Matariki day morning
Sun and the new year dawning
An R-zero of six
May I throw in the mix
Six times more passings mourning
17 July 2020
Many in New Zealand are currently celebrating Matariki, the Maori New Year. See:
Sadly no-one from New Zealand appears to have looked at my online findings about COVID-19 since 9 July, despite emails I sent recently to the NZ Government (The NZ Prime Minister), The NZ Director-General of Health, The Prime Minister’s Chief Science Advisor, and University researchers about my work.
I realise WordPress stats may not necessarily be accurate.
COVID-19 appears to be more than twice as infectious as most researchers believe. I hope some influential people will view my work soon.
I have estimated one person infected with COVID-19 may on average infect 6 others. See:
I have received some automatic acknowledgements of receipt of my emails.
My results (Means) from a spreadsheet (over 1.8 million non-empty cells on one spreadsheet) produced close to my calculated Ro = 6.0468.
For SQRT(2), P = 1/SQRT(2), a run produced:
We now use our simple model to verify our calculation Re = 5.8.
We modify the model so that we have a 2-cycle infection period.
For i>2 using C = 8 and C = 5, for given constant values of LAMBDA and Re, we can calculate
C[i] = Re * ( LAMBDA * C[i-1] + (1 – LAMBDA) * C[i-2] )
Where C[i] is the estimated number of new cases in cycle i.
The number of new cases follows the sequence , 8, 44, 231.
We need to find LAMBA so that
231 = 5.8 * (LAMBDA * 44 + (1 – LAMBDA) * 8)
LAMBDA = ( 231/5.8 – 8 )/36.
However the value of LAMBDA obtained gave a value near 233 instead of 231, so we recalculated LAMBDA using 229.5 instead of 231.
We obtained LAMBDA = 0.87691571 and rounded this value to use
LAMBDA = 0.877 and (1 – LAMBDA) = 0.123 to generate the Table below:
Each estimate for the number of new cases is obtained by multiplying the previous number of cases (previous cycle) by LAMBDA and the number of cases in the cycle before by (1-LAMBDA).
The value 5 (highlighted in red) is only used to start the sequence. Hence we obtain
44.26 = 5.8 * (LAMBDA*8 + (1 – LAMBDA)*5) and
230.84 = 5.8 * (LAMBDA*44.26 + (1 – LAMBDA)*8).
The values obtained are very close to the actual case numbers.
We earlier used Goal Seek in Excel to get LAMBDA = 0.876. The above value gave better results.
We see the 5-day value (0.877) above and the cumulative 5-day value (0.8432) for the 2-cycle CSAW Excel scenario are reasonably close (r~1.4) considering the first calculation has two levels and the second has ten levels with over half the infections below occurring in the first two days and over three-quarters in the first four days:
A better approach is to estimate LAMBDA to fit the values 5, 8, 44, 231.
We start with Re = 5.8 (the cell highlighted in gold below) and use Goal seek in Excel to set the number of new cases for 26 March to 231 by changing Re. We obtain Re ~ 5.7115:
The values match.
We also calculate Re in the CSAW model using the formula
Re = 10 * r^11 * (r-1)/(r^10 – 1)
Using values of r in the table below:
For r = SQRT(2) [~1.4142] we obtain
Ro ~ 6.0468.
We therefore estimate Re up to 26 March to be in the range 5.7 to 5.9 and estimate Ro = 6.
By 31 March the above tables estimate totals of 1489 and 1497 cases.
This number of cases was not achieved until the beginning of May.
Let C[i] be the number of new cases in cycle i.
Let R[i] = C[i]/C[i-1] and r[i] = R[i]^(1/5]
Then R[i] is the ratio of the number of new cases in cycle i divided by the number of new cases in the previous cycle, and r[i] is the daily rate.
In April for Re = 5.8, R[i] becomes 5.223183, and r[i] becomes 1.391833.
In April, for Re = 5.7115386, R[i] eventually is 5.254786 and r[i] is 1.393513.
The above are the limits.
We consider the above close to our estimated r = 1.4.
These results help validate our calculated Re = 5.8 for the CSAW model.
We conclude that in New Zealand up to 26 March, Re = 5.8 and Ro = 6.
Our values for Ro are likely to be conservative. The CSAW modelling uses mean values. Isolation of identified cases in New Zealand is likely to underestimate calculations for Ro.
Reminder. For more detailed findings see:
Below are two emails I sent recently.
