Herd immunity: how it really works

There are few concepts as trivial yet as widely misunderstood as herd immunity. In a sense, it’s not all that surprising, because frankly, there’s something almost magical about it – herd immunity means that in a population, some people who are not or cannot be immunized continue to reap the benefit of immunization. On its own, this may even be counter-intuitive. And so, unsurprisingly, like many evidently true concepts, herd immunity has its malcontents – going so far as to condemn the very idea as a ‘CDC lie’ – never mind that the concept was first used in 1923, well before the CDC was established.1

Now, let’s ignore for a moment what Dr Humphries, a nephrologist-turned-homeopath with a penchant for being economical with the truth when not outright lying, has to say – not because she’s a quack but because she has not the most basic idea of epidemiology. Instead, let’s look at this alleged ‘myth’ to begin with.

Herd immunity: the cold, hard maths

Our current understanding of herd immunity is actually a result of increasing understanding of population dynamics in epidemiology, towards the second half of the 20th century. There are, on the whole, two ways to explain it. Both are actually the same thing, and one can be derived from the other.

The simple explanation: effective $R_0$$R_0$ depletion

The simple explanation rests on a simplification that makes it possible to describe herd immunity in terms that are intelligible at the level of high school maths. In epidemiology, $R_0$ (pron. ‘arr-nought‘, like a pirate), describes the basic reproduction rate of an infectious disease.2 To put it at its most simplistic: $R_0$ is the number of cases produced by each case. The illustration on the side shows the index case (IDX) and the first two generations of an infection with $R_0 = 3$.

Now, $R_0$ is a theoretical variable. It is usually observationally estimated, and don’t take measures intended to reduce it into account. And that’s where it gets interesting.

Consider the following scenario, where a third of the population is vaccinated, denoted by dark black circles around the nodes representing them. One would expect that of the 13 persons, a third, i.e. about. 4 , would remain disease-free. But in fact, over half of the people will remain disease-free, including three who are not vaccinated. This is because the person in the previous generation did not pass on the pathogen to them. In other words, preventing spread, e.g. by vaccination or by quarantine, can affect and reduce $R_0$. Thus in this case, the effective $R_0$ was closer to 1.66 than 3 – almost halving the $R_0$ by vaccinating only a third of the population.

We also know that for infections where the patient either dies or recovers, the infection has a simple ecology: every case must be ‘replaced’. In other words, if the effective $R_0$ falls below 1, the infection will eventually peter out. This happens quite often when everyone in a population is dead or immune after an infection has burned through it (more about that later).

Thus, the infection will be sustainable if and only if

$R_{0} \geq 1$

Under the assumption of a 100% efficient vaccine, the threshold value $\bar{p_V}$ after which the infection will no longer be able to sustain itself is calculated as

$\bar{p_V} = 1 - \frac{1}{R_0}$

Adjusting for vaccine efficacy, $E_V$, which is usually less than 100%, we get

$\bar{p_V} = \frac{1-\frac{1}{R_0}}{E_V} = \frac{R_0 - 1}{R_0 E_V}$

For a worked example, let’s consider measles. Measles has an $R_0$ around 15 (although a much higher value has been observed in the past, up to 30, in some communities), and the measles vaccine is about 96% effective. What percentage of the population needs to be vaccinated? Let’s consider $\bar{p_V}$, the minimum or threshold value above which herd immunity is effective:

$\bar{p_V} = \frac{R_0 - 1}{R_0 E_V} = \frac{15-1}{15 \cdot 0.96} = \frac{14}{14.4} \approx 97.22\%$

The more complex explanation: $\mathcal{SIR}$$\mathcal{SIR}$ models

Note: this is somewhat complex maths and is generally not recommended unless you’re a masochist and/or comfortable with calculus and differential equations. It does give you a more nuanced picture of matters, but is not necessary to understand the whole of the argumentation. So feel free to skip it.

