COVID Odyssey. COVID-19 NZ: [In NZ Ro ~ 3, 4, or 6? See Update] Re~5.8 & Ro~6?

COVID Odyssey

COVID-19 in New Zealand

Findings

Alan Neil Grace

[Updates will be available soon only in an ebook;
Contact Alan Grace for details about the ebook]

Introduction
[Ro ~ 3, 4 or 6? See Update] Re ~ 5.8 and Ro ~ 6?

Please read Updates:
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 Ro update: Occam’s Razor~ A close shave: In NZ is Ro ~ 3, 4, or 6? What are Ro values worldwide?

Originally: COVID-19 is more than twice as virulent as most researchers believe. One person may on average infect 6 other people. The number of cases may double every two days. We explore an Excel spreadsheet simulation, CSAW (“SeeSaw”): COVID-19 Sampling Analysis Worksheets, generating random samples of the spread of COVID-19 in New Zealand.

Update: Historically here we built an incubation period into the simulations. This is no longer considered necessary since incubation periods are already built into the data.

We investigate whether one infected person may on average infect 6 others in New Zealand up until Lockdown Level 4 on 25 March at 11.59. First we investigate whether one infected person may on average infect 5.3782 (more than 5 and 3/8) other people in New Zealand using a simple model.

We collate information from posts when this site was a blog and examine the spread of COVID-19.

Starting 1 August (or earlier), updates will only be found in the companion ebook (currently under development). Preliminary results can be found in this companion site (the page you are reading). Historically the site has been a blog (over 100 COVID-19 posts); now it has become essentially a website.

We divide this section into six Chapters:

  • Chapter 1: Scenario A. Infectious period One cycle (5 days)
  • Chapter 2: Scenario B. Infectious period Two cycles (10 days)
  • Chapter 3: The CSAW Excel spreadsheet
  • Chapter 4: Worst case scenarios
  • Chapter 5: Results
  • Chapter 6: Conclusions

We apologise for the presentation of this material. This material was originally part of a blog. Accordingly we do not follow academic writing protocols or referencing. Mathematical symbols including superscripts and subscripts were not available.

For a wider-page (old) version of the page you are reading, see:
https://alangraceblog.wordpress.com/2020/05/27/covid-19-nz-is-re-5-3782/
(note that the above page is an older version, particularly for conclusions, but may be more readable)

In the first wave of COVID-19 we found that cases increased by a factor of 1.4 each day from Day 18 (16 March) up until Lockdown Level 4. This suggested that Re = 5.37824 for a five day cycle.

The factor a=1.4 clearly fits the actual (cumulative) number of cases well:

nzcases300

Note:
Ro is used when there is no quarantine. i.e. when the virus is unrestrained.
Re is used when there is quarantine (Re is the effective reproduction rate).
We may use R in general discussion.
For an explanation about the basic reproduction number Ro
(pronounced R-Nought or R-Zero) see:
https://aaamazingphoenix.wordpress.com/2020/05/26/covid-19-nz-is-ro-likely-to-be-2-9/

In our discussions, the reader will need to decide where it may be appropriate to replace Ro by Re and vice versa.

The factor 1.4 means that the number of cases almost doubles every two days since 1.4^2 = 1.96.

If Ro = SQRT(2) then this would mean doubling every two days. The cases would follow the pattern 5, 7, 10, 14, 20, 28,  … where every two days the number has doubled.

This would suggest that Ro is

SQRT(2)^5 ~ 5.65685

if infectivity is the same for each day once a case is symptomatic.

In Chapter 3 we look at the CSAW simulation spreadsheets where infectivity decreases (decays) by the same factor each day.

We find that Ro may be over 6 or even 6.3 or more.

New Zealand was very lucky during the first wave since New Zealand was almost too late going into Lockdown Level 4.

Scenario A and Scenario B both give the same estimated values for the total number of cases. We assume an infectious period of one cycle (Scenario A) or two cycles (Scenario B).
We adopt a cycle length of 5 days, and an incubation period one cycle long.

In Chapter 2 and Chapter 3, we use Excel to simulate an outbreak. We generate over 16,100 cases for each simulation (sample), and 40 simulations for each dataset, calculating the mean number of cases infected by each person in each simulation and dataset.

Where no source is indicated below or elsewhere on this site, data can be found at: https://www.worldometers.info/coronavirus/

We have found that

  • One infected person may on average infect 5.3782 other people
    (more than 5 and 3/8 and possibly 6) in New Zealand in March (even 5.3782 is twice as many people as most experts expect).
  • In reality a 5-day cycle proves useful
  • The total number of cases can be expected to almost double every two days (1.4^2 = 1.96). Note: ‘^‘ means ‘to the power of’

We collate information from previous posts.

We generate a curve fitting the total (cumulative) daily case numbers.
See the table of data and graph below.

Since our estimates for Ro are based on the actual New Zealand situation where there has always been at least some form of self-isolation, our estimates for Ro may only be high estimates for Re since the virus has not been unrestrained.

For an explanation about the basic reproduction number Ro
(R-Nought or R-Zero) see:
https://aaamazingphoenix.wordpress.com/2020/05/26/covid-19-nz-is-ro-likely-to-be-2-9/

We assume the daily total number of cases increases by a factor of 1.4 (for the increase in the number of cases) since this fits the actual data well at least for most of March.

Chapter 1: Scenario A.
Infectious period One cycle (5 days)

In summary:

  1. The total number of cases increases daily by close to 40%
    (by a factor of 1.4).
    We start with an estimated 10 cases (total) on 16 March 2020
    (actual number of cases = 8) and find a graph plotting an increase by a factor of 1.4 daily up to 26 March (the day after New Zealand went into Lockdown Level 4 at 11.59 pm on 25 March) fits the data well.
  2. Therefore the daily number of new cases follows the same increase (factor 1.4).
  3. The spread follows 5-day cycles (5-day incubation period followed by one 5-day infectious period).
  4. The 5-day cycles have a factor of 1.4^5 = 5.3782.
  5. One infected person can therefore infect on average close to 5.4 other people (Re ~ 5.4 and Ro is therefore at least 5.4). This is at least double what most experts expect.
  6. We conclude that Ro = 5.65685 is also possible.
  7. We also see that Ro may be in the range 6 to 6.5.
  8. We looked at a two-cycle infectious period (with Re near half the size) but have concluded the a one-cycle period appears to work better.
  9. Re =5.3782 suggests cumulative case numbers almost double every two days (1.4^2 = 1.96).

A curve using a factor of 1.4 for total case numbers fits well for most of the second half of March (see table at the end of this post). i.e.

C[D] = 1.4 * C[D-1] for D > 18

where C[D] is the estimated (total) number of case numbers on Day D and C[18] = 10.

We also see that

C[D] = SQRT(2) * C[D-1] for D > 16

is possible where C[D] is the estimated (total) number of case numbers on Day D and C[16] = 5.

We found that a 5 day cycle worked well for the cycles up to 26 March and generated the following table:

modelcases14tablefinal-7b

The column before the black vertical line is generated first, starting with the value 10 for the estimated number of cases on 16 March.

The rest of the numbers in the last column before the black vertical line are generated by multiplying by 5.3782 (the daily factor is 1.4; a 5-day cycle therefore has a factor of
1.4 ^ 5 = 5.3782).

The penultimate column before the black vertical line contains the differences of the last column.

This (penultimate) column is therefore the estimated daily number of new cases.

We see this (penultimate) column also has a factor of 5.3782.

It should not be surprising that the daily number of new cases also has a factor of a = 1.4.

The total number of cases starting on Day D on successive days are
C[D], a * C[D], and (a^2) * C[D].

This means that the daily number of new cases on the latest two days are

a * C[D] – C[D] = (a-1) * C[D]

and

(a^2) * C[D] – a * C[D] = a * (a-1) * C[D]

This is a factor of a times the number of new cases on the previous day.

When the total number of cases increases by a factor a then the daily number of new cases also increases by a.

i.e. The number of new cases also increases daily by a = 1.4.

This means that each new case infects on average 5.3782 other people.

This factor (in this column) means that Re ~ 5.3782 (= 1.4 ^5) from  16 March (8 actual cases) to 26 March (283 actual cases).

Rounding this number provides an estimate of at least 5.4 for Ro in New Zealand and hence worldwide for about the first four weeks after the first case appears.

We conclude that Ro is at least 5.4.

Initially a value half this amount was expected.

However after searching last night:
The COVID-19 Coronavirus Disease May Be Twice As Contagious As We Thought
‘A single person with COVID-19 may be more likely to infect up to 5 or 6 other people, rather than 2 or 3, suggests a new study of Chinese data from the CDC. It’s not clear if this higher number applies only to the cases in China or if it will be similar in other countries.’ See:
https://www.forbes.com/sites/tarahaelle/2020/04/07/the-covid19-coronavirus-disease-may-be-twice-as-contagious-as-we-thought/#4eecbbc729a6

Excerpts from the article are at the bottom of this post.

We see below how the curve with  a daily factor of a = 1.4 fits the actual data:

nzcases300

Below is the same graph up to Day 35 (2 April):

nzcases3500

Note that New Zealand went into Lockdown Level 4 on 25 March (Day 27) at 11.59 pm.

New cases on the next day(s) are likely to have been infected  5 days prior to being identified as a case.

Below is the table of data for March.

Note that New Zealand went into Lockdown Level 4 at 11.59 pm on 25 March.
Perhaps New Zealand’s curve flattened more than expected.
Perhaps New Zealand may have had by the end of March over 900 more cases (1,555 instead of 647; 2.4 times as many) if we had not gone into Lockdown L4.

This could have resulted in a total of over 3100 cases (2 x 1,555 cases, double the current 1504 cases) assuming a midpoint on 31 March (Day 33).

A midpoint two days later on Day 35 (2 April) may have meant a total of over 6,000 cases (2 x 3,049 cases) if the red curve had remained accurate.

Note that the curve is flipped (values up to the mid-point are rotated by 180 degrees) at the midpoint to create an S- Shaped curve (“S-Curve) to estimate the remaining values.

We obtain the graph below (estimated values are approximate after midpoint):

NZcases6500Chart2

Note that the values are the same for the top two graphs up to Day 33 (31 March).

Note also that instead of doubling single values, historically two days’ values have been added together.

The graph below shows the result of “flipping” the actual cases at Day 34.5 (1-2 April) with the case midpoints giving a total case value 1505 = 708+797

Ideally we want our estimated curves to be higher than the actual data in most places so that some flattening on the actual curve may be indicated.

