# COVID Odyssey: CSAW simulation ~ form[ul]ation explanation

CSAW

The CSAW simulation
Needs no more validation
Tho I’m so nigh certain
Next’s show’s final curtain

Alan Grace
8 September 2020

We provide an explanation of the CSAW (COVID-19 Sampling Analysis Worksheets) model (“SeeSaw” model).  Please also see:
COVID Odyssey: CSAW simulation ~ formulation verification

First we randomly simulate an outbreak of COVID-19. Once we have generated one outbreak (sample), we generate 39 more (40 outbreaks altogether) and provide the mean number infected for Days 11 – 13 (individually and overall) for each outbreak (sample) and overall statistics for the 40 outbreaks (samples).

We look at over 16,000 cases and randomly determine when each case (on Day 11-13) was infected and when this is done for all the cases, we count up the number of people (directly) infected by each case.

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).

We use a daily rate of increase of r = 1.4 (40% increase per day) and adopt a 10-day infectious period (when a case can infect others).

Below there are two tables (combined) that we use to generate each case/outbreak:

Each case can infect other people for the next 10 days.

We start our generation on Day 12 (Cases 290-404). This allows us to provide statistics for Day 11 later.

We also create stats for Day 13. This means we need to look at the next 10 days (to Day 23) to see how many are infected from Day 13 cases.

We see on Day 23 we are up to almost 16,400 cases. Starting from Case#290 this means we need to look at over 16,100 cases.

First we look at Case#290. We randomly select a day (from 1-10) days ago when Case#290 was infected.

We use a random number (from 0 to 1) to determine which day Case#290 was itself infected.

Look at the Cumulative column in the left table above. We see there is an almost 30% chance (0.2959456) of Case#290 being infected “one day” ago. Note that the incubation period is already built in to the actual case numbers.

From the table on the right we see the total number of cases for the selected day and randomly select one of the cases to infect Case#290. For example we have 83 cases on Day 11. If Case#290 ends up being infected one day ago (on Day 11) we choose one of these cases to infect Case#290, say Case#270.

The case number is recorded (alongside Case#290).

We repeat this process for the next approximately 16,100 cases (until the end of Day 23) so that we know which case infected each of these cases.

We count up the number of cases infected by Case#290 and each of the other cases.

Below is part of a run (reduced later to 4 columns):

We see that 6 others have been infected by Case#290.

Below is part of another run (the headings are the same as above).
We see on Day 13, Case#560 infected Case#16372.

We calculate the mean number of people infected by each case on Day 11, 12, and 13 for the first outbreak and for the other 39 outbreaks simulated.

The results for the mean are similar to the values obtained for r = 1.4 in the formula for Ro:

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

where we obtained (using r = 1.4), Ro  = 4.143238436

The results look like this in CSAW v2:

Maximum Infected means the number of people directly infected by 1 case (i.e. not by another person).

Below are worldwide estimates for Ro (with some previous results divided by r):
COVIDWorldAvNewRanked4r
COVIDWorldAvNewAlpha4r

You may also like to look at:
COVID Odyssey by Alan Grace: New Year Fare ~ Contents

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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