The slightly more complex explanation relies on a three-compartment model, in which the population is allotted into one of three compartments: $\mathcal{S}$usceptible, $\mathcal{I}$nfectious and $\mathcal{R}$ecovered. This model makes certain assumptions, such as that persons are infectious from the moment they’re exposed and that once they recover, they’re immune. There are various twists on the idea of a multicompartment model that takes into account the fact that this is not true for every disease, but the overall idea is the same.3 In general, multicompartment models begin with everybody susceptible, and a seed population of infectious subjects. Vaccination in such models is usually accounted for by treating them as ‘recovered’, and thus immune, from $t = 0$ onwards.

Given an invariant population (i.e. it is assumed that no births, deaths or migration occurs), the population can be described as consisting of the sum of the mutually exclusive compartments: $P = \mathcal{S}(t) + \mathcal{I}(t) + \mathcal{R}(t)$. For the same reason, the total change is invariant over time, i.e.

$\frac{d \mathcal{S}}{d t} + \frac{d \mathcal{I}}{d t} + \frac{d \mathcal{R}}{d t} = 0$

Under this assumption of a closed system, we can relate the volumes of each of the compartment to the transition probabilities $\beta$ (from $\mathcal{S}$ to $\mathcal{I}$) and $\gamma$ (from $\mathcal{I}$ to $\mathcal{R}$), so that:

$\frac{d \mathcal{S}}{d t} = - \frac{\beta \mathcal{I} \mathcal{S}}{P}$

$\frac{d \mathcal{I}}{d t} = \frac{\beta \mathcal{I} \mathcal{S}}{P} - \gamma \mathcal{I}$

$\frac{d \mathcal{R}}{d t} = \gamma \mathcal{I}$

Incidentally, in case you were wondering how this connects to the previous explanation: $R_0 = \frac{\beta}{\gamma}$.

Now, let us consider the end of the infection. If $\mathcal{S}$ is reduced sufficiently, the disease will cease to be viable. This does not need every individual to be recovered or immune, however, as is evident from dividing the first by the third differential equation and integrating and substituting $R_0$, which yields

$\displaystyle \mathcal{S}(t) = \mathcal{S}(0) e^{\frac{-R_0 (\mathcal{R}(t)-\mathcal{R}(0))}{P}}$

Substituting this in, the limit of $\mathcal{R}$, as $t$ approaches infinity, is

$\displaystyle \lim_{t\to\infty}\mathcal{R}(t) = P - \lim_{t\to\infty}\mathcal{S}(t) = P - \mathcal{S}(0) e^{\frac{-R_0 (\mathcal{R}(t)-\mathcal{R}(0))}{P}}$

From the latter, it is evident that

$\displaystyle \lim_{t\to\infty}\mathcal{S}(t) \neq 0 \mid \mathcal{S}(0) \neq 0$

In other words, once the infection has burned out, there will still be some individuals who are not immune, not immunised and not vaccinated. These are the individuals protected by herd immunity. This is a pretty elegant explanation for why herd immunity happens and how it works. There are three points to take away from this.

First, herd immunity is not unique to vaccination. The above finding in relation to the nonzero limit of $\lim_{t\to\infty}\mathcal{S}(t)$ holds as long as $\mathcal{S}(0) \neq 0$, but regardless of what $\mathcal{R}(0)$ is. In other words, herd immunity is not something artificial.

Two, for any $i \in \mathcal{S}$ (that is, any susceptible person) at time $t$, the probability of which compartment he will be in at $t+1$ depends on whom he encounters. That, statistically, depends on the relative sizes of the compartments. In this model, the assumption is that the sample $i$ will encounter will reflect the relative proportions of the individual compartments’ sizes. Thus if $i$ meets $n$ people at time $t$, each compartment will be proportionally represented, i.e. for any compartment $\mathcal{C}$, the proportion will be $\frac{\mathcal{C}(t)}{P-1}$ for all $\mathcal{C} \neq \mathcal{S}$, for which the proportion will be $\frac{\mathcal{S}(t) - 1}{P - 1}$, since one cannot meet oneself. Given that the transition probability $\beta_{i}(t)$ is assumed to equal the probability of meeting at least one element of $\mathcal{I}$, the following can be said. $i$‘s risk of infection depends on the relationship of $n$ and $\mathcal{I}(t)$, so that $i$ is likely to get infected if

