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We Investigate whether in New Zealand Ro ~ 3, 4, or 6 and calculate values using a formula for Ro worldwide. Based on a daily increase in cases of r = 1.4 (40%), modelling is used to determine the best value for Ro.
You may also like to look at:
COVID Odyssey: [Pre-]Summer Summary~ How many people may one person infect on average?
COVID Odyssey: Don’t underestimate Ro ~ How many people may one person infect on average?
COVID Odyssey: CSAW simulation ~ form[ul]ation explanation
COVID Odyssey: CSAW simulation ~ form[ul]ation verification
Note: Ro=3 means that on average one person with COVID-19 will infect 3 other people.
We update our estimates for Ro and first investigate if, in New Zealand, Ro~2.9 [later Ro~4.1]. Previous modelling has suggested Ro = 6.
For worldwide values for Ro (if for NZ, Ro~2.9, 4 or 6) see:
COVIDWorldAvNewRanked
COVIDWorldAvNewAlpha
COVIDWorldAvNewRanked4
COVIDWorldAvNewAlpha4
COVID Odyssey: Vir[tu]al World Tour ~ How many people can one person infect in your country?
Occam’s Razor
When you put it to the test
Occam’s Razor we suggest
Is the one stressed you knew
You may truly pursue
The simplest answer’s the best
Alan Grace
12 May 2020
Ro=2.9 and Ro=4 means that in New Zealand one person may infect on average 2.9 (close to 3) and 4 other people (respectively).
We eventually end up with this table (see below):
We modify the CSAW (COVID-19 Sampling Analysis Worksheets) model (“SeeSaw” model) to help determine Ro in New Zealand where r ~ 1.4.
First we try removing a factor of r*r from our formula (created previously) and we obtain the formula Ro = 4.8r – 3.8 to estimate Ro, where r is a high daily rate of increase with the formula matching closely actual case numbers over several days (see below). For the estimation of r see:
COVID Odyssey: Vir[tu]al World Tour ~ How many people can one person infect in your country?
When we simulated the spread of COVID-19, we added in a gap-day and a factor of r^2 = r*r for an incubation period in a formula.
However when simulating using actual cases the incubation period is already factored in.
Hence the gap-day should not have been included.
We first try removing a factor of r*r from the formula (to adjust for the removal of the extra incubation gap-day).
The formula becomes:
Re#2* = 10r^9 * (r – 1)/(r^10 – 1)
or
Ro = 10r^9 * (r – 1)/(r^10 – 1)
[same formula as above]
or
Ro = 4.8r – 3.8 (see below).
To convert from the old values for Ro to the new values we need to divide the old values by (r*r).
The tables below include updated values:
COVIDWorldAvNewRanked
COVIDWorldAvNewAlpha
We sort above on Av r and use the formula Ro~ 4.8r – 3.8 (see below) to estimate Ro, where r is a high daily rate for the spread of COVID-19 estimated for countries worldwide using algorithms/heuristics developed in:
COVID Odyssey: Vir[tu]al World Tour ~ How many people can one person infect in your country?
Below is a graph of the above formulae for estimating Ro:
The straight line below the red curve provides a reasonable approximation for the curve.
Perhaps this should not be surprising since we have used a model in our simulations where a daily rate of r is “offset” against a daily decay rate of 1/r.
The straight line goes through (1,1) and the intersection of the two curves and has the equation y = mr +c where
m ~ 4.77574 and c ~ -3.77574
We round these values to
m = 4.8 and c = -3.8 to obtain the equation
y = 4.8r – 3.8
All equations go through (1,1).
The charts are so close; it is hard to see the difference:
We obtain the following table (the values for 4.8r-3.8 and Re#2* are reasonably close; we only need values to 1 d.p.) :
We can also try
y = 5r – 4
The chart (graph) becomes:
We see the new equation is closer to the red curve up until about r = 1.35.
If we need to have a closer match to the red curve, we can use two straight lines:
y = 5r – 4 up until r = 1.3
and
y = 4.5r – 3.35 for r > 1.3.
The graph becomes:
We compare estimations for Ro in the table below:
We leave it to you to decide if Ro = 5r – 4 is suitable over the entire range. Values start to get a little high after r = 1.4 if this is done, particularly if values are rounded to one decimal place.
However the method used in the algorithms/heuristics and the averaging of results suggest that there may be an underestimation of Ro; hence Ro = 5r – 4 may provide a simple suitable formula for estimating Ro provided that values calculated using r provide estimates that match actual cumulative cases over several days (preferable two cycles) early in an outbreak within a country.
Since only two countries listed below have r>1.42, we adopt the formula
Ro = 5r – 4
COVIDWorldAvNewRanked
COVIDWorldAvNewAlpha
We conclude that one person with COVID-19 may on average infect three other people. When r = 1.4 (e.g. after rounding in New Zealand), using the formula Ro = 5r – 4,
Ro = 3
[5×1.4 – 4 = 3].
We need to modify our simulation (remove the incubation gap-day) to confirm our results. The current linear (straight line) result may be surprising.
We may find we need to add in an extra factor of r. If we need to add in an extra factor of r, we may try using these (quadratic) curves to estimate Ro:
y = r(5r – 4) up until r = 1.3
and
y = r(4.5r – 3.35) for r > 1.3
The formula becomes:
Re#2* = 10*r^10*(r-1)/(r^10-1)
We would have the following table (note the extra factor of r):
We note that in the formula
Re#2* = 10*r^10*(r-1)/(r^10-1)
r^10 is not much bigger than (r^10-1) for the values we are using for r.
r^10/(r^10-1) is just over 1. For r = 1.4, we obtain the value 1.03581 for the ratio.
Therefore
Re = 10*(r-1)
Provides a lower bound for Ro.
