We modify the way we calculate Ro in three ways.
We use the formula we have already developed to calculate Ro based on r, the daily increase in total case numbers.
When r = 1.4 as in New Zealand early on in the first 2020 outbreak, r^2 = 1.96 which is close to 2. We would like to include r = SQRT(2) in our estimate for Ro so that case numbers have the potential to double every two days.
We suggest two ways that this may be done.
See the WELCOME menu for the formula and background.
We have always found that a daily increase of r = 1.4 for estimating COVID-19 cases in New Zealand works well up until the day after Lockdown Level 4 started at 11.59 pm on 25 March 2020:
We have the following table for estimating Ro (see last column):
When r = 1.4 we want Ro = 6.05, the value for Ro calculated when r = SQRT(2).
Let Re be the value for Ro for r = 1.4.
Re = 5.8 and 6.05 = 5.8 + 0.25 so our range for Ro could be
Ro is in the range Re to Re + 0.25.
We could do this for all values of r (just add on 0.25 as an upper-bound for all values when calculating Ro).
As an alternate method we could adjust our formula as below cf:
- To[i] be the total number of case numbers for Day i without isolation (i = 1 to 10).
- Co[i] be the number of cases for Day i without isolation.
- ro be the daily increase without isolation.
Ro be the average number of people one person may infect on average over a ten day period without isolation.
Similarly we define Te[i], Ce[i], re, and Re to be the effective numbers with isolation.
Where there is no confusion, we refer to ro and re simply as r.
The cases Ce[i] are isolated when they are discovered.
To calculate Ro, we need to include their spread as if they had not been isolated.
Ce = 10
Te = 10
For i = 1, 2, …, 10
Te[i] = re x Te[i-1]
Ce[i] = Te[i]-Te[i-1]
To = Te
For i = 1, 2, … , 10
To[i] = To[i-1] x re + Ce[i] / (re^10)
ro = (To / To ) ^ (1/10)
We calculate Ro and Re from our formula.
We obtain the following table:
In the table the value for Ro when r = 1.4 comes close to the desired figure.
We could add on 0.02 to our estimate for Ro to get close to 6.05 when r = 1.4.
We leave that decision to you.
The PDFs below shows a larger range of values:
We think it is probably easier just to calculate Ro by adding on 0.25 to all values of Re.
However when r = 1 we want Ro = 1.
So define Ro =1 when r =1.
To also get an extra 0.25 when r = 1.4 we could use a linear approach and instead when r is between 1 and 1.4 add on to Ro an extra amount d where
d = (r – 1)*0.25/0.4
when r = 1, d = 0 and when r = 1.4, d = 0.25 as required.
When r > 1.4 we could simply add on 0.25 to Ro or use the calculation above.
To match (almost) exactly so that r = 1.4 gives Re as calculated for r = SQRT(2), use
d = (r – 1)*0.61573
We obtain these results for r = 1 to r = 1.4:
If the same linear approach calculation was extended to r = 2.01 we would estimate Ro at just under 21 (20.94177). See:
When r = 1.4 one person may on average infect 6 other people.
When r = 2.01 one person may infect on average almost 21 other people.
We stay with our original formula presently.