NZcases6500ChartFlipped

Using Day 33.5 (instead of 34.5) we may need to add on an extra 10% to account for the expected very long tail.

Day 33.5 gives a total case value of 647 + 708 = 1355.

Adding on 10% gives 1355 x 1.1 = 1490.

From the bar chart below Day 33.5 appears to be a likely midpoint (31 March – 1 April):

NZ16Bar

The bar chart is bimodal. It has two peaks and cannot be represented very closely after the first peak (even when the curve is flipped) by a curve which is increasing.

Here is another graph:

NZcases6500ChartFlipped2

Finally we take Day 34 as the midpoint and double it to get 2 x 708 = 1416

Here is the graph:

NZcases6500ChartFlipped3

It is difficult to work out the midpoint even when the outbreak appears over.

D Actual a
Date Day# Cases 1.4
07/03/20 9 5 C=
08/03/20 10 5 10a^(D-18) Diff
09/03/20 11 5
10/03/20 12 5
11/03/20 13 5
12/03/20 14 5
13/03/20 15 5
14/03/20 16 6
15/03/20 17 8
16/03/20 18 8 10.00
17/03/20 19 12 14.00 4.00
18/03/20 20 20 19.60 5.60
19/03/20 21 28 27.44 7.84
20/03/20 22 39 38.42 10.98
21/03/20 23 52 53.78 15.37
22/03/20 24 66 75.30 21.51
23/03/20 25 102 105.41 30.12
24/03/20 26 155 147.58 42.17
25/03/20 27 205 206.61 59.03
26/03/20 28 283 289.25 82.64
27/03/20 29 368 404.96 115.70
28/03/20 30 451 566.94 161.98
29/03/20 31 514 793.71 226.78
30/03/20 32 589 1,111.20 317.49
31/03/20 33 647 1,555.68 444.48
01/04/20 34 708 2,177.95 622.27
02/04/20 35 797 3,049.13 871.18
03/04/20 36 868 4,268.79 1,219.65

Any difference in the estimated/actual totals up to 26 March is caught up on the next day(s).

The spread in New Zealand in March was worse than in the animated gif below which shows five cycles with R = 2.6.

New Zealand reached the same number of cases (368 as shown in the gif below) on 27 March one day after two cycles! Note that instead of 5 cases we started with 10 cases as our estimate in  our model which provided a good fit for the actual number of cases in New Zealand.

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-oNtbreak-get-predicting-the-outbreak-figures-f0b8e8b61991

spread

We now look at abstract of the article referred to above:

Sanche S, Lin YT, Xu C, Romero-Severson E, Hengartner N, Ke R. High contagiousness and rapid spread of severe acute respiratory syndrome coronavirus 2. Emerg Infect Dis. 2020 Jul [date cited]. https://doi.org/10.3201/eid2607.200282

DOI: 10.3201/eid2607.200282

Original Publication Date: April 07, 2020

https://wwwnc.cdc.gov/eid/article/26/7/20-0282_article?deliveryName=USCDC_333-DM25287

Volume 26, Number 7—July 2020
Research
High Contagiousness and Rapid Spread of Severe Acute Respiratory Syndrome Coronavirus 2

Steven Sanche1, Yen Ting Lin1, Chonggang Xu, Ethan Romero-Severson, Nick Hengartner, and Ruian KeComments to Author
Author affiliations: Los Alamos National Laboratory, Los Alamos, New Mexico, USA

Abstract
Severe acute respiratory syndrome coronavirus 2 is the causative agent of the 2019 novel coronavirus disease pandemic. Initial estimates of the early dynamics of the outbreak in Wuhan, China, suggested a doubling time of the number of infected persons of 6–7 days and a basic reproductive number (R0) of 2.2–2.7. We collected extensive individual case reports across China and estimated key epidemiologic parameters, including the incubation period. We then designed 2 mathematical modeling approaches to infer the outbreak dynamics in Wuhan by using high-resolution domestic travel and infection data. Results show that the doubling time early in the epidemic in Wuhan was 2.3–3.3 days. Assuming a serial interval of 6–9 days, we calculated a median R0 value of 5.7 (95% CI 3.8–8.9). We further show that active surveillance, contact tracing, quarantine, and early strong social distancing efforts are needed to stop transmission of the virus.

In New Zealand Re = 5.3782 indicates that the number of cases almost doubles every two days (1.4^2 = 1.96).

If you look at the two-day ratio from 16 to 24 March (days 18, 19, … ,26) of

(total) number of cases two days ahead
current (total) number of cases

i.e.

C[D+2]/C[D] for D=18, 19, … , 26

The average is 2.053157507.

To get the daily increase we take the square root.
The SQRT of 2.053157507 is 1.432884331

This gives a 5-day increase of
1.432884331^5 = 6.040260494

We note that a two-day increase using the factor 1.4 is

1.4^2 = 1.96

We conclude that a factor of SQRT(2) is also possible giving a 5-day increase of

SQRT(2)^5 = 5.65685

We adopt Ro = 5.65685.

This means that it is possible for the total number of cases to double every two days.

We see that

C[D] = SQRT(2) * C[D-1] for D > 16

is possible where C[D] is the estimated (total) number of case numbers on Day D and C[16] = 5.

This also gives C[18] = 10 as we had with the factor 1.4.

We also consider a 5.2 day cycle.
‘The mean incubation period was 5.2 days (95% confidence interval [CI], 4.1 to 7.0), with the 95th percentile of the distribution at 12.5 days.’ See:
https://www.worldometers.info/coronavirus/coronavirus-incubation-period/

A 5.2 day cycle has an increase of
1.432884331^5.2 = 6.490795035

This suggests that Ro in the range 6 to 6.5 is possible in New Zealand.

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)

Chapter 2: Scenario B.
Infectious period Two cycles (10 days)

In a previous post we saw that

C[D] = 1.14333333 * ( C[D-2] + C[D-3] ) for D > 20
(note C[D-1] is missed out)

with starting values C[18] = 10, C[19] = 14, and C[20] = 19.6

i.e. a modified Fibonacci sequence.

See:
https://aaamazingphoenix.wordpress.com/2020/05/25/covid-19-nz-using-a-fibonacci-sequence-to-estimate-total-case-numbers-v/

For i>2 using C[2] = 8 and C[1] = 5, for given constant values of LAMBDA and Re, we can calculate

C[i] = Re* ( LAMBDA * C[i-1] + (1 – LAMBDA) * C[i-2] )

Where C[i] is the estimated number of new cases in cycle i.

See Chapter 6: Conclusions for calculations.

For Re = 5.8 we obtain:

COVIDTable58r58

In the above model each case infects people over two cycles (10 days).

Hence each cycle contains new cases infected by cases infected themselves during the two previous cycles.

Chapter 3:
The CSAW Excel Spreadsheet

We use Excel to simulate a (10-day) scenario with infectivity decreasing (decaying) each day.

We randomly generated sets of 40 samples and their means in Excel using a factor of 1.4 each day for the increase in the number of cases.

Below are two sets of samples (the Mean is actually the mean of the means for 40 samples): The above tables used an infection rate P=0.50489 which halved on each successive day. The value of P was calculated so that in 10 days the sum was 1.

The values above suggest that Ro may be over 3.5 (see dataset for P=0.5) in this scenario.

However we understand that infection is highest on the first day and diminishes on successive days.

A scenario where the infection rate was the same each day (P=0.1), produced means over 13.7.

Below are the calculations for P= 0.75.

v= 3
w= 4
v/w= 0.75
w/v= 1.333333 Factor
1/Sum= 0.019891
10 1 0.019891
9 1.333333 0.026522
8 1.777778 0.035362
7 2.37037 0.04715
6 3.160494 0.062866
5 4.213992 0.083822
4 5.618656 0.111762
3 7.491541 0.149017
2 9.988721 0.198689
1 13.31829 0.264918
Sum= 50.27318 1

We work backwards, assigning a value of 1 on day 10, then multiplying each day by the inverse, in this case by 1/0.75 = 1.333333. We scale the values by the inverse of the Sum (see bottom row), so that the daily factors sum to 1.

On the spreadsheet each case was assigned a number, and each new case was randomly assigned an infection day using a table like the above (the day number adjusted up to 10 days prior), and a case from this day was randomly selected as the case causing infection. The number of people infected by each case was then counted, and the mean calculated for each new case infected on Day 12 (so that there were 10 prior days’ cases causing infections on each day for future development).

The means were calculated for the 40 samples, and the overall mean and standard deviation calculated.

Below we look at sample runs for P = 1/2 (0.5), 7/10 (0.7), 5/7 (0.714286), and 3/4 (0.75).

Mean= 3.207675 Mean= 5.50636
Std Dev= 0.175297 Std Dev= 0.25242
Max= 3.605263 Max= 6.236842
Min= 2.921053 Min= 5.008772
P= 0.5 P= 0.7
Sample# Mean Sample# Mean
1 3.605263 1 5.807018
2 3.131579 2 5.307018
3 3.464912 3 5.570175
4 3.061404 4 5.535088
5 3.307018 5 5.622807
6 3.447368 6 5.438596
7 3.491228 7 5.438596
8 3.149123 8 5.403509
9 3.096491 9 5.912281
10 3.087719 10 5.368421
11 3.22807 11 5.868421
12 3.298246 12 5.666667
13 3.315789 13 5.675439
14 2.921053 14 5.508772
15 3.105263 15 5.482456
16 2.921053 16 5.482456
17 3.175439 17 6.236842
18 3.377193 18 5.684211
19 3.061404 19 5.429825
20 3.175439 20 5.526316
21 3.192982 21 5.526316
22 3.394737 22 5.184211
23 3.052632 23 5.236842
24 3.245614 24 5.491228
25 3.192982 25 5.5
26 2.947368 26 5.324561
27 2.929825 27 5.561404
28 3.254386 28 5.350877
29 3.438596 29 5.008772
30 3.166667 30 5.175439
31 3.333333 31 5.140351
32 2.938596 32 5.421053
33 3.061404 33 5.280702
34 3.131579 34 5.807018
35 3.035088 35 5.622807
36 3.254386 36 5.675439
37 3.263158 37 5.061404
38 3.45614 38 5.27193
39 3.175439 39 5.833333
40 3.421053 40 5.815789
Mean= 5.8 Mean= 6.555263
Std Dev= 0.223263 Std Dev= 0.260832
Max= 6.359649 Max= 7.04386
Min= 5.394737 Min= 5.72807
P= 0.714286 P= 0.75
Sample# Mean Sample# Mean
1 6.22807 1 6.535088
2 6.359649 2 6.403509
3 5.815789 3 6.929825
4 5.780702 4 6.473684
5 5.561404 5 6.526316
6 5.95614 6 6.798246
7 5.780702 7 6.412281
8 5.508772 8 6.614035
9 5.842105 9 6.649123
10 5.447368 10 6.315789
11 5.807018 11 6.649123
12 6.008772 12 6.307018
13 6.192982 13 6.333333
14 5.982456 14 7.04386
15 5.649123 15 6.447368
16 5.973684 16 6.622807
17 5.868421 17 6.412281
18 5.982456 18 6.54386
19 6.008772 19 6.552632
20 5.45614 20 6.964912
21 5.710526 21 6.885965
22 5.72807 22 6.605263
23 5.666667 23 6.394737
24 5.807018 24 5.72807
25 5.675439 25 6.263158
26 5.798246 26 6.473684
27 5.842105 27 6.736842
28 5.394737 28 6.54386
29 5.447368 29 6.184211
30 6.008772 30 6.973684
31 5.552632 31 6.394737
32 5.631579 32 6.631579
33 5.912281 33 6.587719
34 5.807018 34 6.947368
35 5.833333 35 6.859649
36 5.570175 36 6.596491
37 5.885965 37 6.649123
38 6.122807 38 6.666667
39 5.719298 39 6.438596
40 5.675439 40 6.114035

We have seen scale factors of P = 1/2 (0.5), 7/10 (0.7), 5/7 (0.714286), and 3/4 (0.75).