Preliminary CSAW version 2 (“SeeSaw v2”) Excel simulation results suggest for r=1.4, Ro~4.14 (4.1432) as in the above table:
Maximum Infected means the number of people directly infected by 1 case (i.e. not by another person).
In the above tables the mean etc are based on the means of 40 samples.
i.e. The mean is the mean of 40 sample means.
The graph below uses the formula Re#2* = 10*r^10*(r-1)/(r^10-1) to estimate Ro:
The updated files (calculations using Av r) are:
COVIDWorldAvNewRanked4
COVIDWorldAvNewAlpha4
The above table suggests one person may infect on average at least four others when r=1.4.
To convert from original estimates for Re to the new estimates, divide by r.
We may assume Ro for new Zealand is between three and four. i.e. one person infected with COVID-19 may infect on average three to four other people within a ten-day (two-cycle) symptomatic infectious period and any pre-symptomatic infectious period. Recall that we originally estimated Ro = 6.
We tentatively now conclude Ro = 4 in New Zealand.
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:
https://academic.oup.com/jtm/article/27/2/taaa021/5735319
(Journal of Travel Medicine, Volume 27, Issue 2, March 2020)
“Countries” with a population of less than three million may be affecting the rankings.
These countries may need to be checked:
CountriesPopulation
CountriesPopulationAlpha
Source: https://www.worldometers.info/coronavirus/
For an introduction to methods used to calculate Ro by epidemiologists see:
COVID-19: Methods of Analysis
R0history
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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
| CSAWv3 | Maximum Infected | |||
| Alan Grace | 14 | (by 1 case) | ||
| alan99nz@gmail.com | ||||
| Copyright © Alan Grace June 2020 | ||||
| Cases: | 83 | 115 | 162 | 360 |
| Day | Day | Day | Day 11-13 | |
| 11 | 12 | 13 | 99 | |
| Mean= | 4.107229 | 4.158478 | 4.143210 | 4.139792 |
| Std Dev= | 0.206364 | 0.156181 | 0.158615 | 0.077888 |
| Max= | 4.469880 | 4.400000 | 4.469136 | 4.261111 |
| Min= | 3.493976 | 3.747826 | 3.759259 | 3.947222 |
| P= | 0.714286 | 0.714286 | 0.714286 | 0.714286 |
| Sample# | Mean | Mean | Mean | Mean |
| 1 | 3.915663 | 4.347826 | 4.123457 | 4.147222 |
| 2 | 3.746988 | 4.252174 | 4.265432 | 4.141667 |
| 3 | 4.012048 | 4.400000 | 3.901235 | 4.086111 |
| 4 | 4.180723 | 4.043478 | 4.358025 | 4.216667 |
| 5 | 4.120482 | 4.147826 | 4.179012 | 4.155556 |
| 6 | 4.084337 | 4.095652 | 4.055556 | 4.075000 |
| 7 | 3.879518 | 4.260870 | 4.382716 | 4.227778 |
| 8 | 4.277108 | 4.060870 | 4.327160 | 4.230556 |
| 9 | 4.337349 | 4.208696 | 4.092593 | 4.186111 |
| 10 | 3.951807 | 4.365217 | 3.870370 | 4.047222 |
| 11 | 4.108434 | 4.052174 | 4.000000 | 4.041667 |
| 12 | 4.253012 | 3.965217 | 4.302469 | 4.183333 |
| 13 | 4.072289 | 4.234783 | 3.993827 | 4.088889 |
| 14 | 4.385542 | 3.895652 | 3.950617 | 4.033333 |
| 15 | 3.939759 | 4.069565 | 4.246914 | 4.119444 |
| 16 | 4.108434 | 3.913043 | 3.938272 | 3.969444 |
| 17 | 4.132530 | 4.234783 | 4.345679 | 4.261111 |
| 18 | 3.493976 | 4.260870 | 4.191358 | 4.052778 |
| 19 | 4.421687 | 4.400000 | 3.759259 | 4.116667 |
| 20 | 4.192771 | 4.208696 | 4.240741 | 4.219444 |
| 21 | 4.144578 | 4.252174 | 4.191358 | 4.200000 |
| 22 | 4.072289 | 4.269565 | 4.074074 | 4.136111 |
| 23 | 4.144578 | 3.747826 | 3.987654 | 3.947222 |
| 24 | 4.469880 | 4.217391 | 4.024691 | 4.188889 |
| 25 | 4.409639 | 4.182609 | 4.154321 | 4.222222 |
| 26 | 3.939759 | 3.991304 | 4.104938 | 4.030556 |
| 27 | 3.759036 | 4.347826 | 4.037037 | 4.072222 |
| 28 | 4.120482 | 3.895652 | 4.265432 | 4.113889 |
| 29 | 4.313253 | 4.086957 | 4.030864 | 4.113889 |
| 30 | 4.337349 | 4.147826 | 4.166667 | 4.200000 |
| 31 | 3.867470 | 4.165217 | 4.222222 | 4.122222 |
| 32 | 3.951807 | 4.400000 | 4.271605 | 4.238889 |
| 33 | 3.879518 | 4.147826 | 4.469136 | 4.230556 |
| 34 | 4.132530 | 4.252174 | 4.259259 | 4.227778 |
| 35 | 4.349398 | 4.069565 | 4.345679 | 4.258333 |
| 36 | 4.204819 | 3.956522 | 4.265432 | 4.152778 |
| 37 | 4.156627 | 4.321739 | 3.944444 | 4.113889 |
| 38 | 4.036145 | 4.226087 | 4.111111 | 4.130556 |
| 39 | 4.277108 | 4.034783 | 4.154321 | 4.144444 |
| 40 | 4.108434 | 4.208696 | 4.123457 | 4.147222 |
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