COVID Odyssey: NZ New Year Fear 42 Post #969~ Worldwide windup~ Verification of our formula for Ro

We verify our formula for Ro:


For definitions and background please see the WELCOME menu.

We also use the same formula to estimate Re.

Re is the effective reproduction number when there is isolation.

We assume that n =10 and that infectivity reduces by a factor of 1/r each day where r is the daily increase in case numbers and Ro is the average number of people one person with COVID-19 may infect without isolation.

First we define

Sn = 1 + r + r^2 + . . . + r^(n-1)



where ^ means ‘to the power of’.

This gives the same result as our formula (see the table below)

Ro = n * r^(n+1) * (r – 1)/(r^n – 1)
(providing we define Ro = 1 when r = 1 so that Ro is defined when r = 1).

We have just summed the geometric series (Sn) to obtain our formula.

We use n =10

We now check the calculations.

When r = 1 we require Ro = 1.

When n = 10 and r = 1, Sn = 10.

Hence Ro =1 as required.

We also see the values for Re in the table below are very close to the results of simulations in this post:

COVID Odyssey: NZ New Year Fear 41 Post #968~ Worldwide windup~ More simulations to confirm our formula

This is all that is required to verify our formula.


The last two columns can be ignored but for completeness are explained below.

Note: For r = 1.4, we have also estimated Ro as high as 7, see:

COVID Odyssey: NZ New Year Fear 14 Post #941~ COVID-19: A table comparing Ro and Re

COVID Odyssey: NZ New Year Fear 17 Post #944~ COVID-19: Calculating Ro based on Re Case numbers

When r = 1.4, r^2 = 1.96.

This suggests that when r = 1.4, Ro is close to the value calculated for r = SQRT(2).

The effect of isolating cases is likely to have reduced the value for r if it was calculated without any isolation.

To also get the extra amount when  r = 1.4 (so that the value for Ro is calculated to be the same value as when r = SQRT(2) using our formula) we could use a linear approach and add on to Ro an extra amount d.

To match (almost) exactly so that r = 1.4 gives Re as calculated for r = SQRT(2), use

d = (r – 1)*0.61573

For 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.


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