The scale factors have themselves been scaled so that the sum for each (column) is 1.

Scales: 1/2 7/10 5/7 3/4
1 0.500489 0.308721 0.295946 0.264918
2 0.250244 0.216104 0.21139 0.198689
3 0.125122 0.151273 0.150993 0.149017
4 0.062561 0.105891 0.107852 0.111762
5 0.031281 0.074124 0.077037 0.083822
6 0.01564 0.051887 0.055026 0.062866
7 0.00782 0.036321 0.039305 0.04715
8 0.00391 0.025424 0.028075 0.035362
9 0.001955 0.017797 0.020053 0.026522
10 0.000978 0.012458 0.014324 0.019891

This effect of this scaling can be seen in the graph below:

ScaleFactors

We use the sample below on the next page:

Mean= 5.818421
  Std
Dev=
0.244262
Max= 6.307018
Min= 5.342105
and P= 0.75

We normalise the values (subtract the mean from each and divide by the standard deviation) then, after rounding, create a frequency table.

As expected the graph roughly looks like a normal curve. The sampling distribution of the means is normal even when the underlying distribution is not normally distributed.

Sample# Mean N. Mean Round Dev Frequency
1 6.192982 1.834647 2 -4 0
2 5.95614 0.541293 1 -3 0
3 5.692982 -0.89577 -1 -2 3
4 5.824561 -0.17724 0 -1 8
5 6 0.780803 1 0 17
6 5.675439 -0.99157 -1 1 8
7 5.491228 -1.99751 -2 2 4
8 6.026316 0.924509 1 3 0
9 6 0.780803 1 4 0
10 5.745614 -0.60836 -1 Checksum 40
11 6.017544 0.876607 1
12 5.72807 -0.70416 -1
13 5.719298 -0.75206 -1
14 5.596491 -1.42269 -1
15 6.166667 1.690941 2
16 5.912281 0.301783 0
17 5.921053 0.349685 0
18 5.807018 -0.27304 0
19 5.578947 -1.51849 -2
20 5.859649 0.014371 0
21 5.780702 -0.41675 0
22 5.657895 -1.08738 -1
23 6.04386 1.020313 1
24 5.780702 -0.41675 0
25 5.824561 -0.17724 0
26 5.701754 -0.84787 -1
27 5.780702 -0.41675 0
28 5.868421 0.062273 0
29 5.833333 -0.12934 0
30 6.219298 1.978353 2
31 5.903509 0.253881 0
32 5.464912 -2.14122 -2
33 5.789474 -0.36885 0
34 5.859649 0.014371 0
35 5.789474 -0.36885 0
36 5.842105 -0.08143 0
37 6.140351 1.547235 2
38 5.938596 0.445489 0
39 6.087719 1.259823 1
40 6.061404 1.116117 1

Here are graphs from the data above:

SampleMeans

SampleMeansBar

It should now make sense if we talk in more depth about the spreadsheet.

We start Day 1 on 16 March when there were 8 cases in New Zealand. Day 1 is therefore 17 days after New Zealand identified its first case of COVID-19 on 28 February 2020.

On the spreadsheet each case was assigned a number, and each new case was randomly assigned an infection day using a table like the above (the day number adjusted up to 10 days prior), and a case from this day was randomly selected as the case causing infection.

The number of people infected by each case was then counted, and the mean calculated for each new case infected on Day 12 (so that there were 10 prior days’ cases causing infections on each day). The spreadsheet also includes calculations for Day 10 and 11.

We assume spread by a factor of 1.4 on each day. i.e. the total number of cases increases by 1.4 each day.

The means were calculated for 40 samples, and the overall mean and standard deviation calculated.

For each sample 16,108 cases were generated randomly in Excel.
This was later increased to over 20,000 cases.

For the 115 cases infected on Day 12, the number of new cases infected by each case was counted and the mean calculated.

Samples were randomly generated 40 times, providing a set of 40 sample means.
The main spreadsheet contained over 1.44 million non-empty cells.
This was later increased to over 1.8 million non-empty cells.

Once a case became infectious the infection rate diminished over the next 10 days at the same factor (P) each day. Once a value of P (a fraction) was chosen, daily factors were calculated so that over 10 days the sum of the daily factors was 1 and each day the factor decreased by the factor P.

The infection rate was not uniform each day since we understand that infection is highest on the first day and diminishes on successive days.

A scenario where the infection rate was the same each day (P=1), produced means over 13.7.

We build in a two- day incubation period (48 hours; any more made the numbers too big).

We explain the process using Case #290 (the first case generated on Day 12).

First we find (generate) who infected Case #290 (so that we can include results for Days 10 and 11). To do this we need to randomly choose a day number from 1 to 10 when Case #290 was infected. This is like throwing dice, we need to use a random number for this, generated so that it is equally likely to be any decimal number between 1 and 10.

We use this random number to choose a day using the scale factor as we have already discussed.

This infection day, numbered from 1 to 10, needs to be adjusted by adding on the difference between the current day number and 10. Case #290 was infected in Day 12 so we add 2 days to the number we have obtained already. The random day number from one to ten becomes relative to Day 12.

We use another random number to choose anyone infected on the chosen (adjusted) day to infect Case #290, using tables we have already created. This table was generated (once) starting with 10 cases on the first day (Day 1 has Case #1 to Case #10, and each day generating more to get 1.4 times as many on the next day, and repeating the process for each successive day. Day 2 would have 4 new Cases Case #11 to Case #14, Day 3 would have new Case #15 to Case#19 and so on.

Day 12 has new Case #290 to Case #404 (115 new cases).

The first case on Day 12 is Case #290. We record the case number of the person infecting #290 with the details for Case #290.

We then do the same for Case #291 and all the other cases for day 12, then for the other days.

At the end we have done this for over 16,100 (16,108) cases, Case #290 to Case #16398.

Once we have done this for all these cases (over 1,400 cases that may be infected by Case #290 over 10 days), we can count up the number infected by each case, starting with Case #290.

For Case #290, we can count how many other cases were infected by Case #290. During the process above anyone infected by case #290 had Case #290 recorded alongside their details. If Case #700, #987, and #11392 were infected by Case #290, they have case #290 (290) recorded alongside each of their case numbers so we only need to search for who has the number 290 next to their case number.

We can then do the same, for Case #291 and each of the other cases infected on Day 12.

We have now counted the number of people infected by Case #290 up to Case #404.

We then find the mean of all these numbers to get the average number of people infected in this sample.

This gives us a mean for Sample #1. We do exactly the same for Sample #3 up to Sample #40.

All the samples generate different results each time for Case#290 up to Case #404.

Each of these samples generates different results and hence we have different means for each sample.

Finally we find the overall mean and standard deviation for all of our (40) sample means.

This gives us the results for one execution (run) of the spreadsheet.

We can repeat the process over and over again using the same scale factor (fraction) as before or trying out different fractions. We prefer to use 5/7 since 7/5 = 1.4.

Chapter 4: Worst case scenarios

New Zealand currently has 22 deaths from COVID-19 (and 1504 cases) on June 12.

In a final version of a report dated 24 March commissioned by the NZ Ministry of Health, a worst case scenario estimated 27,600 deaths in New Zealand by early July. This report uses Ro = 3.5 in the scenario. See bottom of p4 in the (link to the) report in this post:
https://aaamazingphoenix.wordpress.com/2020/05/16/covid-19-nz-could-the-results-in-this-model-occur/

You may be able to get it directly below:
https://www.health.govt.nz/system/files/documents/publications/report_for_chief_science_advisor_-_health_-_24_march_final.pdf

The report includes this graph:

ModelReport

The peak appears to be on Day 104 = 13 June (104 Days from 1 March).

Using 5 July as an “early July” peak, the peak should instead be on Day 126.

Let’s look at the CFR (Case Fatality Risk/Ratio).

The report estimates that 0.83% of sick people die. i.e. CFR = 0.083.

The actual New Zealand CFR figure  is 22/1504 ~ 1.46% (22 Deaths from 1504 cases).

In the report, the estimated number of deaths in a worst case scenario is 27,600 deaths in early July.

However to adjust the length (number of days) to the New Zealand midpoint, we need to divide by 2 twice to correct the number of days and hence take the square root twice (of 27,600) and double the answer. This calculation is OK since the estimate of 12.9 deaths before doubling is greater than the actual number (1 death).

This gives us a midpoint on Day 31.5 and an estimated total of 25.8 deaths.

This is a good estimate for the actual number of deaths (22).

However the estimated number of deaths on Day 31.5 is 12.9, a factor of 10 from the real number (1).

Using the report CFR (0.0083), the number of cases on Day 31.5 would be 12.9/0.0083 ~ 1,550 cases, about double the actual number of cases (after the adjustment to day numbers below), at least close to the right ballpark.

Using the actual New Zealand CFR (.0146), the estimated number of cases for 12.9 deaths is 12.9/0.0146 ~ 880 cases, closer to the actual case figure and only a few days out (after the adjustment to day numbers below). The actual figure is 858 cases on 3 April (although New Zealand still only had one death on this date).

