In this post we develop the formula we have used for calculating Ro, the number of people one person may infect on average over a 10-day symptomatic period (n = 10) given a daily increase r in the number of cases and a daily decrease of 1/r in infectivity.
Note: The formula in this post has been updated. See:
Mostly over the last year we have used this table (no longer preferred):
Originally (and now) we prefer to use this table:
We use this table to develop the above formula:
The total down the bottom of the second column is the sum of
1 + r + r^2 +r^3 + … + r^(n-1)
This is a geometric series with a total equal to (r^n – 1)/(r – 1)
In the third column we divide by this number (e.g. 69.81366 for r = 1.4).
This is the same as multiplying by the inverse, (r – 1)/(r^n – 1)
Since n = 10, over an 11 day period (including an extra incubation day) the number of cases increases by r^(n+1) producing the partial formula:
r^(n+1)*(r – 1)/(r^n – 1)
To get the result in our simulations we need to multiply this by n to get
Ro = n * r^(n+1)*(r – 1)/(r^n – 1)
This is the formula in the second table.
For an infectious period of n = 10 days, we obtained Re = 5.8 for r = 1.4 and Ro = 6 for r = SQRT(2) .
These are the values we obtained in our simulations.
The formula and our simulations are also correct for r = SQRT(3).
Note: we multiply or divide by r to get (change) the results in the first two tables. i.e. to convert between the two tables.
By definition we let:
- r denote the effective Reproduction rate of COVID-19 for one day
- Ro (R0; R-Zero; R-Nought) denote the Reproduction number for COVID-19 without any quarantine or isolation
- Re denote the effective Reproduction number for COVID-19
(Re assumes isolation/quarantine is happening)
- a case be defined as a person diagnosed as having COVID-19
Note that Ro and Re are numbers (not rates), the number of people one person with COVID-19 may infect on average without quarantine or isolation (Ro) and with quarantine or isolation (Re).