COVID Odyssey: Vir[tu]al World Tour ~ How many people can one person infect in your country?

Please read Update:
COVID Odyssey Ro update: Occam’s Razor~ A close shave: In NZ is Ro ~ 3, 4, or 6? What are Ro values worldwide?
COVIDWorldAvNewRanked4
COVIDWorldAvNewAlpha4
COVIDWorldAvNewRanked4r
COVIDWorldAvNewAlpha4r

We estimate Ro (see below) by applying the analysis undertaken in New Zealand to the rest of the world. Note that data for the analysis below is for confirmed cases (only) downloaded from the website:
https://ourworldindata.org/covid-cases

We assume a homogeneous population and an S-I-R model early in a COVID-19 outbreak where everyone is (equally) susceptible, daily case numbers indicate the number of infectious people, and no people have recovered.

We will first look at data up until the end of April, then data up until early August.

First, we only look at cases up to the end of April. Consequently there will be some countries that only had a low number of cases up to the end of April, and these countries may end up with low scores for r, Re, and Re#2 (see below).

We use techniques and results to demonstrate a proof of concept.

We analyse and rank over 190 countries (later 208) using estimates for r, Re, and Re#2 to estimate Ro.

Note: New Zealand estimates for Re and Re#2 are for the second half of March up to 26 March.

For background information see:
COVID Odyssey. COVID-19 NZ: Re#2~5.8 (see below) & Ro~6
https://aaamazingphoenix.wordpress.com/2020/05/27/covid-19-nz-is-re-5-3782/
(data for the above link includes confirmed and probable cases).

Hence results below using only confirmed cases are likely to be less than those in the link above.

We have looked at the following simulation several times showing the outspread of COVID-19 (Coronavirus) from 5 to over 360 people in 5 cycles with Re = 2.6:

CoronaSpread

Source: New York Times.

How can a Coronavirus out-spread from 5 to 368 people in 5 Cycles (Credit: The New York Times)?

If 5 people with new coronavirus can impact 2.6 others each, then 5 people could be sick after 1 Cycle, 18 people after 2 Cycles, 52 people after 3 Cycles and so on. See:
https://towardsdatascience.com/how-bad-will-the-coronavirus-outbreak-get-predicting-the-outbreak-figures-f0b8e8b61991

We assume a cycle length of 5 days.

For any day D, let:

  • C[D] denote the number of cases on day D
  • Re denote the effective Reproduction rate of COVID-19 for one cycle
  • Re#2 denote the effective Reproduction rate of COVID-19 for two cycles
  • r denote the effective Reproduction rate of COVID-19 for one day
  • Ro denote the Reproduction rate without any quarantine or isolation

Re and Re#2 estimate the average number of people infected by one person with COVID-19.

We want to use high values for Re, r and Re#2 to estimate Ro.

We assume Ro is a constant for COVID-19 for a country. We assume that we can use Ro values for countries to estimate a range of values for Ro world-wide.

Our analysis indicates Ro is likely to be close to 6.

For COVID-19 in NZ we found that Re#2~5.8 & Ro~6. See:
https://aaamazingphoenix.wordpress.com/2020/05/27/covid-19-nz-is-re-5-3782/

We could calculate r by using the formula r = C[D+1]/C[D], however this could create quite large fluctuations from day to day.

Instead for each day D, we first calculate

Re = C[D+5] / C[D]

then calculate

r = Re ^ (1/5)

where ^ means ‘to the power of’.

We now calculate Re#2 using the formula

Re#2 = 10 * r^11 * (r-1) / (r^10-1)

We tabulate the results in three pdf files (up to the end of April):
COVIDWorld
COVIDWorldAll
COVIDWorld1

The first file (6 pages) contains the daily results where Re at least 4.9 (and cumulative number of cases at least 8) for all countries in the world.

The second file (195 pages) contains the daily results for all values of Re at least 1 (and cumulative number of cases at least 5) for all countries in the world.

The third file (26 pages) contains the daily results for all values of Re at least 1, cumulative number of cases at least 5, and r between 1.2 and 1.43 for all countries in the world. This equates to Re ~ 2.5 to Re ~ 6 in the table below.

Below is a table correlating Re, r, and Re#2. New Zealand results are highlighted in the right column. Results for the simulation above  correspond to r ~ 1.21 (Re ~ 2.6 and Re#2 ~ 3)

ReTable

Below is a chart comparing the above values:

ChartRe2

We see that the curves cross when r > 1.5. We note that this happens after Re > 7.5 (Re ~ 7.758).

No-one believes Re > 7 so we do not need to worry about this cross-over. Values of Re greater than 7 (r > 1.475773) can be ignored.