Since we are considering a worst case scenario we expect actual figures to lag behind the estimates.

Deaths usually lag behind cases by perhaps at least one cycle since obviously most deaths occur several days after the corresponding case is identified.

New Zealand had a total of 13 deaths on 21 April.

The model in the report uses Ro = 3.5 (i.e. no isolation).

However the low CFR in the report (0.0083 vs 0.0146), may (or may not) mean Re ~ 2 after adjusting for the actual CFR.

In the report model the CFR is assigned 0.83% but since the model assumes only 67% of infections lead to sickness (are the rest assumed asymptomatic?), we may need to add 50% and change the CFR to 1.25%. This may depend on whether a “case” in the CFR definition means an infected case (which may be asymptomatic) or a symptomatic case.

Day 31.5 assumes a starting date of 1 March.

The first death in New Zealand occurred on 28 February, two days before 1 March.

Therefore after adjusting Day 31. 5 to Day 33.5 to account for the actual start date, this is close to the actual mid-point.

The worst case scenario is close to the actual scenario when the length of time to the midpoint is adjusted to be close to the actual midpoint.

Sadly the actual model in the report estimates a total of 27,600 deaths, over 5,500 deaths per one million of population which is extremely unrealistic.

Below are the countries with the largest number of deaths per one million of population on 10 June:

Country, Total Total Tot Cases/ Deaths/ Population
Other Cases Deaths 1M pop 1M pop
San Marino 688 42 20,279 1,238 33,927
Belgium 59,437 9,619 5,130 830 11,586,764
Andorra 852 51 11,028 660 77,258
UK 289,140 40,883 4,260 602 67,865,632
Spain 289,046 27,136 6,182 580 46,753,788
Italy 235,561 34,043 3,896 563 60,466,601
Sweden 45,924 4,717 4,549 467 10,095,634

At the time of writing:

The USA has 345 deaths per one million of population.

China has 3 deaths per one million of population.

New Zealand has about 4.4 deaths per one million of population.

When we were estimating the number of deaths in New Zealand, we used Norway to estimate an upper limit. See:
https://aaamazingphoenix.wordpress.com/2020/04/06/covid-19-nz-how-many-cases-and-deaths-may-we-expect/

Below are today’s (10 June) statistics for Norway:

Country, Total Total Tot Cases/ Deaths/ Population
Other Cases Deaths 1M pop 1M pop
Norway 8,576 239 1,583 44 5,418,762

Norway currently has ten times as many deaths per one million of population as New  Zealand (New Zealand has about 4.4 deaths per one million of population) and over five times as many cases per one million of population (1,583 vs 301 cases) and less than 10% more population (5,418,762 vs 5,000,000).

We consider Norway’s figures represent a realistic worst case scenario for New Zealand.

The deaths in Norway per one million of population (44) are less than one percent (0.8%) of the deaths estimated in New Zealand in the report (5,520 = 27,600/5).

Perhaps a cumulative normal distribution is also useful in predictions and case estimates. It is easy to read estimates for the total number of cases from cumulative graphs.

The mean (midpoint) and standard deviation apply to the number of Days and the cumulative normal distribution is scaled to fit actual cumulative case numbers in March.

We can roughly fit a scaled normal curve (cumulative normal distribution) to the New Zealand Case Numbers (total = 1504):

NormalGraph

Below is the data table.

Date Day Normal 34.5,7.2 Normal 36.5,6.5 Actual Cases
28-Feb 1 0.00 0.00 1
29-Feb 2 0.00 0.00 1
1-Mar 3 0.01 0.00 1
2-Mar 4 0.02 0.00 1
3-Mar 5 0.03 0.00 1
4-Mar 6 0.06 0.00 3
5-Mar 7 0.10 0.01 3
6-Mar 8 0.18 0.02 4
7-Mar 9 0.30 0.03 5
8-Mar 10 0.50 0.07 5
9-Mar 11 0.83 0.13 5
10-Mar 12 1.34 0.24 5
11-Mar 13 2.12 0.44 5
12-Mar 14 3.32 0.79 5
13-Mar 15 5.09 1.39 5
14-Mar 16 7.66 2.37 6
15-Mar 17 11.34 3.98 8
16-Mar 18 16.49 6.52 8
17-Mar 19 23.56 10.46 12
18-Mar 20 33.10 16.41 20
19-Mar 21 45.72 25.20 28
20-Mar 22 62.07 37.87 39
21-Mar 23 82.88 55.73 52
22-Mar 24 108.85 80.28 66
23-Mar 25 140.64 113.28 102
24-Mar 26 178.81 156.57 155
25-Mar 27 223.77 212.05 205
26-Mar 28 275.72 281.49 283
27-Mar 29 334.59 366.36 368
28-Mar 30 400.04 467.69 451
29-Mar 31 471.42 585.83 514
30-Mar 32 547.78 720.37 589
31-Mar 33 627.90 869.99 647
1-Apr 34 710.37 1032.51 708
2-Apr 35 793.63 1204.92 797
3-Apr 36 876.10 1383.55 868
4-Apr 37 956.22 1564.29 950
5-Apr 38 1032.58 1742.92 1039
6-Apr 39 1103.96 1915.33 1106
7-Apr 40 1169.41 2077.85 1160
8-Apr 41 1228.28 2227.47 1210
9-Apr 42 1280.23 2362.01 1239
10-Apr 43 1325.19 2480.15 1283
11-Apr 44 1363.36 2581.48 1312
12-Apr 45 1395.15 2666.35 1330
13-Apr 46 1421.12 2735.79 1349
14-Apr 47 1441.93 2791.27 1366
15-Apr 48 1458.28 2834.56 1386
16-Apr 49 1470.90 2867.56 1401
17-Apr 50 1480.44 2892.11 1409
18-Apr 51 1487.51 2909.97 1422
19-Apr 52 1492.66 2922.64 1431
20-Apr 53 1496.34 2931.43 1440
21-Apr 54 1498.91 2937.38 1445
22-Apr 55 1500.68 2941.32 1448
23-Apr 56 1501.88 2943.86 1451
24-Apr 57 1502.66 2945.47 1456
25-Apr 58 1503.17 2946.45 1461
26-Apr 59 1503.50 2947.05 1469
27-Apr 60 1503.70 2947.40 1469
28-Apr 61 1503.82 2947.60 1472
29-Apr 62 1503.90 2947.71 1474
30-Apr 63 1503.94 2947.77 1479
1-May 64 1503.97 2947.81 1485
2-May 65 1503.98 2947.82 1487
3-May 66 1503.99 2947.83 1487
4-May 67 1504.00 2947.84 1486
5-May 68 1504.00 2947.84 1487
6-May 69 1504.00 2947.84 1486
7-May 70 1504.00 2947.84 1488
8-May 71 1504.00 2947.84 1489
9-May 72 1504.00 2947.84 1490
10-May 73 1504.00 2947.84 1492
11-May 74 1504.00 2947.84 1494
12-May 75 1504.00 2947.84 1497
13-May 76 1504.00 2947.84 1497
14-May 77 1504.00 2947.84 1497
15-May 78 1504.00 2947.84 1497
16-May 79 1504.00 2947.84 1498
17-May 80 1504.00 2947.84 1498
18-May 81 1504.00 2947.84 1499
19-May 82 1504.00 2947.84 1499
20-May 83 1504.00 2947.84 1503
21-May 84 1504.00 2947.84 1503
22-May 85 1504.00 2947.84 1503
23-May 86 1504.00 2947.84 1504
24-May 87 1504.00 2947.84 1504
25-May 88 1504.00 2947.84 1504
26-May 89 1504.00 2947.84 1504
27-May 90 1504.00 2947.84 1504

The standard deviations are chosen to fit the actual data for as much of March/April as possible.

Let’s suppose we would like to investigate the effect of a scale factor of 1.4 per day.

The last column in the table above shows a curve with a mean two days later (36.5).

The new scale factor is 1504 x 1.4^2 ~ 2948.

We chose this value simply to try out scaling of the cumulative normal distribution to estimate cumulative case values.

We saw above in Chapter 1 that we could have had 1,555 case numbers by Day 33 giving a total of over 3100 cases.

In the table below using a factor of 1.4 we see that we could have had the same number of cases by Day 33.

Below is the graph (red curve) with a standard deviation chosen to roughly fit early actual case number values.

We also look at values to give a total close to 3008 (=1504 x 2) cases.

NormalGraphs

We also look at values to give a total close to 3008 (=1504 x 2) cases.

Total Cases 1504 2947.84 3008 1504
Date Day Normal 34.5,7.2 Normal 36.5,6.5 Normal 36.5,6.5 Actual Cases
28-Feb 1 0.00 0.00 0.00 1
29-Feb 2 0.00 0.00 0.00 1
1-Mar 3 0.01 0.00 0.00 1
2-Mar 4 0.02 0.00 0.00 1
3-Mar 5 0.03 0.00 0.00 1
4-Mar 6 0.06 0.00 0.00 3
5-Mar 7 0.10 0.01 0.01 3
6-Mar 8 0.18 0.02 0.02 4
7-Mar 9 0.30 0.03 0.04 5
8-Mar 10 0.50 0.07 0.07 5
9-Mar 11 0.83 0.13 0.13 5
10-Mar 12 1.34 0.24 0.25 5
11-Mar 13 2.12 0.44 0.45 5
12-Mar 14 3.32 0.79 0.81 5
13-Mar 15 5.09 1.39 1.41 5
14-Mar 16 7.66 2.37 2.42 6
15-Mar 17 11.34 3.98 4.06 8
16-Mar 18 16.49 6.52 6.66 8
17-Mar 19 23.56 10.46 10.67 12
18-Mar 20 33.10 16.41 16.75 20
19-Mar 21 45.72 25.20 25.71 28
20-Mar 22 62.07 37.87 38.65 39
21-Mar 23 82.88 55.73 56.86 52
22-Mar 24 108.85 80.28 81.92 66
23-Mar 25 140.64 113.28 115.59 102
24-Mar 26 178.81 156.57 159.77 155
25-Mar 27 223.77 212.05 216.38 205
26-Mar 28 275.72 281.49 287.23 283
27-Mar 29 334.59 366.36 373.84 368
28-Mar 30 400.04 467.69 477.24 451
29-Mar 31 471.42 585.83 597.79 514
30-Mar 32 547.78 720.37 735.07 589
31-Mar 33 627.90 869.99 887.75 647
1-Apr 34 710.37 1032.51 1053.59 708
2-Apr 35 793.63 1204.92 1229.51 797
3-Apr 36 876.10 1383.55 1411.78 868
4-Apr 37 956.22 1564.29 1596.22 950

For comparison for Days 26 to 28 we reproduce part of the table below which uses the daily factor 1.4 starting with 10 cases on Day 18 (actual values are in the middle column):

24/03/20 26 155 147.5789
25/03/20 27 205 206.6105
26/03/20 28 283 289.2547

In the table below, we also look at what happens when we change the standard deviation to 6.55 (for 2947.84 column only) and also when we increase the total cases to 3110.