$\displaystyle n \frac{\mathcal{I}(t)}{P-1} \geq 1$

This elucidates two risk factors clearly, and the way to reduce them: reduce interactions (quarantine/self-quarantine), thereby reducing $n$, and reduce the proportion of infectious cases ($\frac{\mathcal{I}(t)}{P-1}$). The latter is where herd immunity from immunisation comes in. Recall that for a constant $n$, $i$‘s risk of infection at $t$ rises as $\mathcal{I}(t)$ rises.4 Recall also that while susceptible cases can turn into infectious cases, recovered (or vaccinated) cases cannot. And so, as $\mathcal{R}(0)$ converges to $P-1$,5 $i$‘s risk of infection at any time $t$, denoted by $\beta_{i}(t)$, falls. In other words,

$\displaystyle \lim_{\mathcal{R}(0) \to P-1} \beta_{i}(t) = 0$

Or to put it simply: the more are vaccinated at the start, the lower the probability, all things being equal, to meet someone who can pass on the infection.6

A final point to note is that this is primarily a model of statistical dynamics, and deals with average probabilities. It does not – it cannot – take account of facts like that some some susceptible people are just darn unlucky, and bump into a flock of unvaccinated, shiny-eyed snowflakes. Equally, in some places, susceptible people and infected people congregate, creating a viral breeding ground, also known as a Waldorf school. There are agent based models, which are basically attempts at brute force hacking reality, that can take account of such disparities. The takeaway is that herd immunity does not mean no susceptible individual will get infected. What it does mean is that their probability of getting infected is going to be significantly lower, for two reasons. First given a constant number of encounters ($n$), the likelihood of one of them being with an infectious individual is going to be much lower. More importantly, however, because of herd immunity, the disease is going to be able to persist in the population for a far shorter time – eventually it will burn through the small number of ‘accessible’ susceptible persons. Since the cumulative risk $\beta_{i}^T$ for $i \in \mathcal{S}$ for an infection that dies out after time $T$ is defined as

$\beta_i^T = \displaystyle \int\limits_0^T \beta_{i}(t) \, \mathrm{d}t$

– the sooner the infection dies out, the smaller the likelihood that $i$ will be infected. With that mathematical basis, let’s tackle a few of the myths about herd immunity.

Myth #1: herd immunity only works with naturally acquired immunity

This argument goes roughly along the following lines: herd immunity does exist, but it only exists if and where the immunity is acquired the ‘natural’ way, i.e. by surviving the disease. Case in point:

The \$64,000 question, of course, is what the difference is between the residual immunity from a vaccine and the residual immunity from having survived the illness. A vaccine effectively ‘simulates’ the illness without actually causing the pathological symptoms. From the perspective of the immune system, it is largely irrelevant whether it has been exposed to an actual virus that can damage the body, or merely a capsid protein that is entirely harmless but will nonetheless elicit the same immune reaction. That should suffice to bust this myth, but it’s worth considering immunity quantitatively for a moment. As we have seen above, the source of immunity doesn’t matter. In fact, it doesn’t even have to be immunity: culling every animal except one in a herd is an entirely good way to reduce disease transmission. So is sealing oneself away from the rest of society and spending the evenings telling sexually explicit stories, as the heroes and heroines of Boccaccio’s Decameron have done, since we know that

$\displaystyle n \frac{\mathcal{I}(t)}{P-1} \geq 1$

Boccaccio’s crowd of assorted perverts knew nothing of all this, of course, but they did know that if they reduced $n$, the number of contacts with possibly infected persons, their chances of surviving the plague would increase. As it indeed did. Score one for medieval perverts. The bottom line is that it is entirely immaterial how immunity was obtained.

Myth #2: Herd immunity is a concept deriving from animals. It doesn’t work on humans.

This is one of the more outlandish claims, but shockingly, it actually has a tiny kernel of truth.

Now, much of the above is a veritable storehouse of insanity, but the point it makes in the beginning has some truth to it. In human populations, herd immunity sometimes behaves anomalously, because humans are not homogenously distributed. This is true a fortiori for humans who decide not to vaccinate, who – for better or worse – tend to flock in small groups. The term of venery for a bunch of anti-vaxxers is, in case you were wondering, a ‘plague’.7

Herd immunity was, in fact, observed in a range of species. Humans are different as we can knowingly and consciously decide to create herd immunity in our population and protect our fellow men, women and children, the last of whom are particularly susceptible to infectious diseases, from some of the worst killers.