We look at COVIDWorld  to estimate Re and hence r, or r directly, from which we can calculate Re#2.

We only need to estimate r to 2 decimal places.

For New Zealand we get r ~ 1.38 which for compatibility with our previous results we round to r = 1.4. This gives Re#2 = 5.8.

We could use 5.8 as an estimate for Ro. However we note that 1.4^2 = 1.96.
This is close to 2 so we think that r = SQRT(2) may give a better estimate for Ro.

This gives Re#2 = 6 and we adopt the estimate Ro = 6 from New Zealand Data.

We may get different estimates for Ro for various countries which we can perhaps use to obtain an overall value for Ro. Ro is not intended to be variable.

We are interested in Re#2 to estimate Ro because most experts agree that COVID-19 is infectious for  two cycles (and maybe longer).

As you scroll through COVIDWorld you will see we have highlighted some values of interest in estimating Re and hence r.

We remind you that COVIDWorld only contains days when Re > 4.9 ( r > 1.375).

Not all countries meet this criterion.

For example Australia only has one day when r is slightly over 1.3 (Re ~ 3.7, and Re#2 ~ 4.2 from the above table) and therefore is not in the file COVIDWorld.

We would have to look in the file COVIDWorldAll instead to analyse values of Re, r, and Re#2 for Australia.

We can also look at COVIDWorld1 to include Australia.

For Australia we average the top 4 values for Re#2 from 12-21 March to get Re#2 ~ 3.88 from which we estimate Ro =4.

If an estimate ends up being rounded down, perhaps average the estimate obtained for Re#2 and the next one in the table.

Even when a country is in COVIDWorld, we should still look at COVIDWorldAll to confirm results.

For each country we look for sequential days when Re (or r) is relatively stable with relatively high values. This usually happens about from the middle of March to towards the end of March (earlier for China).

We may continue with more analysis below at a later date.

We may also look at automated the process of estimating r and Re#2.

Perhaps we may consider using moving averages for this.

Here are our preliminary results automatically ranking almost 200 (198) countries (explanation below):
COVIDWorldSorted

Until then we leave it to you to estimate values for Re, r, and Re#2 in your country using the two files above.

World rankings are based on Re#2 values for each country calculated from values of r* (in a new column) as follows:

  1. When we come to a new country r* is assigned the value 1.
  2. An estimated value for r for a given day is valid if r is between 1 and 1.43 (Re ~ 6) and the cumulative (total) number of cases is at least 8.
  3. If r is not valid then r* is assigned the previous day’s value for r*.
  4. If r is valid then r* becomes the average of r and the previous day’s value for r*.
  5. When the values for r* are calculated for all days up to the end of April for all countries, the maximum value for each country of r* is obtained and a new value for Re#2 calculated again using this value.
  6. Each country is ranked according to its new (maximum) value calculated for Re#2.

We expect the value calculated for Re#2 for each country to be slightly lower than it should be because:

  1. Only confirmed cases are used for each day (probable cases were not included in the dataset).
  2. Each day’s value for r* is an average over two days (whenever r is valid).

The averaging means that r* is less than the highest value of r.
Hence Re#2 underestimates Ro.

Hence Re#2 is a conservative estimation of Ro.

Countries up to #10 (Lithuania) all have values for Re#2 slightly over 6 (and also higher as well). See: COVIDWorld

For New Zealand using the above method r* was obtained as 1.3756 whereas previously we calculated a value for r of 1.4 (using confirmed and probable cases).

The last column (MA) in the table below shows how conservative the values for r* are in comparison to r (the bottom right value, 1.37556, is the maximum for r* in NZ):

ReNZtableMA

This resulted in an estimated value for new Zealand for Re#2 of 5.3882 (used for sorting using r = 1.3756) instead of 5.8005 (calculated using 1.4).

The values are also slow to come down. Quite often the lag between r and r* may only be one day. Compare the values for r* (MA column) at the bottom right of both tables with the preceding value for r in the row above two columns over. The table for NZ continues:

ReNZtableMA2

We also create a table with actual confirmed and probable NZ cases.
We apply the same formula to create column r* (MA in previous tables).
We obtain a maximum value for r* of 1.40190 (c.f. our estimate  r=1.4 in NZ).

TableReNZ4

Looking at the highlighted values in the Re#2 column, we estimate again that Ro = 6 in NZ.

We also note that when on 22 March r = 1.41014, r^2 ~ 1.988. This would tend to help validate our previous use of r = SQRT(2) to obtain Re#2 = 6.04683.

