# COVID Odyssey: Don’t underestimate Ro ~ How many people may one person infect on average?

Ro

Still forever be fraught
Will we still wonder
It’ll be under
‘Til true value be sought

Alan Grace
9 September 2020

In New Zealand we find Ro ~ 4.1 (4.143238) with a 10-day infectious period. We find 10-day moving averages work well to help confirm the value for Ro (~ 4.1) given a daily increase in new case numbers close to 40% over several days (r ~ 1.4) in theory and using actual NZ cases.

If a case may be infectious for 15 days (three 5-day cycles) we find that Ro may be 6.04, close to the value we previously considered for a 10-day (two 5-day cycles) infectious period.

Ro is by definition the reproduction number in a totally naïve population (with no quarantine or isolation). e.g. If Ro = 4 then one person with COVID-19 will on average infect four other people.

We also revise estimates for Ro worldwide. We assume a 10-day infectious period.

The estimation of Ro for COVID-19 is fraught with underestimation. All estimations of Ro taken over a long period of time that involve averages (means), moving averages, least-squares approximation or rely on actual case numbers over a long period are likely to underestimate Ro.

Cumulative case numbers are used (here) in analysing Ro. A daily cumulative rate also means the same rate for daily new case numbers. We have already proved this.

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).

In New Zealand we have found a daily rate of increase of r = 1.4 (40% increase per day) in March and assume a 10-day infectious period (when a case can infect others).

We assume a daily increase of r (e.g. r = 1.4) and a daily decay of infectivity of 1/r over a 10-day infectious period for each case.

COVID Odyssey: [Pre-]Summer Summary~ How many people may one person infect on average?

We have r ~ 1.4 (a = 1.4) giving a good fit 16 March to 26 March (the first day of Lockdown Level 4) :  Since cases each day produce new cases over a 10-day period, the number of new cases on each successive day is contributed towards by cases over ten days. With a daily increase of r and a decay of 1/r, we end up with the same contribution from each day (r*1/r = 1).

The table below relates the various values for r we have used, and estimates for Ro using the formula at the top of the table, and similar formulae We estimated in New Zealand a value for r of 1.4 to calculate Ro.

Since r*r is 1.96 (close to 2) we also consider r = SQRT(2).

Our original values for Ro are in the last column. Hence when you see values near these elsewhere on this site, you will need to divide by r to get our current estimate for Ro ~ 4.1.

Previously we considered Ro ~ 5.8 and 6 (see last column in the above table).

We note that a median value for Ro of 5.7 was found in this article:
20-0282

For various estimates for Ro (some over 6), you may also like to look at:
taaa021 An estimate for r in the article, r ~ 1.22, was estimated by us as r ~ 1.4 (see graph above).

Note that r = SQRT(2) is also included on the graph.

With r ~ 1.22 we would have estimated Ro ~ 2.5;
with r ~ 1.4 we estimate Ro ~ 4.1.

Since cases are usually isolated or quarantined once they are identified, our estimate of r ~ 1.4 may be low and maybe r ~ SQRT(2) should be considered. This would make Ro almost 4.3 (4.28).

The last row in the table above shows that a 2% increase in r (from r ~ 1.4 to r = SQRT(2) may produce a 3% increase in estimating Ro.

Since some will consider our estimate for Ro ~ 4.1 high, we retain this estimate and do not increase it.

We have previously estimated Ro ~ 3, 4, and 6 in New Zealand.

These values correspond to (note the change in the first power of r below):
Ro=10 * r^9 * (r-1)/(r^10 -1)
Ro=10 * r^10 * (r-1)/(r^10 -1)
Ro=10 * r^11 * (r-1)/(r^10 -1)

using r = 1.4 or r = SQRT(2).

Previously we saw that

Ro=10 * r^10 * (r-1)/(r^10 -1)

gave exact values for case numbers when r = 1.4 using the calculations below (first adding the totals for the previous 10 days, then multiplying the result by Ro/10): Hence we adopt Ro = 4.1
[4.1432 (4 d.p.)].

r  = SQRT(2) also works (and other values): We can estimate the total number of Cases (C) for a day (the next day) as

C = AV * Ro

where AV is the average number of cases over the preceding n days (e.g. n = 10).

AV is a moving average.

In the penultimate table(s) above, Sum10*Ro/10 can be re-arranged as (Sum10/10) * Ro = AV * Ro.

This is only likely to work reasonably accurately over many weeks near our ideal situation where the number of cases increases by close to r every day.

