COVID Odyssey: CSAW simulation ~ form[ul]ation verification

CSAW

The CSAW formulation
Needs more verification
Tho I’m so nigh certain
Next’s show’s final curtain
Please seek a neat foundation

Alan Grace
5 September 2020

We look at the formula that we developed from the CSAW
(COVID-19  Sampling Analysis Worksheets) model (“SeeSaw” model). 
We establish a table to verify the formula and use the table to confirm
Ro for New Zealand. We issue a caveat for using this methodology for
other countries without graphing the results.

We assume a person with COVID-19 has a 10-day infectious period
(when the disease can be transmitted to others). Any actual period
longer than this would produce a larger value for Ro.

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 have developed the formula

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

where Ro is the number of people that may on average be infected by one
person with COVID-19 and r is a high daily rate of increase for case numbers.

For new Zealand we established r = 1.4  and hence Ro ~ 4.1.

The formula

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

has the same value in each of 10 days (hence the factor of 10) since the daily increase (r) is offset by a daily rate of decay in infectivity of 1/r.

We establish a table of data where the number of cases each day is generated using r  = 1.4.

The second column is the sum of the previous number of cases over the last 10 days (Sum10).

We divide the value for Ro (~4.1432) by 10 (to get a daily figure) and multiply this value by Sum10.

As you can see below, the result is the number of cases for the next day.

This verifies the formula.

This also suggests that if

C[i] is the total number of Cases on Day i

and

MA[i] is the moving average over the previous 10 days (before Day i)

then

Re ~ MA[i]/C[i]

We can use consistently large values for Re to estimate Ro.

This should be obvious from the table below since Ro=C/MA gives C=Ro*MA:

We can also try the method in the first table in reverse for a country.

Below is New Zealand data.

Once again Sum10 is the sum of the number of cases over the last 10 days.

This time we divide the number of new cases (C) by this number and multiply the result by 10 to estimate Ro.

We see Ro  is just over 4.1 for most of the highest values (ignore 4.55).

We use 4.1 as an estimate for Ro and multiply Sum10 by Ro/10 (see last column).

We see that for many days (6 out of 8 days) up to 26 March, the value is close to the actual number of cases.

This helps verify Ro ~ 4.1 in New Zealand.

We try the same  for the 12 worst countries in Africa and estimate Ro = 2.6 in the table below.

We note that when r =  1.22, using our formula Ro ~ 2.56 (close to 2.6).

This (r = 1.22) was the estimate in this article published 16 July 2020:
Africas40249-020-00718-y. See:
COVID Odyssey: The pain in Spain 50% gain ~ Spanish Inquisition

We need to confirm this with a graph. We see that r = 1.22 (top curve) does not fit the data well during the dates in the graph.

We see that once again r = 1.4 is a better fit (red curve):

We conclude that r = 1.4 and Ro = 4.1 for these 12 African countries (combined).

The result r = 1.4 occurs before the table starts calculating Sum10!

The values 2.37 (in the article) and 2.6 (above) appear to be just estimates for Re.

In the next post we may explain our simulation model in detail.

Below is a table showing values for Ro (cf Re#2*) based on the formula Ro=10*r^10*(r-1)/(r^10-1) and estimates using more simple calculations: 

In the Graph below y is a good estimate for Ro where

y = r(5r – 4) up until r = 1.3

and

y = r(4.5r – 3.35) for r > 1.3

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

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