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
Please seek neat explanation

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.

Let

  • r denote the effective Reproduction rate of COVID-19 for one day
  • Ro denote the Reproduction rate without any quarantine or isolation
  • Re denote the effective Reproduction rate of COVID-19
    (Re assumes isolation/quarantine is happening)

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:

CSAWcases

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):

CSAWv3Sampling

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.

CSAWv3Sampling2

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

See:
COVID Odyssey: CSAW simulation ~ formulation verification

The results look like this in CSAW v2:

CSAW   CSAW2

CSAW3   CSAW4

CSAW5  CSAW6

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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