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The math behind Herd Immunity

With the resurgence of preventable diseases in the US due to anti-vaccination campaigns, one wonders how dangerous is it to be unvaccinated in these regions. The answer is that it depends. To get a general idea for the risk of unvaccinated individuals either for medical reasons or by choice, we need to consider Herd Immunity, which is when a large enough portion of the population is immune to protect people who are not immune. This is because an individual who is not immune is much less likely to come in contact with a diseased individual, since most people will not become infected. For herd immunity to be effective, a critical fraction of the population must be immune to the disease.

In order to calculate this critical fraction, we need to use the basic reproduction number. The basic reproduction number tells us on average how many people that one infected person will infect. For diseases that can cause epidemics, the basic reproduction number is scarily high. For example, measles has a reproductive number of about 12. We need to somehow lower the basic reproduction number below 1 to stop the spread of the disease. The basic reproduction number, R­0 is calculated by multiplying the probability of transmission, p, by the number of people that the infected individual will come into contact with while contagious, k. However, not everyone that the infected person contacts can contract the disease. If a certain fraction of people are vaccinated, f, then they will not contract or spread the disease assuming the vaccine is 100% effective. Then the number of at risk individuals is (1-f)*k. The adjusted reproduction number is now p*k(1-f) or R­0*(1-f). Using some algebra and setting the adjusted reproduction number to 1 gives us the critical fraction of the population that must be immunized to stop the spread of the disease.

f = 1-(1/R­0)

Using this formula, and the reproductive number for measles, the critical fraction of the population that needs to be immune to keep at risk individuals safe is 1-1/12 =.92. In certain communities where the anti-vaccination population is high, the fraction of individuals vaccinated might dip down below the critical fraction and start a preventable outbreak. Herd immunity is effective, but it is only effective when a significant part of the population is immune. Although getting vaccinated is an individual choice, it can affect an entire community by lowering herd immunity and putting everyone who is not immune at risk.

I used information from this site as well as some of the models from class.

Here is a fun animation showing how different vaccination rates can hinder outbreaks.


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