Except for the above table, we are only using data up to the end of April (almost 14,500 rows of data out of 35,000 rows). Consequently values for r for some countries may not have peaked particularly for low values in the above table. Nevertheless the concept is proven. Below is a version of the file with some countries removed (ranking for countries stays the same as before):
COVIDWorldGood

Below are the countries with the highest values (r=r*) from the above file:

COVIDtop20

Recall that historically we estimated that Ro = 6 in New Zealand.

New Zealand is ranked 12th in the list (15th, 18th, 19th, and 23rd on later lists).

The above list confirms Ro is 6 (at least) for COVID-19.

The above is a proof of concept.

We conclude from the COVIDWorldSorted file that conservatively Ro is at least 6 for COVID-19. i.e. one person with COVID-19 may on average infect 6 other people.

We also ran the data using all dates up to 6 August; almost 35,000 (over 34,800) rows of data.

An estimated value for r for a given day is valid if r is between 1 and 1.44 (Re ~ 6.19) and the cumulative (total) number of cases is at least 8.

New Zealand dropped to 15th, despite data for the whole world being deleted, and Australia to 67th (from 64th). See:
COVIDWorldSortedNew

Below are the countries with the highest values (r=r*) from the above file:

COVIDWorldTop20New

We have already seen that Re#2 underestimates Ro.

The three highest values for Re#2 confirm that Ro is at least 6 for COVID-19, especially since we already know that Ro=6 from analysis of New Zealand case data.

Note: Anguilla had too few cases (3) and Hong Kong had no data (perhaps cases included with China?).

Here are the two lists side by side:

COVIDWorldComp20

Next we defined an estimated value for r for a given day to be valid if r is between 1 and 1.45 (Re ~ 6.41) and the cumulative (total) number of cases is at least 8.

New Zealand dropped to 19th and Australia to 69th:

COVIDWorldRanked145

Here is the data file:
COVIDWorldRanked145

Finally we defined an estimated value for r for a given day to be valid if r is between 1 and 1.46 (Re ~ 6.634) and the cumulative (total) number of cases is at least 8.

COVIDWorldRanked146

New Zealand has risen one place. Here is the data:
COVIDWorldRanked146

Note that when r = SQRT(2) [ ~1.41421], Re ~ 6.0468.

At first we suspected some extreme values may be affecting some rankings:

Spain is now at the top with Re#2~6.43 and Switzerland is #5 (formerly #35).

Below is data for Spain:

SpainRe

Here is data for Switzerland:

SwitzerlandRe

We also removed the restriction on having at least 8 cases. We obtained:

COVIDWorldRanked146A

Here is the data file:
COVIDWorldRanked146A

Australia has gone down to #116 (from being in the top 75).

We note that New Zealand is close to our estimate of r = 1.4 (1.393832).

Recall that for New Zealand we estimated when calculating Ro that r = SQRT(2)  [~1.41421], from which we obtained an estimate for Ro of 6.05. See:
https://aaamazingphoenix.wordpress.com/2020/05/27/covid-19-nz-is-re-5-3782/
(also see the first table in the present post).

We concluded previously that Ro~6.

We revise our conclusion and conclude Ro is in the range 6 – 6.4.

This is within the range in this graph:

RoChart

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)

Our model where a constant daily rate of increase of r [ e.g. r = 1.4 or r = SQRT(2) ] is offset by a constant daily rate of decay of infectivity of 1/r over a ten-day [ two-cycle ] period appears to work well. See:
https://aaamazingphoenix.wordpress.com/2020/05/27/covid-19-nz-is-re-5-3782/

We have used the formula

Re#2 = 10 * r^11 * (r-1) / (r^10-1)

Re#2 is an estimate for Ro for a 10-day infectivity period.

At least when n is close to 10, for an n-day infectivity period, we can estimate Ro using the formula

Ro = n * r^(n+1) * (r-1) / (r^n-1)

When using r = SQRT(2) [~1.41421], with n = 10, we have obtained

Ro ~ 6.05

We calculated this value for New Zealand. We conclude that all countries ranked higher than New Zealand in the above tables also have at least this value for Ro.

Hence the top 23 countries (including New Zealand) in the above table also have a value for Ro of at least 6.05.

We assume that n = 10. To allow for a margin of error we conclude that Ro is between 6 and 6.4.

We believe that Ro is likely to be near the lower end of this range (around 6.05).

We conclude that world-wide one person with COVID-19 may infect on average 6 other people. This is more than double many other estimates, include estimates from the WHO.

We also note that when r = SQRT(2), the number of COVID-19 cases has the potential to double every two days.

Environmental factors may result in the lower Ro estimates for lower ranked countries.

We review our methodology (repeated below) for estimating Ro based on Re#2.

Recall we changed the value for r in step 2 up to between 1 and 1.46.

We ignored values outside this range (see step 3).

