The term “herd immunity” elicits strong responses from some in the anti-vaccine camp. Perhaps some do not like the use of the word “herd” because of its association with sheep, or other such animals that do the bidding of their herders. Or perhaps, and this is my belief, it is because the very idea of herd immunity rests upon the premise that vaccines are effective at stopping disease progression, and vaccine efficacy is one of the major things anti-vaxers deny.

Many times they demand proof that herd immunity exists. In and of itself, that request is not unreasonable. After all, scientific proof is the standard by which we measure claims; so what sort of support is there for the idea of herd immunity?

**What is herd immunity?**

First and foremost let us quickly define what herd immunity is. Basically the idea of herd immunity says that in large groups of individuals, in regards to contagious diseases (those that spread from individual to individual), if a large enough number of individuals is immune to the disease, the chances that a chain of disease transmission will be interrupted are very high, resulting in self-contained, small outbreaks that will die out quickly. Thus, even individuals that are not immune will be protected by the wall that is set up by the vaccinated ones.

**The herd’s immunity shields those that have no individual immunity**.The herd immunity threshold (the number of immune individuals in a herd at which level the disease cannot persist) varies depending on how contagious the disease in question is, how efficacious the vaccine is, the exposure rates etc. For example, the herd immunity threshold for pertussis is 92-94%, whereas for diphtheria it is 85%.

Here is a nice infographic from the National Institute of Allergy and Infectious Diseases showing the concept of herd immunity at work:

**The Idea of Herd Immunity Makes Sense**

Let us think of two extreme scenarios.

*Scenario 1 – No One is Immune*

If 100% of the population has no immunity, the disease will spread rapidly. Everyone who is exposed will get sick, and besides those individuals who are completely isolated from society,

**everyone will get sick**.*Scenario 2 – Everyone is Immune*

If 100% of the population is immune, the disease will not spread and it will die off quickly. Every person who is exposed to it will be safe and

**no one will get sick**.*Reality*

In reality, the level of immunity in a given population, a.k.a herd, will be somewhere in between 0% and 100%, and the level of disease spread will be somewhere in between “everyone will get sick” and “no one will get sick”. In other words some will get sick. The closer you are to either extreme scenario the closer the “some” will be to that extreme’s outcome. Therefore the less people are vaccinated, the more the disease will spread. Vice-versa, the more people are vaccinated the less the disease will spread.

So from a logical point of view, if you accept that vaccines are efficacious in protecting against disease, you must reach the conclusion that herd immunity must exist, to some extend.

But what kind of proof do we have that in reality such protective effects do exist whenever we vaccinate large numbers of individuals?

**The Math of Herd Immunity**

Now, some folks can doubt the idea of herd immunity, but I think they will not doubt math. We can infact use math to come up with a formula for herd immunity. Here are the details:

*R0**(The Basic Reproduction Number of the disease) -*The average number of other individuals each infected individual will infect in a population that has no immunity to the disease. (for example, each infected individual on average will infect 13 others)

**S**- The proportion of the population who are susceptible to the disease, i.e. neither immune nor infected (for example, 15 % of the population)

In order for a disease not to die off, at the very least each infected individual should infect another individual. Now, R0 tells us the number of people that our infected person would infect,

**if everyone he came in contact with had no immunity**. However in real life some of his contacts will be immune, and only**S**of those contacts will have no immunity. So our infected individual will come in contact with**S**vulnerable people, of which only**R0**will get infected. Since we said that we need at least 1 new infection to keep the disease spread alive that means that:**R0 x S = 1**

Anything above 1 and the disease will grow and become an epidemic; anything below 1 it will eventually die off.

Remember, what we’re trying to calculate is the herd immunity threshold. Let us denote that by

**HI**. Keep in mind that HI denotes the percentage of the population that is immune to the disease (for example 78% or 0.78). Also keep in mind that S denoted the percentage of the population that is not immune to the disease (for example 22% or 0.22). Therefore, if we add HI and S up we get the full population, which in percentage terms is 1.**HI + S = 1**

Or, alternatively:

**S = 1 – HI**

Substituting (1-HI) for S in our first equation you get:

**R0 x ( 1 – HI ) = 1**

Solving for HI in this equation give us:

**HI = 1 – 1/R0**

So let’s assume that for disease X, R0 = 10. In other words if everyone was not immunized, an infected person could be expected to infect another 10 people. This formula would tell us that the Herd Immunity threshold would have to be:

**HI = 1 – 1/10 = 0.9 or alternatively 90%**

If we assume that vaccine efficacy is 95%, this would mean that we would have to vaccinate at least 95% of the population to reach the required herd immunity threshold (because 0.95 * 0.95 = 0.90).

And there you have it, an admittedly over-simplified, bare bones, dirty calculation of the herd immunity threshold, and how it depends on the reproduction number of the given disease. In real life, scientists must take many more variables into account, things such as Average Age at which the disease is contracted, Average Life Expectancy etc, but this gives you a good idea of the general way in which those herd immunity levels are calculated.

**The numbers are calculated based on existing statistical information; they are not pulled out of thin air**.So far we’ve shown that the concept of Herd Immunity makes sense and can be derived mathematically. But we still need to show that it works in the real world. Do we have examples, studies showing the herd immunity at work?

**The Scientific Evidence**

**Infant Rotavirus Vaccination May Provide Indirect Protection to Older Children and Adults in the United States,**The Journal of Infectious Diseases, Published by Oxford University Press on behalf of the Infectious Diseases Society of America 2011. DOI: 10.1093/infdis/jir492

This is a study which looked at rotavirus discharge and cause-unspecified gastroenteritis hospital data from a sample of 1,000 hospitals in 42 states in the U.S. Pre-vaccine era (2000-2006) discharge numbers were compared with the first full vaccination year (2008) data. Here are some of the key findings, but keep in mind that, as of that time,

**the only age group with any significant vaccine coverage was the < 1 years old**. Coverage for older children was negligible, a.k.a**the other age groups were unvaccinated**, thus providing an excellent test sample for herd immunity.**Rotavirus discharges**saw significant reductions. Specifically by age group:- 0-4 – 78% reduction
- 5-14 – 71% reduction
- 15-24 – 65% reduction
**Cause-unspecified gastroenteritis discharges**also saw a statistically significant reduction. Specifically by age group:- 0-4 – 39% reduction
- 5-14 – 29% redictopm
- 14-24 – 8% reduction

How do we interpret these results? This study

**suggests**that vaccinating children aged 1 year or less against rotavirus, provides indirect protection to other age groups not only towards rotavirus itself but also towards all-cause gastroenteritis. It appears that vaccinating young children reduced disease spread in the other, non-vaccinated groups, therefore this study provides direct support to the idea of herd immunity.Of course, since they had only one full year of vaccination data to work with (2008) it is possible that it could have been a fluke year, an unusually low rotavirus activity year to begin with, so we must be careful not to say that this study proved the protective effects of herd immunity. As we know, one study never proves anything, and these results will need to be replicated (with respect to the rotavirus vaccine). Readers are welcome to leave links in the comments to other studies that have looked at herd immunity.

**Conclusion**

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.

Lastly, a study was presented that showed increased protection on unvaccinated populations after introduction of the rotavirus vaccine.

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.

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