Will your search for R-Nought
Still forever be fraught
Will we still wonder
It’ll be under
‘Til true value be sought
9 September 2020
For COVID-19 we assume 5-day cycles with one cycle for the pre-symptomatic incubation period and a two-cycle (10-day) symptomatic infectious period. Also see:
COVID Odyssey: Fall Flashback ~ Summary 2020 revisited ~ How many people may one person infect on average?
In New Zealand we find one person with COVID-19 (one symptomatic case) may infect on average 4.1 other people (Ro ~ 4.1) with a 10-day (two 5-day cycles) infectious period with r = 1.4.
If a case may be infectious for 15 days (three 5-day cycles) we find that one person with COVID-19 may infect on average 6 other people (Ro ~ 6.04), close to the value we originally obtained for a 10-day (two 5-day cycles) infectious period when we added in an extra one day/ two days to the incubation period (subject to interpretation) in the original CSAW simulation when r = SQRT(2).
A person with COVID-19 may infect other people two days before they show symptoms. This indicates that a 15-day (3-cycle) infectious period may be possible since the 10-day (2-cycle) symptomatic infectious period is only essentially extended by three days (if we include a two-day pre-symptomatic infectious period).
Historically others have considered a 6-day cycle. This indicates that a 15-day infectious period may be possible since 6*2 + 2 = 14, close to 15 days. Also less than 3% (2.833%) of infections for a 15-day (3-cycle) infectious period occur in the last 5 days.
Our estimates for Ro are based initially on the actual number of cases in New Zealand during the first outbreak in 2020.
We extend our results to provide estimates for Ro for over 200 countries.
Our estimates for Ro may be low since once a case is identified, isolation usually takes place.
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)
- a 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).
We assume a person may be infectious for n days once they become symptomatic (show symptoms). A person may be infectious for at least two days prior to becoming symptomatic.
We assume a 5-day cycle where the incubation period is one cycle long (until symptoms appear) and the infectious period is two cycles long.
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 in infectivity of 1/r over a 10-day infectious period for each case.
When r = 1.4
r*r = 1.96
This means that the number of new cases almost doubles every two days.
A case is most infectious on the first day a person is able to transmit COVIT-19. Infectivity reduces on each successive day.
In general, for an n-day infectious period (e.g. n = 10), we obtain the formula:
Ro=n * r^n * (r-1)/(r^n -1)
where ^ means ‘to the power of.’
When r = 1.4
Ro ~ 4.1 (4.143238)
We therefore adopt the formula
Ro=10 * r^10 * (r-1)/(r^10 -1)
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 the 10-day moving average (MA over 10 days).
We obtain the estimates below using actual NZ cases:
We include the value for the 10-day Moving Average (denoted MA and AV) below:
We conclude that up until the first full day of Lockdown Level 4 in New Zealand (26 March 2020), 10-day moving averages of actual case numbers and the formula for Ro (with n=10 and daily increase r = 1.4), the above table produces good estimates for projected case numbers.
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.
We also consider r = SQRT(2) which means r*r = 2. In this situation the number of cases will double every two days.
When r = SQRT(2)
Ro ~ 4.3 (4.275753)
The number of cases doubling every two days is analogous to the story about the invention of chess where each square represents two days. The story says the inventor of chess asked his ruler for a reward of one grain of rice for the first square on the chessboard, two for the second, four for the third, doubling on each successive day. The total number of grains after for all 64 squares (and cases after 128 days excluding some initial incubation periods maybe until there are double digit case numbers) is
2^64 – 1 or
clearly more than the entire population of the world (7,794,798,739 on 1 July 2020) according to:
The entire population of the world would be infected after less than 33 squares are filled (66 days excluding some initial incubation periods maybe until there are double digit case numbers).
Anyone still want to consider herd immunity?
When an outbreak starts, there will be many infected people that have not been identified (are not yet officially cases). This will result in a high increase in the daily rate (r) of newly discovered cases.
Once there are very few infected people that have not been identified/discovered (very few people who are not yet officially cases), there will be a short period where the daily rate (r) of new cases each day remains relatively constant (before r then starts to consistently decrease). We can use this time to estimate Ro.
Ro is officially the reproduction rate in a naïve community (where there is no isolation or quarantine). Our estimates for Ro are likely to be low since the estimates are based on cases which are discovered and therefore likely to be quarantined or isolated or self-isolated so that they are unlikely to infect other people.
If Ro is consistently less than one an outbreak will die out. If Ro is consistently greater than one an outbreak is likely to spread.
We conclude that the value for Ro depends on the average number of days (n) a person may on average be infectious (n = 10 to 15 days). For r = 1.4, we obtain values for Ro in the range 4.1 to 6 (see last table). One person infected with COVID-19 may infect on average 4.1 to 6 other people.
For more information please see:
COVID Odyssey: Don’t underestimate Ro ~ How many people may one person infect on average?
COVID Odyssey by Alan Grace: New Year Fare ~ Contents
COVID Odyssey: CSAW simulation ~ form[ul]ation explanation
COVID Odyssey: CSAW simulation ~ form[ul]ation verification
COVID Odyssey Ro update: Occam’s Razor~ A close shave: In NZ is Ro ~ 3, 4, or 6? What are Ro values worldwide?
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