# COVID Odyssey: Winter Windup 4 ~ The formula for calculating Ro

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:

COVID Odyssey: Winter Windup 5 ~ Conclusions for 2020 ~ Back to the past: Re = 5.8 and Ro = 6

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.

Recall:

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)
• 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).

See:

Search Results for “windup” – COVID Odyssey by Alan Grace: My vir[tu]al COVID-19 journey (wordpress.com)

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