The Simple Math of Herd Immunity

I have often come across claims that wish to contest the existence of herd immunity. I find these puzzling. First of all, they are usually offered without proof. But mostly what befuddles me is that the argument approaches the topic backwards. It starts at the wrong end. Let me explain.

There is a simple way of breaking down the factors that make up herd immunity.

Herd immunity, also known as community immunity, is a population-scale name for a phenomenon in the context of disease that can be passed from person to person, and it follows from two factors: vaccine efficacy and vaccination rate. Firstly, vaccines convey immunity to disease. Immunised individuals have antibodies that will neutralise germs when they come in contact with them, making it much less likely to pass on to others – people who are immune don’t get sick and don’t spread the disease. Secondly, if nearly everyone is immune, then almost no one will spread the disease. Thus, even people who have not been vaccinated (and those whose vaccinations have become weakened or whose vaccines aren’t fully effective) often can be shielded by the herd immunity because vaccinated people around them are not getting sick. Herd immunity is more effective as the percentage of people vaccinated increases. For more, see the World Health Organisation on herd immunity.

The vaccination rate that is critical for stopping the spread of disease depends on how infectious the disease is. For measles for instance, which is highly contagious, approximately 95% of the people in the community must be protected by a vaccine to achieve sufficient herd immunity. You can watch a great animation of measles infection spread in populations with different vaccination rates over at The Guardian: Watch how the measles outbreak spreads when kids get vaccinated – and when they don’t. For Hib in Gambia and Navajo populations on the other hand, less than 70% vaccine coverage was sufficient to eliminate the disease, as reported by the WHO.

People who are not immunized increase the chance that they and others will get the disease. It is important to note that there will always be some people who rely on herd immunity rather than individual immunity to stop disease, such as:

The math of herd immunity

The simple math of the concepts for herd immunity is this:  vaccines make people immune + majority of people are vaccinated = community is protected from the spread of disease.

Saying “there is no herd immunity” or “herd immunity does not work” is a roundabout way of saying either that “vaccines don’t protect from disease” or “vaccination rates are not high enough”. That would be a better way of starting the discussion. Then we (and the claimant) would directly know what they were in fact talking about.

When a disease incident occurs, of the kind of infectious disease which (in the absence of immunity) can be passed from person to person, then the situation can be analysed with the four scenarios presented in the table above.

For anyone interested in the the finer details of the real math and biology behind herd immunity, you can read “Herd Immunity”: the rough guide published in the journal of Clinical Infectious disease, or Wikipedia on the mathematics of mass vaccination. I’ll just mention that the necessary vaccination coverage is calculated with the so called reproduction number (R0) of the particular disease, a factor simply for how many people will be infected on average by the person with the disease. What is known as the “herd immunity threshold” or “the critical immunisation threshold” (denoted qc) can then be calculated with the reproduction number:

qc = 1 − 1/R 

Calculated this way, qc gives the percentage of fully immune people required to stop the spread of the disease. In real life scenarios, neither disease- nor vaccine-conveyed immunity is ever quite 100%, and this has to be accounted for by introducing the factor E – vaccine efficacy, or the percentage of people who received the vaccine and will be immune. For measles, which has a reproduction number 12 (or 11-18), and when we factor in vaccine efficacy E, which for measles is about 97 %, we get the vaccine coverage (Vc) necessary for herd immunity, (qc), by the following formula:

Vc =qc/E

Vc = (1 – 1/12) / 0.97 = 0.945

In other words, roughly 94.5 % measles vaccine coverage is necessary to stop the spread of measles in a population.

To recap: herd immunity is a direct effect of vaccine efficacy, that is, the fact that vaccines protect against disease. If one wishes to contest herd immunity, it’s not the term “herd immunity” wherein the problem lies. The problem is either with the vaccination rate (as we are seeing in many places) or with vaccine-induced immunity. The claim of “no herd immunity” really is this: “vaccines do not prevent disease”. This, already, is a more useful starting point for the discussion. All we have to do is look at the evidence – do vaccines prevent the disease they are meant to prevent, and how well?

Vaccines work

There is an incredibly informative site that has graphs (with sources to the raw data) that illustrate the effects on population level of the introduction of vaccines against measles, polio, diphtheria, pertussis, Hib, and Hep B in the United States as well as in the United Kingdom. Please take a look at their great graphs. I’ll only include one  here – a graph of measles vaccine introduction in England and Wales. This graph I find particularly interesting because it shows an effect similar to one known in research as the “dose-response” relationship – a very robust method of determining causality.

The change illustrated by this data isn’t a one-time occurrence, a single time-point at which the vaccine was introduced and the disease disappeared simultaneously – for many vaccines, where vaccine coverage was efficient from the start (people were eager to make sure they got their shots as soon as they got available) the disease disappeared rapidly. Instead, here we can follow the effect of gradually increasing vaccination rates (the dotted line), and see that the more people that got the vaccine, the fewer the cases of measles. The graph also shows the dip in the vaccination rates, as they venture below 90% soon after Mr Wakefield began fuelling the fears of the anti-vaccine campaigners around the beginning of the century. The resulting epidemics in the UK have so far only reached over a thousand cases per season, and are thus luckily but a slight bump at the end of the red curve:

Here a comment from federal health resource site in the same context:

If the drop in disease were due to hygiene and sanitation, you would expect all diseases to start going away at about the same time. But if you were to look at the graph for polio, for example, you would see the number of cases start to drop around 1955 – the year the first polio vaccine was licensed. If you look at the graph for Hib, the number drops around 1990, for pneumococcal disease around 2000 — corresponding to the introduction of vaccines for those diseases.