Total Cases 1504 2947.84 3110 1504
Date Day Normal 34.5,7.2 Normal 36.5,6.55 Normal 36.5,6.5 Actual Cases
28-Feb 1 0.00 0.00 0.00 1
29-Feb 2 0.00 0.00 0.00 1
1-Mar 3 0.01 0.00 0.00 1
2-Mar 4 0.02 0.00 0.00 1
3-Mar 5 0.03 0.00 0.00 1
4-Mar 6 0.06 0.00 0.00 3
5-Mar 7 0.10 0.01 0.01 3
6-Mar 8 0.18 0.02 0.02 4
7-Mar 9 0.30 0.04 0.04 5
8-Mar 10 0.50 0.08 0.07 5
9-Mar 11 0.83 0.15 0.14 5
10-Mar 12 1.34 0.27 0.25 5
11-Mar 13 2.12 0.49 0.47 5
12-Mar 14 3.32 0.87 0.84 5
13-Mar 15 5.09 1.52 1.46 5
14-Mar 16 7.66 2.58 2.51 6
15-Mar 17 11.34 4.29 4.20 8
16-Mar 18 16.49 6.98 6.88 8
17-Mar 19 23.56 11.12 11.03 12
18-Mar 20 33.10 17.34 17.31 20
19-Mar 21 45.72 26.47 26.59 28
20-Mar 22 62.07 39.57 39.96 39
21-Mar 23 82.88 57.92 58.79 52
22-Mar 24 108.85 83.04 84.70 66
23-Mar 25 140.64 116.64 119.51 102
24-Mar 26 178.81 160.54 165.18 155
25-Mar 27 223.77 216.60 223.71 205
26-Mar 28 275.72 286.51 296.97 283
27-Mar 29 334.59 371.71 386.52 368
28-Mar 30 400.04 473.16 493.42 451
29-Mar 31 471.42 591.16 618.06 514
30-Mar 32 547.78 725.27 760.00 589
31-Mar 33 627.90 874.18 917.85 647
1-Apr 34 710.37 1035.72 1089.31 708
2-Apr 35 793.63 1206.94 1271.20 797
3-Apr 36 876.10 1384.23 1459.65 868
4-Apr 37 956.22 1563.61 1650.35 950

We also used Goal Seek in Excel (Data tab, What-If-Analysis).

Using Goal Seek and least squares up to Day 28 we found the best fit was obtained using a standard deviation of 6.68 days.

NormalGraph6

We also obtained this graph using the same mean and standard deviation:

NormalGraphDaily

Note that this is not the actual mean and standard deviation.

Since the curve (for the New Zealand actual case data) is not close to being a normal curve, the actual mean and standard deviation are not useful for estimating case numbers. The normal curve does not represent the high values well.

The real mean and standard deviation are shown in the graph below:

CasesNormalActual

Here are both graphs showing the normal curves side-by-side:

Later we also considered a standard deviation of 6.5.

The actual daily cases up to Day 37 are in the table below. A more complete table up to 27 May has already been listed above earlier in this section.

We look further at what may have happened if the original daily factor of 1.4 (starting with 10 cases on Day 18) had continued.

In the table below all columns except the factor 1.4 column have 1504 cases total (in the complete table).

Date Day Normal 34.5,6.68 Normal 34.5,7.2 Factor 1.4 Actual Cases
15-Mar 17 6.62 11.34 8
16-Mar 18 10.17 16.49 10.00 8
17-Mar 19 15.29 23.56 14.00 12
18-Mar 20 22.54 33.10 19.60 20
19-Mar 21 32.57 45.72 27.44 28
20-Mar 22 46.13 62.07 38.42 39
21-Mar 23 64.06 82.88 53.78 52
22-Mar 24 87.25 108.85 75.30 66
23-Mar 25 116.58 140.64 105.41 102
24-Mar 26 152.85 178.81 147.58 155
25-Mar 27 196.72 223.77 206.61 205
26-Mar 28 248.60 275.72 289.25 283
27-Mar 29 308.59 334.59 404.96 368
28-Mar 30 376.44 400.04 566.94 451
29-Mar 31 451.47 471.42 793.71 514
30-Mar 32 532.60 547.78 1,111.20 589
31-Mar 33 618.41 627.90 1,555.68 647
1-Apr 34 707.14 710.37 1,053.59 708
2-Apr 35 796.86 793.63 1,229.51 797
3-Apr 36 885.59 876.10 1,411.78 868
4-Apr 37 971.40 956.22 1,596.22 950

For simplicity, in the table below we double the daily estimate (see last column) to get an estimated total number of cases. i.e. We assume the midpoint may be at each date.

Remember that the last two columns almost double (1.4^2 = 1.96) every two days.

If isolation (including self-isolation) did not already have an effect on slowing the rate (factor 1.4; see actual new cases after 26 March), we could have expected 2222 cases (30 March estimate total) since new cases on 30 March would are likely to have been infected before Lockdown Level 4 started at 11.59pm on 25 March.

It is likely there could have been a total of 3,000 to 6,000 cases even with Lockdown Level 4 starting at 11.59pm on 25 March.

New cases could easily have been infected at least five to eight days earlier (or more with a cycle length greater than five Days or an infection period two cycles long).

If Level 4 Lockdown had been delayed, we can look at the figures at least five to eight days later to see how this may have affected the total number of cases.

In the table below we double the daily estimate (see last column) to get an estimated total number of cases. i.e. We assume the midpoint may be at each date.

The table below also has the data for our worst case scenarios.

Note: SQRT(2) ~ 1.414213562

Date Day# (D) Actual Cases a = SQRT(2) a = 1.4 Estimated Total
14-Mar 16 6 5.00
15-Mar 17 8 7.07
16-Mar 18 8 10.00 10.00
17-Mar 19 12 14.14 14.00
18-Mar 20 20 20.00 19.60
19-Mar 21 28 28.28 27.44
20-Mar 22 39 40.00 38.42
21-Mar 23 52 56.57 53.78
22-Mar 24 66 80.00 75.30
23-Mar 25 102 113.14 105.41
24-Mar 26 155 160.00 147.58
25-Mar 27 205 226.27 206.61
26-Mar 28 283 320.00 289.25
27-Mar 29 368 452.55 404.96
28-Mar 30 451 640.00 566.94
29-Mar 31 514 905.10 793.71 1587.429547
30-Mar 32 589 1280.00 1111.20 2222.401365
31-Mar 33 647 1810.19 1555.68 3111.361911
1-Apr 34 708 2560.00 2177.95 4355.906676
2-Apr 35 797 3620.39 3049.13 6098.269346
3-Apr 36 868 5120.00 4268.79 8537.577084
4-Apr 37 950 7240.77 5976.30 11952.60792
5-Apr 38 1039 10240.00 8366.83 16733.65109
6-Apr 39 1106 14481.55 11713.56 23427.11152
7-Apr 40 1160 20480.00 16398.98 32797.95613
8-Apr 41 1210 28963.09 22958.57 45917.13858
9-Apr 42 1239 40960.00 32142.00 64283.99401


Now we look at a normal curve which has a total of 27,600 cases (not deaths!).

Normal2700

The curve stays above the actual number of cases which means the cumulative number of cases in the normal curve is also above the actual cumulative number of cases.

At the peak in the above graph there are over 600 (629) cases per day.

Clearly even the number of cases in the normal curve is huge compared to the actual case numbers and extremely unlikely; let alone the same number of deaths.

If 27,600 cases did occur in New Zealand before early in July, the estimated number of resulting deaths using the report CFR of 0.083 would be

27,600 x 0.083 = 2,290.80 deaths.

This would mean almost 460 deaths per one million people in New Zealand.

Only eight countries in the world have 450 or more deaths per one million of population.

Therefore even 2,290.80 deaths (450 per million) is very unlikely in New Zealand even in a worst case scenario.

However since at least 450 deaths per million have occurred in eight countries we need to accept that this number is possible in a worst case scenario.

The report estimated 27,600 deaths in New Zealand, which is 5,520 deaths per one million of population. We saw above that the highest number of deaths per one million of population in the world is still less than 1,300 (1,238) deaths, less than one quarter of 5,520 deaths.

New Zealand’s total number of cases (1,504 cases) is less than the 2,290.80 death estimate with no active cases and no new cases for the last 20 days on 12 June.

Even the USA with over 2,000,000 cases and over 115,000 deaths has 353 deaths per one million of population, much less than 460.

The peak in the graph above is at the end of April.

The report mentioned a peak early in July.

Here is the normal curve in the report again:

ModelReport

We place the two curves side by side:

Most of the March data below was available before the final report was submitted.

Date Day Normal Curve Actual Daily Cases
28-Feb 1 1.07 1.00
29-Feb 2 1.31 0.00
1-Mar 3 1.60 0.00
2-Mar 4 1.94 0.00
3-Mar 5 2.36 0.00
4-Mar 6 2.85 2.00
5-Mar 7 3.43 0.00
6-Mar 8 4.12 1.00
7-Mar 9 4.93 1.00
8-Mar 10 5.88 0.00
9-Mar 11 6.99 0.00
10-Mar 12 8.28 0.00
11-Mar 13 9.78 0.00
12-Mar 14 11.52 0.00
13-Mar 15 13.52 0.00
14-Mar 16 15.81 1.00
15-Mar 17 18.43 2.00
16-Mar 18 21.42 0.00
17-Mar 19 24.81 4.00
18-Mar 20 28.65 8.00
19-Mar 21 32.97 8.00
20-Mar 22 37.81 11.00
21-Mar 23 43.23 13.00
22-Mar 24 49.26 14.00
23-Mar 25 55.95 36.00
24-Mar 26 63.34 53.00
25-Mar 27 71.47 50.00
26-Mar 28 80.39 78.00
27-Mar 29 90.12 85.00
28-Mar 30 100.70 83.00
29-Mar 31 112.16 63.00
30-Mar 32 124.52 75.00
31-Mar 33 137.78 58.00
1-Apr 34 151.96 61.00
2-Apr 35 167.05 89.00
3-Apr 36 183.05 71.00
4-Apr 37 199.92 82.00
5-Apr 38 217.63 89.00

Below is the cumulative normal distribution graph:

Normal27000Cum2

We are now ready to produce our own worst case scenario.

The data for the graph below (up to Day 39 = 6 April) is two tables above- the table with these words above it:
Note: SQRT(2) ~ 1.414213562

We consider two daily factors a = 1.4 and a = SQRT(2).

These factors correspond to five day (cycle) factors of

Re = 5.37824 and
Ro = 5.656854249

We obtain the graph below:

ChartWorst

Note:

Day 1 is 28 February (first NZ case).

Day 27 is 25 March (Lockdown Level 4).

Day 35 is 2 April.

Day 65 is 2 May.

Below is the Daily Graph showing the estimated daily number of new cases.

Note that the mean (midpoint) differs by one half-day.

Could New Zealand have coped with 3350 to 4250 new cases daily around Day 39 (6 April)?

Even our previous normal curve had over 600 new cases for several days near the peak.

The actual number of cases was  89 the previous day.

Compare this with the actual number of cases on the graph (maximum 89 cases).

The worst case scenarios estimated 23,427.11 to 24,721.55 cases with over 23,000 cases in two worst case scenarios by 16 April (Day 49). Recall that one report commissioned by the NZ Ministry of Health estimated more deaths than this (27,600 deaths) by early July.

The graph below shows the number of daily cases for the above worst case scenarios.

ChartWorstDaily

Note: Peak for blue curve is not shown (the half-day value is not plotted at the midpoint). We use the values plotted (3347). The actual value for the blue curve at the midpoint (39.5 days) is 3960.

We will use the midpoint new case values (3,350 – 4,250 cases) as an estimate for the total number of deaths. See table below.

The midpoints are at Day 39 and Day 39.5 (near Day 40) in the graph above.
Recall that for actual case values we calculated the mean (midpoint) at 35 days and a standard deviation of 9.155 days in Excel.

So long as the the Case Fatality Risk/ Ratio (CFR), the number of cases per million (CpM), and deaths per million (DpM) are realistic, the total number of deaths can be up to at least three times the maximum daily number of new cases in a worst case scenario. See the USA graph below.

We consider if our daily maximum number of new cases may be acceptable to be an estimate for the number of deaths.

Look at the table below comparing NZ worst case scenarios with statistics for Belgium.

We will assume the curves are symmetrical about the midpoints and adopt the original totals of 23,400 and 24,700.

Our own worst case scenarios estimated 23,427.11 to 24,721.55 cases with over 23,000 cases in two worst case scenarios by 16 April (Day 49). Recall that one report in the last Chapter estimated more deaths than this (27,600 deaths) by early July.

We compare our worst case scenarios with the actual numbers for Belgium
(14 June: Belgium population 11.6 million; NZ population 5 million):

NZ CpM 4685.42 4944.31
Belgium CpM 5181.00 5181.00
NZ Daily max Cases/ Estimated Deaths 3346.73 4241.55 CFR
NZ DpM 669.35 848.31 0.171572875
Belgium Dpm 833.00 833.00 0.160779772

We consider the CpM and DpM acceptable in our scenario.

The worst case NZ CFR (last column) is only a little higher than the Belgium CFR (see last column) so we deem it acceptable.

We note that the NZ C FR estimate is over 11.7 times the actual CFR (0.01462766).

We accept a worst case scenario of 24,700 cases and 4250 (4241.55) as an estimate for the total number of deaths.

Below is a graph of daily case number for Belgium:

Belgium

Source:
https://www.worldometers.info/coronavirus/country/belgium/

The USA graph below demonstrates that the number of cases can take many times the number of days it took to reach the peak to go down to zero.

We now consider a slow drop after the midpoint. i.e. the graph is not symmetrical.

This will increase the number of cases.

The limit to the number of cases depends on what may be acceptable for the number of cases per million (CpM).

We will look at how large the increase may be later.

For now, read the discussion below but consider at this stage that the limit for any increase may be small. i.e. We may already be close to the limit already.

The values in the above table in the symmetrical case may be very close to the limit for a realistic number of cases (the CpM in the table may not allow for much of an increase).

The discussion below may only have theoretical value as far as case numbers in New Zealand are concerned and little if any practical application for case numbers in New Zealand.

We leave the discussion here since it may have a practical application outside New Zealand even if not for New Zealand.

This method may be useful for USA cases.

In round figure let’s suppose the  peak is at Day 40 (actually Day 39 and Day 39.5) and it takes twice as long (80 days) for the number of cases to go down to zero.

Let’s consider the downward slope a triangle (on average).

On Day 40 the lower value has 3350 cases, with a total of about 15,000 (15,060) cases up to Day 40.

We first look at a rectangle 3350 cases high an 80 (2 x 40) days long.

For the rectangle 3350 x 80 = 268,000 cases.

The triangle has half this number, 134,000 cases.

Adding on the 15,000 cases in the first half (up to Day 40) we get a total of 149,000 cases in this scenario.

This shows that 150,000 cases (instead of 23,400 if the curve is symmetrical) is quite possible 120 (=40+80) Days from 28 February (27 June) since we have only considered the scenario with the lower number of cases.

In reality the curve may take far more than double the time to come down.

In a worst case scenario, a figure of 150,000 cases may be conservative.

Instead of 23,400 to 24,700 cases, 150,000 cases or more may be likely.

It is difficult to work out the midpoint even when the outbreak appears over.

New Zealand’s CFR is

22/1504 = 0.01462766.

Using the 24,721 figure above, we may have expected

24,721 x 0.01462766 = 362 deaths, 340 more deaths or over 16 times our present number (22).

With 150,000 cases, this becomes

150,000 x 0.01462766 = 2,194 deaths, over 6 times as many.

Even 362 deaths is 72 deaths per million of population.

In a worst case scenario, New Zealand could easily have over 2,200 deaths.

Our daily case estimate range at the midpoint (3350 to 4250) in a worst case scenario could translate to a (total) death estimate (3350 to 4250) if the curves are not symmetrical. i.e. As with the USA, the curve takes much more than twice as long to come down to zero.

In the USA graph below, the maximum number of daily cases is 39,072 on April 24, yet the total number of deaths is three times this figure on 14 June (117,527).

For lower values for NZ consider again our discussion above using triangular shape for after the midpoint.

Note that how far we can take this discussion depends on what is considered acceptable for the CpM.

If it takes five times longer to come down, the death estimate in New Zealand is almost 4,150 using 3,350 cases and clearly more if we use 4,250.

Fortunately we only have a total of 22 deaths (4.4 per million), 1504 cases (none active), and no new cases for over 20 days.

In our worst case scenarios, New Zealand could have had 3350 to 4250 new cases daily around the midpoint (Day 39 = 6 April). See the graphs above.

Even the lower figure (3350) near the daily peak is more than double our total number of cases (1504).

Would it have been easy for New Zealand to cope with near this many new cases on each day for a number of days?

The actual number of cases in New Zealand was  89 the previous day.

Fortunately the graphs rise and fall sharply but a sharp fall may not happen in reality.

Consider the slow decline in cases in the USA:

UScases

Source: https://www.worldometers.info/coronavirus/country/us/

For using linear regression to create a straight line for the top of a triangle, see:
https://aaamazingphoenix.wordpress.com/2020/05/09/covid-19-usa-could-the-total-number-of-confirmed-cases-exceed-3-million/

Once the line is created, all we need to do is change the constant (c) so that it matches the actual value (add on the difference to c in the equation y=mx+c).

We can then find the x-axis intercept to estimate the number of days it takes to get down to zero cases.

Finally we can use the rectangle/triangle method above to estimate the total number of cases.

You may like to try this out using USA figures.

Chapter 5: Results

Both Scenario A and Scenario B generate the same estimates for the daily total number of cases.

We are confident in our models since the estimates (exactly the same in both scenarios) closely fit the actual data.

Our approach has been mathematical.

Scenario A and Scenario B both give the same estimated values for the total number of cases.

In Chapter 2 and Chapter 3, we used Excel to simulate an outbreak. We generate over 16,100 cases for each simulation, and 40 simulations for each dataset, calculating the mean number of cases infected by each person in each simulation and dataset. We prefer to use P=5/7 [~0.7143] since 7/5 = 1.4. We will also consider P=1/SQRT(2) [~0.7071].

We found that Re>3.2 (usually a mean over 3.2 was achieved with P=0.5; and mean over 4 with P=0.6). Re>5.5 was usually achieved with P=0.7 or less, where each day for 10 days the infectivity for each case reduces by the same (scaled) factor close to P. Recall that P is scaled so that the sum of the daily factors sum to 1. Recall also that where a mean is mentioned, it is an average of 40 sample means.

We note that with P=0.5, on the sixth day the probability of infection is only 0.01564, halving each successive day, so essentially a two-week cycle is in reality just practically one week.

In our discussions, the reader will need to decide where it may be appropriate to replace Ro by Re and vice versa.

We started with estimates for Re of 2.9 in Scenario B and 5.3782 in Scenario A. Even Re = 2.9 is greater than most experts expect.

The spread of COVID-19 in New Zealand is worse than in the above animation (which uses Re = 2.6).

Both estimates for Re and Ro are within the values in the above graph in the article by Sanche et al.

By contact-tracing we hope to reduce the spread to a one-cycle situation.
i.e. With an infectious period two cycles long, we anticipate/ hope that contact tracing will have identified and isolated potentially infected people before the second cycle starts.

We conclude that Re = 5.3782 is possible.

In the abstract for the article by Sanche et al. above  we see that “Assuming a serial interval of 6–9 days, we calculated a median R0 value of 5.7 (95% CI 3.8–8.9).”

We conclude that Ro = 5.65685 is also possible in Scenario A. So is a range of values for Ro from 5.67 to 6.27. in Scenario B.

We note that an infected person may become infectious two days before symptoms appear.

With a daily factor of 1.4, the values almost double every two days (1.4^2 = 1.96). This can significantly change estimates for the total number of cases even if the midpoint date estimate is only one or two days later.

Our daily case estimate range at the midpoint (3350 to 4250) in a worst case scenario could translate to a (total) death estimate (3350 to 4250) if the curves are symmetrical or not. i.e. As with the USA, the curve takes  more than four times as long to come down to zero.

The number of cases at the midpoint could be less than the total number of deaths in a worst case scenario.

In the previous section we saw that in a worst case scenario 24,700 cases were possible in New Zealand with 4,250 deaths.

It is very easy to get it wrong when looking at live updates on each day as the outbreak progresses.

However near the midpoint, we can add two consecutive actual days’ case numbers together or double the case number for single days to get an estimate of the total number of cases.

We may need to add on 10% to allow for a very long tail.

We may have ended up with two or four times the actual total number of cases.

It is likely there could have been a total of 3,000 to 6,000 cases even with Lockdown Level 4 starting at 11.59pm on 25 March. New cases could easily have been infected at least five to eight days earlier (or more with a cycle length greater than five Days or an infection period two cycles long).

New Zealand should consider itself very lucky.

On June 11, New Zealand had not had any new cases for 20 days and had no active cases.

Sadly “Six rest homes had Covid-19 cases, two of which — Rosewood Rest Home in Christchurch and CHT St Margaret’s in Auckland’s Te Atatu — account for 16 of New Zealand’s 22 Covid-linked deaths.”

DeathsRestHomes

Source:
New Zealand Herald 12 June pA2:
“Delays allowed Covid-19 to spread in rest homes, review finds”

New Zealand had 16 significant clusters including the six rest homes above.
It is easy to see that from these clusters the number of deaths in New Zealand could have been double the actual number (22) of deaths from COVID-19  without any extra cases.

New Zealand has 4 deaths (4.4) per one million of population and is ranked #132 (from the top for deaths per one million of population) of the 215 countries etc listed on this website: https://www.worldometers.info/coronavirus/

On 12 June, about 36% have 3 deaths or less per one million of population, 60% have 7 or more, 40% have 15 or more, and 30% have 28 or more.

We leave it to epidemiologists to consider which model best suits the spread of COVID-19.

Chapter 3 shows how even expert predictions may be extremely unlikely (impossible?) in reality. Could New Zealand ever have had 27,600 deaths by early July?

It is easy to have 20/20 hindsight vision.

The spreadsheet has demonstrated that Re>5.3 is also possible in a two cycle scenario. Consider also the maximum and minimum mean values below. For Ro we may need to add three standard deviations to the mean.

For P=1/SQRT(2) we have, for example:

Cases: 59 83 115 257
Day Day Day Day
10 11 12 10-12
Mean= 5.67161 5.659639 5.732609 5.695039
Std Dev= 0.280813 0.219948 0.202852 0.110989
Max= 6.186441 6.084337 6.104348 5.996109
Min= 4.915254 5.120482 5.33913 5.470817
P= 0.707107 0.707107 0.707107 0.707107

Each column uses 40 sample means. Each of the sample means are the mean of the number of new cases (from the number of cases at the top of each column).

For P=1/1.4:

Cases: 59 83 115 257
Day Day Day Day 10-12
10 11 12 99
Mean= 5.722034 5.800602 5.820652 5.791537
Std Dev= 0.340042 0.260213 0.21057 0.123064
Max= 6.457627 6.445783 6.304348 6.062257
Min= 5.016949 5.204819 5.417391 5.509728
P= 0.714286 0.714286 0.714286 0.714286

We note that for P=1/2 we have an infectivity “half-life” of one day and for P=1/SQRT(2) the half-life is two days.

Below is the rest of the table above (using P=1/1.4):

Sample# Mean Mean Mean Mean
1 5.694915 6.228916 5.913043 5.964981
2 5.355932 5.349398 5.704348 5.509728
3 5.745763 5.939759 5.878261 5.867704
4 5.864407 5.891566 5.730435 5.81323
5 6.389831 5.759036 5.495652 5.785992
6 5.830508 5.614458 5.852174 5.770428
7 5.694915 5.86747 5.791304 5.793774
8 5.491525 5.759036 6.304348 5.941634
9 5.389831 5.578313 5.713043 5.595331
10 5.440678 6 6.104348 5.918288
11 5.542373 5.879518 5.913043 5.817121
12 6.084746 5.566265 5.913043 5.840467
13 5.745763 5.891566 5.930435 5.875486
14 6.135593 5.783133 5.826087 5.883268
15 5.677966 6.445783 5.982609 6.062257
16 5.525424 5.373494 6.104348 5.735409
17 6.016949 5.879518 5.730435 5.844358
18 5.847458 5.204819 5.721739 5.583658
19 6.457627 5.86747 5.556522 5.863813
20 5.813559 5.855422 5.747826 5.797665
21 5.101695 6.228916 5.53913 5.661479
22 5.864407 5.819277 5.86087 5.848249
23 5.915254 5.506024 6 5.821012
24 5.254237 6.060241 5.721739 5.723735
25 6.016949 5.927711 5.713043 5.85214
26 5.644068 6 6.113043 5.968872
27 5.813559 5.60241 5.426087 5.571984
28 5.559322 5.759036 5.93913 5.793774
29 5.050847 5.951807 5.869565 5.708171
30 5.474576 5.710843 5.713043 5.657588
31 6.186441 5.73494 5.695652 5.821012
32 5.711864 5.578313 6.017391 5.805447
33 6.101695 5.686747 5.417391 5.661479
34 6.033898 5.457831 5.426087 5.575875
35 5.254237 6.204819 5.817391 5.81323
36 5.542373 6.048193 5.913043 5.871595
37 5.864407 5.638554 5.556522 5.653696
38 5.016949 5.975904 6.017391 5.774319
39 6.033898 5.46988 6.13913 5.898833
40 5.694915 5.927711 6.017391 5.914397

Below are the scale factors for Scenario B (10-day; two cycle infection) for various values of P:

Scales: 1/SQRT(2) 7/10 5/7 3/4
1 0.302341 0.308721 0.295946 0.264918
2 0.213788 0.216104 0.21139 0.198689
3 0.151171 0.151273 0.150993 0.149017
4 0.106894 0.105891 0.107852 0.111762
5 0.075585 0.074124 0.077037 0.083822
6 0.053447 0.051887 0.055026 0.062866
7 0.037793 0.036321 0.039305 0.04715
8 0.026723 0.025424 0.028075 0.035362
9 0.018896 0.017797 0.020053 0.026522
10 0.013362 0.012458 0.014324 0.019891
Sum 1 1 1 1

Note that the sum of the weights is equal to 1.

Below is the graph of the scale factors.

ScaleFactors

If the Scale factor (P) is chosen so that it is a multiple of the inverse of the Daily Factor (F), then the scale factor for each day is constant (using a daily Factor of F=1.4, we can choose P = 1/1.4 and then scale it so that as above the weights sum to 1).

In the table below, the Scaled column is the value for Day 1 in the table above the graph.

The other 9 days end up with the same value.

We multiply this by 10, then as in Scenario B (Chapter 2), we multiply the result by the square of the Factor to get an estimate of the value of Ro (see last row in the table below).

Factor 1.428571 1.4 1.414214
Factor 10/7 7/5 SQRT(2)
Factor Squared 100/49 49/25 2
Factor Inverse 7/10 5/7 SQRT(2)/2
Factor Inverse 0.7 0.714286 0.707107
Factor Scaled 0.308721 0.295946 0.302341
Scaled x 10 x Factor Squared 6.30042 5.800534 6.046828

Let x = Factor
e.g. x = SQRT(2)

Then
FactorScaled = x^9 (x – 1)/(x^10 – 1)
and
Ro = 10x^2 * x^9 (x – 1)/(x^10 – 1) or Ro = 10x^2 * [FactorScaled] or
Ro = 10x^11 * (x – 1)/(x^10 – 1)

For the second row (1.4), we see that 5.8005 is close the mean values in the previous table:

Cases: 59 83 115 257
Day Day Day Day
10 11 12 10-12
Mean= 5.722034 5.800602 5.820652 5.791537

For SQRT(2), P = 1/SQRT(2), a run produced:

Cases: 66 94 132 292
Day Day Day Day
10-12
10 11 12 99
Mean= 6.083333 6.000798 6.032765 6.033904
Std Dev= 0.306238 0.296 0.182605 0.095343
Max= 6.757576 6.606383 6.333333 6.246575
Min= 5.606061 5.457447 5.598485 5.859589
P= 0.707107 0.707107 0.707107 0.707107


Since we have generated distributions rather than sampling from a distribution it is doubtful whether the Central Limit Theorem applies.

Regardless all we wished to do was show that Re around 5.4 or greater was possible in a worst case two cycle infection timeframe. The values in the tables above achieve this.

We conclude that Re = 5.3782 or Re (or Ro) = 5.65685 is possible in a one or two cycle infection timeframe.

We also see that values for Ro over 6 are possible and are in fact likely since the factor 1.4 reflects the actual increase in New Zealand where cases are isolated once they have been identified.

We find it interesting in the bottom row, that the Daily Factor = SQRT(2), with an estimate for Ro of 6.0468, means that the number of infected people doubles every two days.

We have used an incubation period of only two days (we miss out a day) for most of our modelling. This created a good fit for the actual data.

In reality we assume the incubation period is one cycle long. This means the value of Ro is likely to be greater than we have estimated.

We have seen that R0=6.3 or Ro=6.4 is also possible.

Our values for Re and Ro are also twice as big as most experts consider for COVID-19.
This would appear to make Sweden’s attempt at “herd immunity” doomed to failure. A

new study, published in the Emerging Infectious Diseases journal, shifts the R0 for COVID-19 from about 2.2 to about 5.7. With the lower number, only 55% of a population needs to be immune from COVID-19 to stop its spread through herd immunity. Herd immunity refers to enough of a population being immune to a disease that the disease cannot travel through it. …
But if more people get infected from a single person with COVID-19, then more people need to be protected from the disease to stop it from continuing to spread. With an R0 of 5.7, approximately 82% of the population needs to be immune to reach herd immunity and stop the disease from spreading easily through the population, the researchers concluded.
See:
https://www.forbes.com/sites/tarahaelle/2020/04/07/the-covid19-coronavirus-disease-may-be-twice-as-contagious-as-we-thought/#103db97329a6

In January this year the World Health Organization (WHO) estimated the R number of the COVID-19 virus to be between 1.4 and 2.5, whereas researchers in China, Germany and Sweden estimated it to be at least as transmissible as SARS, and calculated it to be 3.28. In China, other researchers have found the R number to be much higher, putting it at 4.7 – 6.6.
See:
https://www.gavi.org/vaccineswork/what-covid-19s-r-number-and-why-does-it-matter

At some point early in the outbreak, some cases generated human-to-human transmission chains that seeded the subsequent community outbreak prior to the implementation of the comprehensive control measures that were rolled out in Wuhan. The dynamics likely approximated mass action and radiated from Wuhan to other parts of Hubei province and China, which explains a relatively high R0 of 2-2.5.
See:
https://www.who.int/docs/default-source/coronaviruse/who-china-joint-mission-on-covid-19-final-report.pdf

Chapter 6: Conclusions

[We will be consulting with colleagues soon and these conclusions may be updated]

We have used an incubation period of only two days in most of our modelling. This created a good fit for the actual data.

The incubation period was created by missing out one day before having transmission occur. Since transmission may occur in reality two days before an infected person becomes symptomatic, this may simulate an incubation period of two extra days.

In reality we assume the incubation period is one cycle long. This means the value of Ro is likely to be greater than we have estimated.

Our Excel simulations use mean values. For better estimates for Re and Ro, perhaps we need to add two or three standard deviations to the means.

We decide to stay with the mean values since these reflect our calculated values.

We are not sure whether our Excel simulations may be called a stochastic process.

We hypothesised that Re = 5.3782 or Re (or Ro) = 5.65685 is possible in a one or two cycle infection timeframe.

We conclude that Re is at least this amount.

Let r be the daily infection rate.

A person becomes less infectious over successive days. A person is most infectious on the first day they can transmit COVID-19.

For any value of r, we can use c/r (chose an appropriate value for the constant c) for the reduction in infectivity (is this the right word?) over successive days.

For r = 1.4, we can choose c/1.4 and for r = SQRT(2), choose c/SQRT(2). This means the daily rate and the reduction in infectivity create a constant value once the incubation period is over.

The reduction in infectivity is inversely proportional to the daily rate in our Excel simulations.

We saw that r = 1.4 fits New Zealand data well until after Lockdown level 4.

We also saw that r = 1.4 leads to Re just over 5.8 when the infectious period (the time when a person can transmit COVID-19) is two cycles long. In reality the infectious period is at least this long.

We adopt Re = 5.8.

Since 1.4 x 1.4 = 1.96, we see that the number of cases almost doubles every two days with r=1.4.

Since SQRT(2) x SQRT(2) = 2, we consider r = SQRT(2) [ = 1.414214] a suitable value for r when estimating Ro. This means that the number of cases will double every two days.

With r = 1.414214 [=SQRT(2)], we obtain a value for Ro just over 6.

Clearly since in practice we have r = 1.4 (with isolation occurring once new cases are identified), for Ro we require r > 1.4.

Hence we consider r = 1.414214 [=SQRT(2)] a suitable value for r when estimating Ro, and we conclude that Ro is at least 6.

We adopt Ro = 6 in New Zealand and hence worldwide. Once we add on three standard deviations for all of Days 10-12 for r=SQRT(2), we commonly get close to or over 6.4.

All the values for Re and Ro are more than twice as big (2.3 times for Ro = 6) as the value used in the animation (2.6).

Here is the animation again:

spread

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

When we looked at calculations for this outspread, we used a 6-day cycle. See:
https://aaamazingphoenix.wordpress.com/2020/05/18/covid-19-nz-can-one-person-infect-2-6-others/

In our New Zealand simulations, we start with 10 cases and use a 5-day cycle length. With Ro = 6, in three cycles (15 days) we have 10, 60, and then 360 cases.

We conclude that in New Zealand up until Lockdown Level 4 (25 March; Day 27), one person may infect on average at least 6 other people if there is no isolation (Ro = 6) and at least 5.8 other people if cases are isolated (Re = 5.8) once they are identified.

Most researchers believe that values are less than half of the above figures. Consequently their models may need to be adjusted once our results are independently verified.

We note that normally once a Lockdown is introduced, we may not expect a fall in the Case number rate (Re) for at least once cycle since new cases could easily have been infected at least five to eight days earlier. In New Zealand there is a much quicker change.

ModelCases14Chart

Perhaps prior isolation was already having an effect?

If the daily rate of increase (r) for case numbers matches the rate of decay of infectivity, then for each day during the infectious period (two cycles = 10 days), the daily contribution towards R is the same.

To calculate R, we can simply multiply by ten the value for the first day of the infectious period.

We add in a factor of x^2 for incubation and realise that
1 + x + x^2 + … + x^10 = (x^10 – 1)/ (x – 1)

For Day i (i=1,2, …, 10) and x = r (the daily rate of increase),  we have a daily factor of

r’ = x^(i+1) * x^(10-i) * (x – 1)/(x^10 – 1)
= x^11 *  (x – 1)/(x^10 – 1)

i.e. The same value for each of the ten days.

Let f(x) = 10x^11 * (x – 1)/(x^10 – 1) then

Ro = f(SQRT(2))
and for the ten days on and before 26 March (the first full day of Lockdown L4)

Re = f(1.4)

i.e. Ro ~ 6 and Re ~ 5.8.

Our values for Re and Ro are twice as big as most experts consider for COVID-19.

We also looked at worst-case scenarios.

We looked at a normal curve which has a total of 27,600 cases (not deaths!).

Normal2700

The curve stays above the actual number of cases which means the cumulative number of cases in the normal curve is also above the actual cumulative number of cases.

We also considered another worst case scenario of 24,700 cases:

ChartWorstDaily

While the above number of cases may be possible in a worst-case scenario, 27,600 deaths (as estimated in a report commissioned by the NZ Ministry of Health) would not appear to be possible by July in New Zealand in a worst-case scenario.

We now use our simple model to verify our calculation Re = 5.8.

We modify the model so that we have a 2-cycle infection period.

For i>2 using C[2] = 8 and C[1] = 5, for given constant values of LAMBDA and Re, we can calculate

C[i] = Re * ( LAMBDA * C[i-1] + (1 – LAMBDA) * C[i-2] )

Where C[i] is the estimated number of new cases in cycle i.

The number of new cases follows the sequence [5], 8, 44, 231.

We note that LAMBDA = 1 gives 231 = Re * 44, Re = 5.25, and r ~ 1.393259.

We need to find LAMBA so that

231 = 5.8 * (LAMBDA * 44 + (1 – LAMBDA) * 8)

i.e.

LAMBDA = ( 231/5.8 – 8 )/36.

However the value of LAMBDA obtained gave a value near 233 instead of 231, so we recalculated LAMBDA using 229.5 instead of 231.

We obtained LAMBDA = 0.87691571 and rounded this value to use
LAMBDA = 0.877 and (1 – LAMBDA) = 0.123 to generate the Table below:

COVIDTable58r58

This table and the table below suggest a total of 7800 to 7900 cases may have been possible by 5 April. New Zealand currently (29 July) has 22 deaths from 1557 cases with a CFR of about 1.4%. This suggests 112 deaths may have been possible by 5 April.

Each estimate for the number of new cases is obtained by multiplying the previous number of cases (previous cycle) by LAMBDA and the number of cases in the cycle before by (1-LAMBDA).

The value 5 (highlighted in red) is only used to start the sequence. Hence we obtain

44.26 = 5.8 * (LAMBDA*8 + (1 – LAMBDA)*5) and

230.84 = 5.8 * (LAMBDA*44.26 + (1 – LAMBDA)*8).

The values obtained are very close to the actual case numbers.

We earlier used Goal Seek in Excel to get LAMBDA = 0.876. The above value gave better results.

We see the 5-day value (0.877) above and the cumulative 5-day value (0.8432) for the 2-cycle CSAW Excel scenario are reasonably close (r~1.4) considering the first calculation has two levels and the second has ten levels with over half the infections below occurring in the first two days and over three-quarters in the first four days:

1 0.2959
2 0.5073
3 0.6583
4 0.7662
5 0.8432
6 0.8982
7 0.9375
8 0.9656
9 0.9857
10 1

A better approach is to estimate LAMBDA to fit the values [5], 8, 44, 231.

We start with Re = 5.8 (the cell highlighted in gold below) and use Goal seek in Excel to set the number of new cases for 26 March to 231 by changing Re. We obtain Re ~ 5.7115:

COVIDTable58Re

The values match.

We also calculate Re in the CSAW model using the formula

Re = 10 * r^11 * (r-1)/(r^10 – 1)

Using values of r in the table below:

TableRer

For r = SQRT(2) [~1.4142] we obtain

Ro ~ 6.0468.

We therefore estimate Re up to 26 March to be in the range 5.7 to 5.9 and estimate Ro = 6.

By 31 March the above tables estimate totals of 1489 and 1497 cases.

This number of cases was not achieved until the beginning of May.

Let C[i] be the number of new cases in cycle i.

Let R[i] = C[i]/C[i-1] and r[i] = R[i]^(1/5]

Then R[i] is the ratio of the number of new cases in cycle i divided by the number of new cases in the previous cycle, and r[i] is the daily rate.

In April for Re = 5.8, R[i] becomes 5.223183, and r[i] becomes 1.391833.

In April, for Re = 5.7115386, R[i] eventually is 5.254786 and r[i] is 1.393513.

The above are the limits.

We consider the above close to our estimated r = 1.4.

These results help validate our calculated Re = 5.8 for the CSAW model.

We conclude that in New Zealand up to 26 March, Re = 5.8 and Ro = 6.

Our values for Ro are likely to be conservative.  The CSAW modelling uses mean values. Isolation of identified cases in New Zealand is likely to underestimate calculations for Ro.

Update:
For an updated value for Ro and worldwide values for Ro, see:
COVID Odyssey Ro update: Occam’s Razor~ A close shave: In NZ is Ro ~ 3, 4, or 6? What are Ro values worldwide?
COVIDWorldAvNewRanked
COVIDWorldAvNewAlpha
COVIDWorldAvNewRanked4
COVIDWorldAvNewAlpha4
COVID Odyssey: Vir[tu]al World Tour ~ How many people can one person infect in your country?

My other COVID-19 posts can be found here:
https://aaamazingphoenix.wordpress.com/tag/coronavirus/

Data for my posts can be found at:
https://www.worldometers.info/coronavirus/
https://en.wikipedia.org/wiki/2020_coronavirus_pandemic_in_New_Zealand
https://www.ecdc.europa.eu/en/publications-data/download-todays-data-geographic-distribution-covid-19-cases-worldwide
https://ourworldindata.org/coronavirus-source-data
https://www.webmd.com/lung/news/20200507/100-days-into-covid10-where-do-we-stand
https://projects.fivethirtyeight.com/covid-forecasts/?
https://ourworldindata.org/coronavirus-source-data

I share my posts at:
https://guestdailyposts.wordpress.com/guest-pingbacks/

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