I have as of late heard multiple people proudly proclaim that they do not “believe in” herd immunity which is… perplexing.

What’s Herd Immunity?

Herd immunity, or community immunity, can have a few different definitions, but basically, it’s a state of protection in a given population where even those who are not immune to an infectious disease are protected because the number of people in the population who are immune is above a critical threshold.

Saying you don’t believe in herd immunity is kind of like saying you don’t believe that 1+1=2. Much like gravity and evolution, it’s not really something that people can opt into or out of believing in… it’s just… how things work. In fact, herd immunity can be shown to exist, at least on a conceptual level, with math alone.

Herd immunity can be derived rigorously through some differential equations but that’s a bit esoteric for most people, so instead we can take a much simpler approach. Herd immunity is an extension of just 2 premises:

1. To get an infectious disease, one has to come into contact with the pathogen responsible for that infectious disease.

2. Individuals who are immune to a given infectious disease will not have the pathogen build up to a level sufficiently high to cause transmission.

Therefore, the more people in a given population are immune, the less likely it is for an individual to come into contact with that infectious agent.

Eventually, as the proportion of individuals who are susceptible continues to drop, we reach a point where even those who are not immune will not get sick because they are not exposed to the infectious disease.

This is the crux of herd immunity. If you don’t care or enjoy science (you’re really missing out) and you want to understand this, this is all you need to understand and you need not read any further- I will only be expanding on this point: the more people are immune to a disease, the less likely people who are not immune to the disease are to get the disease because the less likely an interaction where they are exposed to the pathogen will happen.

Before we begin:

A persistent point of confusion that is worth addressing: elimination in epidemiology means that there is no indigenous transmission of the disease, which is to say, no cases originate inside the region of interest. All cases must be brought in. This term is often conflated with eradication, which means that there are no cases anywhere in the world of the infectious disease, and only 2 diseases have been eradicated thus far: smallpox and rinderpest (a disease of cattle)- both through aggressive vaccination campaigns. However, a number of diseases have been eliminated from certain nations. For instance, polio has been eliminated from all but 3 nations at the time of writing this, and types 1 and 2 have been eradicated so only type 3 remains.

R’s and Naughts

There’s another factor to consider though- how communicable the infectious agent is, and this can vary widely. We quantify this with something known as the basic reproduction number- R0 (pronounced “r-naught”). This is the average number of secondary cases we expect to see from a single primary case in a population of individuals where no one is immune to the infectious disease. Put another way, it tells us the average number of new cases we should expect to see if we take 1 sick person and introduce them to a group of people who have no immunity to that disease. For a pathogen to be able to reliably transmit through a population for an extended period of time, it needs to have at least one infected individual in that population at any given time. This means that every person who has the disease must be able to infect, on average, at least 1 person, or R0 must be greater than or equal to 1. For R0<1, there is no one to replace some of the infected people, so with each generation of the infection, the disease has fewer and fewer infected individuals until there are none. Importantly, R0 does not indicate anything about the severity of the disease in question. In fact, a disease that is rapidly very deadly will typically have a lower R0 than one that is not as the former has less opportunity to cause new infections. R0 can vary widely across different pathogens, and depends on an incredible number of things:

• the mode of transmission e.g. airborne vs. direct contact

• the duration of infectiousness

• the density of the population in question (i.e. are many people living closely together or are they spread out a lot?)

• the frequency of contacts between its members that facilitate the transfer of infection

• the overall health status of the population in question i.e. how many people can be immune and retain that immunity for a given amount of time

• cultural factors e.g. Ebola requires contact with an individual’s bodily fluids to spread and some of the rituals related to the preparation of deceased bodies in endemic nations helped facilitate this transfer resulting in infections.

However, though R0 can vary greatly from outbreak to outbreak, it is still a constant with specific assumptions in mind, detailed later.

Key Periods

One factor in determining R0 is the manner in which a pathogen is transmitted. For instance, measles is transmitted in fine aerosolized droplets, which can remain in the air for several hours. HIV however requires direct contact with infected individuals’ body fluids. Hence measles is much more communicable than HIV. Another factor is how long the infectious period is- that is how long a person who is infected is contagious. This is generally very hard to determine exactly, so we generally consider incubation period to help us. The incubation period is a period of time before the onset of acute symptoms of the infectious disease, during at least parts of which the individual in question can be contagious without any overt signs or symptoms. Note that this is not the same as the similar concept of latent period which is the amount of time it takes for a person to become infectious from the time that they become infected, though these can overlap for many diseases. These concepts are why it is generally not sufficient to “keep [my] child home when they’re sick.” The fact is that for a number of diseases, the infectious period and incubation period overlap. Hence it’s possible to give someone a disease you don’t “have.” For instance, you can transmit influenza for about 1 day before you have any symptoms.

For measles, the incubation period can be as long as 21 days and individuals are generally considered infectious from 4 days before the rash to 4 days after (this typically comes out to 8–10 days)- which creates a very long window for them to pass the disease to others. This is another reason why measles is so communicable. Measles has the highest R0 of any known human pathogen at 12–18 (depending on the specific strain), which, to reiterate, means that for every 1 case of measles we have, we can expect an average of between 12 and 18 new cases. Each of those cases can go on to produce 12–18 more cases, and so on. Hence before the vaccine, EVERYONE got measles! One of the problems with R0 is that it is an average and averages are blind to extreme values. For specific outbreaks R0 for measles has ranged from 3.7 to 203.3 (no, that is not a typo).

