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
In 2020, We found that for a 10-day infectious period, given a daily increase in COVID-19 infections of r = 1.4, one person with COVID-19 may infect on average 4.14 other people and for a 15-day infectious period, one infected person may infect around 6 other people.
Since early calculations (e.g. by WHO) assumed a value for Ro of between 1.4 and 2.5, our results are a significant increase over the original estimates and most current Ro values. See:
In this post we consider again the formula developed last year during simulations for the spread of COVID-19 in New Zealand to calculate (Ro) the number of people one person with COVID-19 may infect on average during the initial outbreak in early 2020 (without quarantine or self-isolation) and consider Re, the effective reproduction rate with quarantine and self-isolation.
We assume a homogeneous population where the spread within the population is similar throughout the population.
New Zealand is relatively isolated and therefore is suitable for estimating the spread of COVID-19 in general in the world.
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. We also consider r = SQRT(2) [~1.4142]
Our findings produced the formula in the table below to calculate Ro, where n is the number of days a person is infectious:
where ^ means ‘to the power of’ and * means multiplication. See:
COVID Odyssey: [Pre-]Summer Summary~ How many people may one person infect on average? – COVID Odyssey by Alan Grace: My vir[tu]al COVID-19 journey (wordpress.com)
We see that r = 1.4 works well in New Zealand until the day after Lockdown Level 4 which started on Wednesday 25 March 2020 at 11.59 pm (starting with 10 cases):
We assume that each day a person’s infectivity reduces by a factor of 1/r:
The various values obtained successively for 1/r are scaled so that the sum is equal to 1.
You can ignore the next two tables if you wish.
Here are the results for n = 10:
Below are the results for n = 15:
When r = SQRT(2), on Day 15 (three 5-day cycles), infectivity would reduce to around 0.23%.
We note that Day 11 to 15 in the last column above only amount to less than 2.6% of the total infections (around 2.8% in the third column) with only Day 11 being close to 1%.
In reality after 10 or more days, likely less, an infected person is likely to be recovered, quarantined or self-isolating, in a serious condition in hospital or dead. This means that in reality the spread of COVID-19 on Days 11-15 is unlikely to be significant at least until hospitals are so overwhelmed that they cannot cater for the number of serious cases.
However Ro (R0; R-Zero; R-Nought) denotes the Reproduction rate without any quarantine or isolation.
When r = SQRT(2), over each two-day period a person may become half as infectious. Perhaps using the analogy of radioactivity, we may consider COVID-19 infectivity has a half-life of two days?
We see from the above table that for a 10-day infectious period when r = 1.4, one person with COVID-19 may infect on average 4.14 other people and for a 15-day infectious period one infected person may infect around 6 other people.
Since early calculations (e.g. by WHO) assumed a value for Ro of of between 1.4 and 2.5, our results are a significant increase over original and most current Ro values .
From the data for the actual number of cases in New Zealand, we saw that the number of cases increased by a factor (r) of approximately 1.4 each day.
When r = 1.4
r*r = 1.96
This means that the number of cases would almost double every two days.
The number of cases would actually double every two days when r = SQRT(2).
We therefore also consider the case when
r = SQRT(2)
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.
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 r and hence Ro are based initially on the actual number of cases in New Zealand during the first outbreak in 2020.
We extended our results to provide estimates for Ro for over 200 countries.
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 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), given a 10-day infectious period
Ro ~ 4.3 (4.275753)
We also looked at the spread of COVID-19 in over 200 countries.
Below are some of our worldwide estimates for Ro using a simple heuristic.
We conclude that in New Zealand (and also worldwide) one infected person may on average infect 4.1 to 6.25 other people depending on the length of the infectious period and the value for r.
Below is the same table from the top of this post again: