The danger of combining Rs
[[https://plus.maths.org/content/problem-combining-r-rates
plus.maths.org/content/problem-combining-r-rates]]
You might think that to get the overall value of R, which applies to the community, hospitals and care homes taken together,
you simply take the average of the two values.
In reality this isn’t the case and leads to a dangerous underestimate of the overall value of R.
It can even lead you to assume that the disease is under control when really it is not.
The reason for this is that the populations, people in the community and people in hospitals, are not completely isolated from each other.^
...
Say the value of R in the community is 2, so on average an infected person in the community infects 2 others in the community,
and the value of R in hospitals is 3, so on average an infected person in a hospital infects 3 others in the hospital.
We also need to take into account that there will be contact between the two groups, so say that on average a person in the community also infects 1 person in a hospital (in addition to the 2 people they will infect in the community).
Similarly, say a person in a hospital infects 1 person in the community on average (in addition to the 3 people they infect in hospital).
Next generation infections
Then as you can see from the diagram above, a person in the community and a person in a hospital together infect 7 people.
This means there are 7/2=3.5 new infections per person on average for this first round of new infections.
As we continue this, we see that the ratio between new infections and infections at the previous step grows to 25/7=3.57 (see the diagram below).
Carrying on further will eventually lead us to an overall value of R of 3.62.
Crucially this is higher than any of the individual values of R, which were 2 and 3.