Effective Reproduction Number

A related concept to R0, the basic reproduction number, is R, the effective reproduction number, sometimes called the net reproductive number or just reproductive number. The key issue with using R0 is that it carries a rather huge assumption: that everyone in the population is susceptible to a disease. This assumption is not typically valid, and when it is valid, it is only temporarily valid. If one considers the introduction of smallpox and other diseases to the New World for instance, it might be appropriate for initial calculations to use R0 to model the infection. But some individuals would survive and gain immunity. If children were born to mothers who had lived through these diseases, then, depending on the nature of the disease, they would have immunity for the first few months of life through transplacental maternal antibodies (or potentially through IgA in breastmilk). Hence we are now no longer in a position where everyone in a population is susceptible and R0 is no longer suitable, so we have to use R, which is essentially what R0 amounts to in the population where only some people are susceptible. A very simple conversion between the 2 exists:

R = sR0

where s is the proportion of individuals in a population that are susceptible. As s approaches 0, R approaches 0, which demonstrates that important point once more: the more immune individuals you have to a disease the harder it is for the disease to spread. For instance, if I take a very communicable pathogen whose R0 = 8, and only half of my population is immune, R drops to 4. Let’s define another quantity: c, the proportion of individuals in the population who must be immune. Then:

1-s = c

1- R/R0 = c

For the case that R < 1, any infection will not be able to sustain itself through a population and will burn out. Thus, the critical proportion of individuals needed to be immune for any infectious disease to not be able to sustain an epidemic is:

1–1/R0 = c = C

Where C is now the critical threshold for herd immunity. At values above C, any infection will not be able to sustain itself through a population because with each successive generation there will be fewer and fewer cases. There’s a non-trivial problem though: generally to have immunity to a disease one must live through it, and living through a disease so that you don’t get the disease is… um… well that doesn’t make much sense. It’s sort of like using pregnancy for birth control. Fortunately, this is where vaccination comes in. Vaccination can be used to effectively and rapidly deplete the susceptibility of a population to infectious disease and therefore bring us to a situation where the immunity of the population is above threshold levels. This requires just a slight adjustment to the math:

(1–1/R0)/V = C

where V is the efficacy of the vaccine. If we assume that everyone gets vaccinated, then a vaccine must have a value of V = C at least to be capable of eliminating a disease from a given population. A very simplified table I made is presented below showing how these variables relate.

This is actually excellent news: the effectiveness of a vaccine need not be anything close to 100% to prevent most common diseases and attain herd immunity, assuming that uptake of the vaccine is high.

“BUT WAIT! NATURAL IMMUNITY AND VACCINE IMMUNITY ARE DIFFERENT!”

I’ll elaborate in a subsequent post on the many ways this idea is incorrect from the perspective of immunology, but in short, whether you have disease-acquired immunity or vaccine-acquired immunity does not matter in this model. The only question is whether or not you are immune, which is really a yes or no question. If you are not immune, you are susceptible, and thus have to rely on those around you not carrying the pathogen, which is much more likely if you are not vaccinated for any vaccine preventable disease. In some cases it’s quite dramatic. For instance unvaccinated individuals are 35 times more likely to get measles than vaccinated ones.

So yes, herd immunity can absolutely be attained through vaccination, provided that the vaccine efficacy is above the threshold for herd immunity. But what if it’s not? Should we just avoid vaccinating then? Some protection will always be better than none at all, and any immunity in the population will make R < R0. A better vaccine should continue to be pursued, however.

Some might protest that we have high vaccination rates above the herd immunity threshold and we still get vaccine-preventable diseases. Herd immunity needs to be examined on a local level. Across the US right now, about 92% of infants are immunized against measles. That number is actually not high enough, unfortunately, based on the simple calculations above. But in the past, when we were above the threshold for herd immunity on the national scale, there was an incident in which a Somali community in Minnesota lost confidence in the measles vaccine after Andrew Wakefield paid them a visit and proceeded to disseminate lies about it as a false authority on the matter. The vaccination rate dropped to 42% and an outbreak occurred. And yet, this would have had virtually no impact on the overall national immunization rate, because compared to the whole US, the size of this community is tiny. Averages are blind to extreme values. That’s why public health experts are concerned primarily with pockets of un- or undervaccinated individuals (though that depends on the nation).

It’s not acceptable to choose not to immunize your child because you plan to hide in the herd. That thinking precipitates the kind of epidemics we are all trying to avoid. By the same token, it is a terribly unfortunate position to be in indeed when you have to rely on others, especially multiple others, to make the right choice for the sake of your own wellbeing, and for a number of individuals there is no such choice because they cannot receive the vaccine.

Key References and Recommended Further Reading

1. BECKER N. MODELING TO INFORM INFECTIOUS DISEASE CONTROL. [S.l.]: CRC PRESS; 2019.

2. NELSON K., WILLIAMS C.M. INFECTIOUS DISEASE EPIDEMIOLOGY. Burlington, MA: JONES & BARTLETT LEARNING; 2014.