We assume that high values for r outside the range indicate a large number of infections within the community which have not been identified as cases. 

We used the following method each day:

World rankings are based on Re#2 values for each country calculated originally from values of r* (in a new column) as follows:

  1. When we come to a new country r* is assigned the value 1.
  2. An estimated value for r for a given day is valid if r is between 1 and 1.43 (Re ~ 6) and the cumulative (total) number of cases is at least 8.
  3. If r is not valid then r* is assigned the previous day’s value for r*.
  4. If r is valid then r* becomes the average of r and the previous day’s value for r*.
  5. When the values for r* are calculated for all days up to the end of April for all countries, the maximum value for each country of r* is obtained and a new value for Re#2 calculated again using this value.
  6. Each country is ranked according to its new (maximum) value calculated for Re#2.

We expect the value calculated for Re#2 for each country to be slightly lower than it should be because:

  1. Only confirmed cases are used for each day (probable cases were not included in the dataset).
  2. Each day’s value for r* is an average over two days (whenever r is valid).

The averaging means that r* is less than the highest value of r.
Hence Re#2 underestimates Ro.

Hence Re#2 is a conservative estimation of Ro.

Step 3 may mean that some countries are ranked lower than they should be.

We could change step 3 so that there is some increase in r*. We could consider an additional threshold value in step 3 for this situation (e.g. 1.2, halfway between 1 and 1.4).

The revised first three steps become:

  1. When we come to a new country r* is assigned the value 1.
  2. An estimated value for r for a given day is valid if r is between 1 and 1.46 (Re ~ 6.6) and the cumulative (total) number of cases is at least 1.
  3. If r is not valid then r* is assigned the the maximum of
    [the previous day’s value for r*; the average of 1.2 and the previous day’s value for r*] .

This would mean that r* could increase even when r is above the range.

The original methodology appears to work to estimate Ro.

Below is the resulting ranking using the revised methodology. New Zealand is now #25, ranked just below Norway which was previously #36. South Africa also moved up above New Zealand to #21:
COVIDWorldRanked146B

We change the threshold value in step 3 to 1.35. New Zealand goes down to #35 and Australia to #155:
COVIDWorldRanked146C

Perhaps we may need to revise our range for Ro to be between 6 and 6.5?

The values for New Zealand in the last three scenarios remain the same.

We compare the results of the last three scenarios (and average the results):
COVIDWorldAvRanked
COVIDWorldAvAlpha

New Zealand is ranked #20 and Australia #137.

We get a range for Ro from 6 to 6.5.

We can also estimate r from C[D], starting with a given day D, and C[D+d], the number of cases d days later, using the formula

r = ( C[D+d] / C[D]  ) ^ (1/d)

We have used this formula for d=5.

We may look at this later using various values for d.

For comparison we sort on the original values of r<1.43 with Re#2 between 1 and 6.5  and the number of cases at least 12. Sorting on r, Re or Re#2 will give the same rankings (ignoring equal values).

Here is the data:
COVIDWorldAvCycle5

New Zealand drops to #43.

For comparison we also sort on the original values of r<1.44 with Re#2 between 1 and 6.5  and the number of cases at least 8.

Here is the data:
COVIDWorldAvCycle5a

New Zealand drops to #50.

Perhaps the best ranking is provided by the solution repeated below using a secondary threshold in step 3 of 1.35. See:
COVIDWorldRanked146C

The revised (first three) steps become:

  1. When we come to a new country r* is assigned the value 1.
  2. An estimated value for r for a given day is valid if r is between 1 and 1.46 (Re ~ 6.6) and the cumulative (total) number of cases is at least 1.
  3. If r is not valid then r* is assigned the the maximum of
    [the previous day’s value for r*; the average of 1.35 and the previous day’s value for r*] .
  4. If r is valid then r* becomes the average of r and the previous day’s value for r*.
  5. When the values for r* are calculated for all days up to the end of April for all countries, the maximum value for each country of r* is obtained and a new value for Re#2 calculated again using this value.
  6. Each country is ranked according to its new (maximum) value calculated for Re#2.

You may also like to look at my analysis for 12 African countries. See:
https://aaamazingphoenix.wordpress.com/2020/08/24/covid-odyssey-african-safari-big-game-shooting-with-a-camera/

Apparently Ro is not regarded as a constant for COVID-19.

All our analysis suggests that Ro is close to 6.

Update:
We update our estimates for Ro and obtain for New Zealand Ro=2.9. For worldwide values for Ro see:
COVID Odyssey Ro update: Occam’s Razor~ A close shave: NZ Ro~2.9; Ro = 4.8r-3.8
COVIDWorldAvNewRanked
COVIDWorldAvNewAlpha

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