R_0 < 1, the disease will eventually die out. The point of herd immunity is to artificially whittle down R_0 to a sub-1 value. If vaccinating \frac{1}{n } of the population reduces effective R_0 to R_0 (1 - \frac{1}{n}), then to reduce R_0 to 1 requires us to vaccinate 1 - \frac{1}{R_0}. Thus for polio, with R_0 = 6, we need to vaccinate

1 - \frac{1}{R_0} = 1 - \frac{1}{6} = \frac{5}{6}

of the population to reach a point where, assuming the vaccine is 100% effective (which it never is – not many things in reality are 100% effective), the disease cannot subsist and spread in the population for long. At that level of vaccination, even unvaccinated people are significantly protected, almost as well as if they were to be vaccinated. In theory, anyway.

Herd immunity in practice

In practice, there are two problems with this. One is that herd immunity doesn’t help you if you, for whatever reason, do things that lead to catching the disease. In other words, it is not a ‘real’ immunity the way getting vaccinated is. It ensures that the likelihood of catching the disease is significantly reduced, and that outbreaks become self-contained, but it doesn’t give you as an unvaccinated individual any immunity per se. The other problem is that herd immunity calculations are premised on 100% vaccine efficacy (which is not the same as efficiency, see the previous post for difference!). The point is that the result of the above calculation is a theoretical minimum, premised on 100% effective vaccines, for herd immunity – reaching it, or even exceeding it, does not yet guarantee herd immunity. And with information about vaccine efficacy not being easy to come by (unlike information about vaccine efficiency, which is available in abundance), it’s hard to determine a correct rate. For these reasons, anyone who could get vaccinated, should get vaccinated.3 Especially for viral diseases, herd immunity is the best protection for those who genuinely cannot get vaccinated. It’s a goal worth striving for as a population – and it costs nothing to confer herd immunity above and beyond getting immunised, which already confers the benefits of individual immunity.

References   [ + ]

1.The situation is different for illnesses where a recovered case may remain infectious, indefinitely or for a given time, e.g. for Ebola virus disease (EVD), where recovered patients continue to have actively infectious virus in their seminal fluid and vitreous humour for several years. For a case study, consider William A Fischer II et al. Ebola Virus RNA Detection in Semen More than Two Years After Resolution of Acute Ebola Virus Infection. Open Forum Infectious Diseases, 2017. DOI: 10.1093/ofid/ofx155/4004818
2.In general, airborne infectious diseases tend to have a higher R_0 than other methods of transmission, such as those that require sexual contact, blood or, as in the case of polio, fecal-oral transmission to spread.
3.A recent meme is claiming or pretending to have the MTHFR mutation and arguing that this should lead to a medical exemption from vaccinations. Most doctors haven’t heard much of MTHFR, and it sounds scary, and when you google it, most of the documentation is on the extremely rare complete MTHFR deficiency, a very rare and very serious illness that has about 70 documented cases in all of medical history. Meanwhile, some degree of MTHFR mutation is present in a good deal – somewhere between 10 to 35% – of the normal population, and is indeed so frequent without any associated clinical syndrome that it’s seen as a harmless variation in genetics. MTHFR (methylenetetrahydrofolate reductase) is an enzyme that catalyses the reduction of 5,10-methylenetetrahydrofolate to 5-methyltetrahydrofolate, which is involved in the methylation of homocysteine, yielding methionine. There is some evidence, from an NIH study with ridiculously small sample sizes, that certain MTHFR SNPs, particularly rs1801133, are associated with adverse events in smallpox vaccination. There are two problems with this. One, I don’t know of this ever having been replicated with any sample size worthy of being drawn conclusions from. Two, given the frequency of MTHFR deficiency in the population, you would expect vast numbers of ‘vaccine injured’ people, much more than the cases in VAERS and definitely more than those that have a credible claim. It makes no epidemiological sense at all. MTHFR polymorphisms are, as the science stands at the moment, not a good reason not to vaccinate.

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