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

We verify our formula for Ro:

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)

and

Ro=n*r^(n+1)/Sn

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:

ReRoCalcsLinearB

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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