Again, you can find all the individual graphs for different vaccines in the link where the measles graph came from, here. In addition to the demonstrative disease prevention data from when standard childhood vaccines have been adopted, you can also find scientific publications that have reported specifically on this naturally following result of vaccine efficacy – the herd immunity. Here a study that shows the herd immunity effect with increased protection of unvaccinated populations after introduction of the rotavirus vaccine, for instance. The same is true for the varicella vaccine that protects from chicken pox, excerpt from a study:

Varicella incidence, hospitalizations, and outbreaks in 2 active surveillance areas declined substantially during the first 5 years of the 2-dose varicella vaccination program. Declines in incidence across all ages, including infants who are not eligible for varicella vaccination, and adults, in whom vaccination levels are low, provide evidence of the benefit of high levels of immunity in the population.

Despite all this data some people still feel cheated by vaccines. They claim that vaccines do not deliver as promised, because vaccine induced immunity wanes with time. What’s the deal with that? Let’s look at it next.

So what about waning immunity? Vaccine-induced vs “natural”?

Are vaccines in fact a “fraud” if immunity against diseases can decrease over time? If you approach the question from the perspective of our natural immunity, this isn’t a “fault” of vaccines, no more than it is a “fault” of the diseases themselves when surviving them does not provide a life-long immunity. In fact, natural immunity does not necessarily provide longer-lasting protection than does a vaccine, as found in this study:

A review of the published data on duration of immunity reveals estimates that infection-acquired immunity against pertussis disease wanes after 4-20 years and protective immunity after vaccination wanes after 4-12 years. Further research into the rate of waning of vaccine-acquired immunity will help determine the optimal timing and frequency of booster immunizations and their role in pertussis control.

It is a common trend for an immunity to a disease to wane over a few years or decades. There are things we can do to achieve a longer lasting immunity – like using adjuvants that allow for a stronger immune reaction at the time of vaccination. For some diseases, booster shots may be necessary. Waning immunity is not a “fault” of neither, the diseases nor the vaccines, it is no more or less than a facet of our immune system.

Thanks to vaccines we can harness our body’s natural reaction to pathogens. Our bodies need information about which threats they should protect us from. With remarkably little side-effects, vaccines manage that great feat of delivering that information, of creating immunity, which, before vaccines, was not possible without first enduring the risk of death and disability (at least once). Back then, being in the lucky group – those who would not need to worry about the next wave of disease – may have been a bleak consolation in comparison to the worry of not knowing which of your siblings, friends and relatives would not survive the next ‘immunisation round’.

For more information on the durations of the vaccine induced immunity of our current vaccines, the New Zealand Immunisation Advisory Centre (IMAC) of The University of Auckland have put together a page with an overview of factors that affect vaccine induced immunity and an informative table summary:

Sadly, it appears to me that talking about herd immunity instead of vaccine efficacy may be an (intentional or) unintentional way of obfuscating the topic – an argument about herd immunity may puzzle the discussion participants and it may not be as clear which kind of evidence we should be turning to in order to get to the bottom of the claim. I think many parents searching for information can become confused by similar claims, and, not easily finding a direct answer, may begin to fear the perceived uncertainties in the vaccine debate. With this blog post I wanted to help clarify the concept.

The most important thing to keep in mind, is that while individual arguments may easily seem convoluted and discussions can feel confusing, the discussion of real importance *from the point of view of the evidence*, is the one conducted in the form of scientific publications. So far all the scientific evidence indicates that risks from vaccines are several orders of magnitude lower than the risks from the diseases themselves – in other words, there is a scientific consensus on vaccine safety. If a body of scientific papers should demonstrate a dramatically lowered efficacy of a vaccine, or a substantial risk for harm from a vaccine, we should all take heed. Note that should such an effect be found, they would be the top-ranking news in the scientific community, and anyone reporting such results (should they turn out robust and confirmable by other studies) would be at the high point of their scientist’s career. No amount of coercion or conspiracy could keep the lid on that kind of knowledge. Being a scientist isn’t a glorious occupation, it’s a tough and relatively poorly paying one. Scientists certainly aren’t evil. They are people just like you and me. Scientists love knowledge and understanding of the natural world above all else, that’s why they’re doing what they’re doing.

To quote the summary on herd immunity by L. Shaka, a parent who used to run a site for information on vaccines:

So how do we know that herd immunity exists? First and foremost, if one accepts that vaccines are effective in stopping disease, it is an inevitable logical conclusion.  If many, many people are immune the disease will not spread too far. Those that are not vaccinated, and are surrounded by a wall of immune people, will most likely not be exposed to the virus as the spread will stop before reaching them.

Secondly, we can show mathematically how to calculate the herd immunity threshold, by taking into account well established concepts such as the basic reproduction number of the disease.

To sum it up the idea of herd immunity is biologically plausible, makes sense logically, can be mathematically modeled based on uncontroversial factors and simple algebra, and is supported by scientific observations and studies.

In other words, to deny herd immunity is to deny biology, logic, common sense, math and science, all in one swoop.

I will round off this post with one last graphic from, illustrating the effect of vaccine efficacy in a population – a.k.a. herd immunity.

You can continue here if you’d like to read more about: Vaccines and Health. If you would like to discuss the topic below, you are very welcome, but please take note of my Commenting policy.

In a nutshell:

  1. Be respectful.
  2. Back up your claims with evidence.