We obtain the estimates below using actual NZ cases: Usually the above calculations using r = 1.4 give a better estimate.
The estimates in the last three columns are all acceptable.

Below are some runs of our CSAW v3 simulation using r = 1.4:
CSAWv3Samples

We conclude that in New Zealand (and elsewhere where r ~ 1.4), one person with COVID-19 may infect on average 4.1 other people (and maybe up to 4.3 (4.27575) using r = SQRT(2)).

Below is a table estimating Ro for other values of r.
See Re#2* (last column): We conjecture the following estimates for Ro for 9-day and 11-day infectious periods for r = 1.4 and r = SQRT(2):
NZRoTable

In general, for an n-day infectious period, we have:

Ro=n * r^n * (r-1)/(r^n -1)

Since a person may infect others up to two days before becoming symptomatic (showing symptoms of COVID-19), we consider infectious periods of n = 10, 11, and 12 days, and get these estimates for Ro: We get these estimates for Ro and for NZ case numbers: We continue up to 15 infectious days (three 5-day cycles): We note that in the last line above, we could have expected close to 400 cases. Clearly isolation/ quarantine was having an effect before the first full day of Lockdown Level 4.

The curve had already started to flatten before Lockdown Level 4 started.

In New Zealand we adopt Ro ~ 4.1 (4.143238) with a 10-day infectious period since cases are infectious for at least 10 days, and all Ro values match the actual number of cases equally well. This means the estimate for Ro is likely to be lower than it should be. i.e. A person infected with COVID-19 may infect on average at least 4.1 other people over a 10-day (two 5-day cycles) infectious period.
Note: Allowing for an extra two infectious days (12 days), we would have obtained Ro ~ 4.9 (4.886185).

All the values for Ro give good estimates for the actual case numbers for
n = 10 to 15.

If a case may be infectious for 15 days (three 5-day cycles) we find that Ro may be 6.04, close to the value we previously considered for a 10-day (two 5-day cycles) infectious period.

When the number of cases each day is exactly r times the previous number, we get an exact match: The top line is the number of infectious days (n) with each column using an n-day moving average. We see that when n = 15, Ro ~ 6.04, close to the value we previously considered. In this scenario a case may be infectious for three 5-day cycles.

We conclude that a person infected with COVID-19 may infect on average at least 4.1 other people over a 10-day (two 5-day cycles) infectious period.

If a case may be infectious for 15 days (three 5-day cycles) we find that Ro may be 6.04, close to the value we previously considered for a 10-day (two 5-day cycles) infectious period.

We compare the 10 and 15-day values (scaled to 1) for a decay rate of 1/1.4. We see that less than 3% (~2.8%) of infections come from Days 11-15 with almost 1% coming from Day#11; infections from Days 1-10 are comparable although almost 3% (~2.8%) less for each day. Epidemiologists will need to consider the likelihood of infections over 15 days.

Ro ~ 4.1 (4.143238) is close to the means we have seen in 10-Day infection simulations. See:
COVID Odyssey: CSAW simulation ~ form[ul]ation explanation

We have a range for Ro of 4.1 to 6.

All these estimates are within the range in this graph: The above graph was obtained from this article: taaa021 (Click to view PDF):
The reproductive number of COVID-19 is higher compared to SARS coronavirus
published 13 February 2020, obtained from here:
(Journal of Travel Medicine, Volume 27, Issue 2, March 2020)

Conservatively we adopt Ro = 4.1.

Ro = 6 may still be possible.

We conclude that our estimates for Ro are likely to be underestimates particularly since the “totally naïve population” requirements are unlikely to have been met. We may need to consider r = SQRT(2).

Ro

Still forever be fraught
Will we still wonder
It’ll be under
‘Til true value be sought

Alan Grace
9 September 2020

In New Zealand we find …

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Appendix
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The CSAW (COVID-19 Sampling Analysis Worksheets; “SeeSaw”) simulations produced summaries like this for a 10-day infectious period in CSAW v2
(See CSAW v3 below these):      Maximum Infected means the number of people directly infected by 1 case (i.e. not by another person).

We end with a sample runs of the CSAW v3 Excel simulation:
10-day period of infectivity; r = 1.4 and the daily rate of decay in infectivity is P = 1/1.4 i.e. 1/r:
CSAWsimV3Sampling

Below are some of our worldwide estimates for Ro:
COVIDWorldAvNewRanked4r
COVIDWorldAvNewAlpha4r

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