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Brief summary
The reproduction ratio, R, of a disease (often called the R number in the media) is the average number of people an infected person goes on to infect. If R>1, the epidemic of the disease will spread, the larger the R the quicker it will spread. And if R<1 the disease will die out.
The general idea behind herd immunity is that if you have a population where many people are immune to a disease, the disease can't take hold and grow into an epidemic, thereby protecting those people who aren't immune. The population (perhaps unfortunately called a herd) protects vulnerable individuals.
Two key concepts in epidemiology the reproduction ratio, R, of a disease and herd immunity. As you will find out in this article, the two are closely related.
The basic reproduction ratio, R0
As we saw in our easy introduction to the reproduction ratio, the value of R, of a disease is the average number of people an infected person goes on to infect. When everyone in the population is susceptible to the disease (say when it first emerges) then this number is also called the basic reproduction ratio, R0. For seasonal strains of flu, the basic reproduction ratio between 0.9 and 2.1, and for measles it is a whopping 12 to 18.
You can see how a large enough
1st generation: 2 new infections
2nd generation: 4 new infections
3rd generation: 8 new infections
4th generation: 16 new infections....
![Exponential growth](/content/sites/plus.maths.org/files/news/2020/herd/chart_exponential.png)
The number of new infections after n generations for R0=2.
Generally, there are
When
1st generation:5,000 new infections
2nd generation: 2,5000 new infections
3rd generation: 1,250 new infections
4th generation: 650 new infections....
Generally, there are
![Growth for R0=0.5](/content/sites/plus.maths.org/files/news/2020/herd/chart_0.5.png)
The average number of new infections (in ten thousands) after n generations for R0=0.5.
What if
The effective reproduction ratio, R
So, given that the
However, once a person has recovered from the disease they will (hopefully) gain some immunity. This means that after a while we're not dealing with a totally susceptible population anymore. Indeed, there may be other reasons why some people in the population aren't susceptible. There may be a vaccine available or perhaps behavioural interventions (such as social distancing or infected people isolating) may be reducing the chance of people catching or passing on the disease.
In most real life situations we should be looking at the effective reproduction ratio of the disease, usually denoted by
As an example, if only half the population is susceptible, so
Herd immunity
What does all this have to do with herd immunity? The general idea behind herd immunity is that if you have a population where many people are immune a disease can't take hold and grow into an epidemic, thereby protecting those people who aren't immune. The population (perhaps unfortunately called a herd) protects vulnerable individuals.
So how many people in a population need to be immune to have herd immunity? Imagine a disease has a basic reproduction ratio
In other words, we need to get the proportion of susceptible people in the population to under
So, to achieve herd immunity we need to make sure that at least a proportion of
The herd immunity calculation for vaccination
As we saw only a proportion of 1-1/R0 of the population need to be immune to achieve herd immunity, where R0 is the basic reproduction number of the disease.
If a vaccine has an effectiveness of x%, then this means that only x% of people who receive it are immune. To achieve herd immunity we therefore require that x% of the proportion of people who are vaccinated translate to a proportion of at least 1-1/R0 of the population. In other words,
This article is based on a chapter from the book Understanding numbers by the Plus Editors Rachel Thomas and Marianne Freiberger.
This article is part of our collaboration with JUNIPER, the Joint UNIversities Pandemic and Epidemiological Research network. JUNIPER is a collaborative network of researchers from across the UK who work at the interface between mathematical modelling, infectious disease control and public health policy. You can see more content produced with JUNIPER here.
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