Dear Prime Minister Jacinda Ardern, Professor Juliet Gerrard (The Prime Minister’s Chief Science Advisor), and Director-General of Health Ashley Bloomfield
——– Forwarded Message ——–
|Subject:||COVID-19 NZ: Re~5.8 & Ro~6 possible using actual NZ case numbers|
|Date:||Wed, 15 Jul 2020 07:17:50 +1200|
Good morning Professor Shaun Hendy
I realise you are a very busy person.
I have been looking at COVID-19 data since February.
I find that Re~5.8 & Ro~6 are possible using actual COVID-19 case numbers in New Zealand up until Lockdown Level 4 in March.
This is twice as much as most researchers consider for Re and Ro.
What range of values have you calculated for Re and Ro up until Lockdown Level 4 in March in New Zealand?
You may be interested in reading my own work. See:
I apologise for the presentation of this material. This material was originally part of a blog. Accordingly I do not follow academic writing protocols or referencing. Mathematical symbols including superscripts and subscripts were not available.
COVID-19 is more than twice as virulent as most researchers believe. One person may on average infect 6 other people. The number of cases may double every two days. We explore an Excel spreadsheet simulation, CSAW (“SeeSaw”): COVID-19 Sampling Analysis Worksheets, generating random samples of the spread of COVID-19 in New Zealand.
We investigate whether one infected person may on average infect 6 others in New Zealand up until Lockdown Level 4 on 25 March at 11.59. First we investigate whether one infected person may on average infect 5.3782 (more than 5 and 3/8) other people in New Zealand using a simple model.
I generate over 20,000 cases in an Excel spreadsheet (for each of the 40 samples), counting how many people each case infects once cases are split into the day they were infected, and calculating the mean for each of 40 samples.I then calculate the mean and standard deviation for each of the 40 sample means.
One CSAW spreadsheet has over 1.8 million non-empty cells.
Worst-case scenarios are also considered.
My calculations suggest that one person may infect on average 6 other people (Ro~6).
Most experts believe each person infects on average two point something other people.
Hence my results are more than twice as big.
Assume infectivity reduces (decays) at the same rate each day during the infectious period.
If the daily rate of increase (r) for case numbers matches the rate of decay of infectivity, then for each day during the infectious period (two cycles = 10 days), the daily contribution towards R is the same.
To calculate R, we can simply multiply by ten the value for the first day of the infectious period.
We add in a factor of x^2 for incubation and realise that
1 + x + x^2 + … + x^10 = (x^10 – 1)/ (x – 1)
For Day i (i=1,2, …, 10) and x = r (the daily rate of increase), we have a daily factor of
r’ = x^(i+1) * x^(10-i) * (x – 1)/(x^10 – 1)
= x^11 * (x – 1)/(x^10 – 1)
i.e. The same value for each of the ten days.
Let f(x) = 10x^11 * (x – 1)/(x^10 – 1) then
Ro = f(SQRT(2))
and for the ten days on and before 26 March (the first full day of Lockdown L4)
Re = f(1.4)
i.e. Ro ~ 6 and Re ~ 5.8.
Our values for Re and Ro are twice as big as most experts consider for COVID-19.
We collate information from posts when this site was a blog and examine the spread of COVID-19.
Historically my site has been a blog (over 100 COVID-19 posts); now it has become essentially a website.We divide this section into six parts.
- Chapter 1: Scenario A. Infectious period One cycle (5 days)
- Chapter 2: Scenario B. Infectious period Two cycles (10 days)
- Chapter 3: The CSAW Excel spreadsheet
- Chapter 4: Worst case scenarios
- Chapter 5: Results
- Chapter 6: Conclusions
The pdf version of this webpage (I am still developing this) is over 60 pages long.
In the first wave of COVID-19 we found that cases increased by a factor of 1.4 each day from Day 18 (16 March) up until Lockdown Level 4. This suggested that Re = 5.37824 for a five day cycle.
The factor a=1.4 clearly fits well the actual (cumulative) case numbers:
Ro (pronounced R-Nought or R-Zero) is used when there is no quarantine. i.e. when the virus is unrestrained.
Re is used when there is quarantine (Re is the effective reproduction rate).
We may use R in general discussion.
In our discussions, the reader will need to decide where it may be appropriate to replace Ro by Re and vice versa.
The factor 1.4 means that the number of cases almost doubles every two days since 1.4^2 = 1.96.
If Ro = SQRT(2) then this would mean doubling every two days. The cases would follow the pattern 5, 7, 10, 14, 20, 28, … where every two days the number has doubled.
New Zealand was very lucky during the first wave of COVID-19 especially since New Zealand was almost too late going into Lockdown Level 4.
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