Myth #3: If herd immunity can be obtained through natural immunity, surely we don’t need vaccines.

This argument has recently been peddled by the illustrious Kelly Brogan MD, who bills herself as a ‘holistic psychiatrist’ who threw away her script pad, which means she tends exclusively to the worried well and those with mild mental health issues where medication does not play as decisive a role as it does in, say, schizophrenia, severe PTSD, crippling anxiety disorders or complex neuropsychiatric post-insult phenomena.8 Here’s her foray into epidemiology, something she vaguely remembers studying in her first year of med school.

In this, Dr Brogan has successfully found almost a century old evidence for what everybody knew, namely that herd immunity can be naturally obtained. To anyone who has read the maths part above, this should evoke a sensation of ‘DUH!’. The problem is twofold. One, the ‘actual virus’ has an unsavoury fatality rate of 0.1%, not including the horribly tragic, heartbreaking late consequence of measles known as SSPE.9 Two, and perhaps more important: you don’t get lifelong, natural immunity if you die. This may have somehow escaped Dr Brogan’s attention, but part of the point of herd immunity is to protect those who would not survive, or would suffer serious sequelae, if they contracted the infection. What we don’t know, of course, how many of that 68% suffered permanent injuries, and how many are not included because they died. What we do know is that all 68% probably had a miserable time. Anyone who thinks measles is so fantastic should start by contracting it themselves.

Myth #4: Herd immunity means 95% need to be vaccinated to prevent a disease.

This one comes courtesy of Sarah aka the Healthy Home Economist,10, who, to what I presume must be the chagrin of her alma mater, states she has a Master’s from UPenn. Suspiciously enough, she does not state what in. I am somehow pretty sure it’s not public health.

The tedious conspiracy theory aside, it is quite evident just how little she understands of herd immunity. No – herd immunity is not based upon11 the idea that 95% must be vaccinated, and it is most definitely not based on the idea that 100% must be vaccinated. Indeed, the whole bloody point of herd immunity is that you do not need to vaccinate 100% to protect 100%. In fact, given the $R_0$ and vaccine efficacy $E_V$, we can predict the threshold vaccination rate for herd immunity quite simply, as demonstrated earlier: the threshold value, $\bar{p_V}$, can be calculated as

$\bar{p_V} = \frac{R_0 - 1}{R_0 E_V}$

As an illustration, the herd immunity threshold $\bar{p_V}$ for mumps, with an efficacy of 88%12 and an $R_0$ of around 5.5, is $\approx 92.98\%$, while for Ebola, which has a very low $R_0$ around 2.0, herd immunity sets in once about 50% are immune.13

And those ‘conventional health authorities’? That’s what we call health authorities whose ideas work.

Myth #5: If vaccines work, why do we need herd immunity?

This argument is delightfully idiotic, because it, too, ignores the fundamental underlying facts of herd immunity. Quite apart from the fact that some people cannot receive some or all vaccines and other people can receive vaccines but may not generate sufficient antibody titres to have effective immunity, sometimes vaccines simply fail. Few things are 100%, and while vaccines are designed to be resilient, they can degrade due to inappropriate storage or fail to elicit a sufficient response for some other reason. Unlike wearing deodorant (or ‘deoderant’, as spelling-challenged anti-vaxxers would say), infections can sometimes be imagined as a chain of transmission. This is a frequently used model to explain the consequences of not vaccinating on others.

In this illustration, an index patient (IDX) is infected and comes in contact with G1, who in turn comes into contact with G2 who in turn comes into contact with G3. In the first case, G1, G2 and G3 are all vaccinated. The vaccine may have a small failure rate – 2% in this case – but by the time we get to G3, his chances of contracting the infection are 1:125,000 or 0.0008%. In the second case, G2 is unvaccinated – if G1’s vaccine fails, G2 is almost guaranteed to also fall ill. By not vaccinating, his own risk has increased 50-fold, from 0.04% to 2%. But that’s not all – due to G2’s failure to vaccinate, G3 will also be affected – instead of the lottery odds of 1:125,000, his risk has also risen 50-fold, to 1:2,500. And this 50-fold increase of risk will carry down the chain of potential transmission due to G2’s failure to vaccinate. No matter how well vaccines work, there’s always a small residual risk of failure, just as there is a residual risk of failure with everything. But it takes not vaccinating to make that risk hike up 50-fold. Makes that deodorant (‘deoderant’?) analogy sound rather silly, right?

Conclusion

Admittedly, the mathematical basis of herd immunity is complex. And the idea itself is somewhat counterintuitive. None of these are fit excuses for spreading lies and misinformation about herd immunity.

I have not engaged with the blatantly insane arguments (NWO, Zionists, Masonic conspiracies, Georgia Guidestones), nor with the blatantly untrue ones (doctors and public health officers are evil and guided just by money as they cash in on the suffering of innocent children). I was too busy booking my next flight paid for by Big Pharma.14 Envy is a powerful force, and it’s a good way to motivate people to suspect and hate people who sacrificed their 20s and 30s to work healing others and are eventually finally getting paid in their 40s. But it’s the myths that sway the well-meaning and uncommitted, and I firmly believe it’s part of our job as public health experts to counter them with truth.15

In every social structure, part of co-existence is assuming responsibility not just for oneself but for those who are affected by our decisions. Herd immunity is one of those instances where it’s no longer about just ourselves. Many have taken the language of herd immunity to suggest that it is some sort of favour or sacrifice done for the communal good, when it is in in fact the very opposite – it is preventing (inadvertent but often unavoidable) harm to others from ourselves.

And when the stakes are this high, when it’s about life and death of millions who for whatever reason cannot be vaccinated or cannot form an immune response, getting the facts right is paramount. I hope this has helped you, and if you know someone who would benefit from it, please do pass it on to them.

References   [ + ]

 1 ↑ Topley, W. W. C. and Wilson, G. S. (1923). The spread of bacterial infection; the problem of herd immunity. J Hyg 21:243-249. The CDC was founded 23 years later, in 1946. 2 ↑ Why $R_0$$R_0$? Because it is unrelated to $\mathcal{R}$$\mathcal{R}$, the quantity denoting recovered cases in $\mathcal{S(E)IR}$$\mathcal{S(E)IR}$ models – which is entirely unrelated. To emphasize the distinction, I will use mathcal fonts for the compartments in compartment models. 3 ↑ I hope to write about SIS, SEIR and vital dynamic models in the near future, but for this argument, it really doesn’t matter. 4 ↑ Technically, as $\frac{\mathcal{I}(t)}{P - 1}$$\frac{\mathcal{I}(t)}{P - 1}$ rises, but since the model presupposes that $P$$P$ is constant, it doesn’t matter. 5 ↑ Since otherwise $\mathcal{R} = P$$\mathcal{R} = P$ and $\mathcal{S} = 0$$\mathcal{S} = 0$, and the whole model is moot, as noted above. 6 ↑ Note that this does not, unlike the $R_0$$R_0$ explanation, presuppose any degree of vaccine efficacy. An inefficiently vaccinated person is simply in $\mathcal{S}$$\mathcal{S}$ rather than $\mathcal{R}$$\mathcal{R}$. 7 ↑ Initially, ‘a culture’ was proposed, but looking at the average opponent of vaccination, it was clear this could not possibly work. 8 ↑ In other words, if you have actual mental health issues, try an actual psychiatrist who follows evidence-based protocols. 9 ↑ Subacute sclerosing panencephalitis is a long-term delayed consequence of measles infection, manifesting as a disseminated encephalitis that is invariably fatal. There are no adjectives that do the horror that is SSPE justice, so here’s a good summary paper on it. 10 ↑ As a rule, I don’t link to blogs and websites disseminating harmful information that endangers public health. 11 ↑ Correct term: ‘on’ 12 ↑ As per the CDC. 13 ↑ Efficacy $E_V$$E_V$ is presumed to be 100% where immunity is not acquired via vaccination but by survival. 14 ↑ Anyone in public health is happy to tell you those things don’t merely no longer exist, they never even existed in our field. 15 ↑ And, if need be, maths.

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