COVID-19 Data and Analysis
NOTE: If you have either dual or multiple screens, on your system, then right click your mouse on a link to a Figure shown below and select the option to "Open link in New Window" as the default is that the figure will open as a "New Tab"
Disclaimer
When viewing this page or any other page associated with my webs, you may see information above the title. For example, any information above the title "Covid-19 Data and Analysis" has nothing to do with my page but is put there by the system that allows me to have a web. In no way do I gain any financial benefit from this additional information in cash or kind. You will notice that the additional information contains references to genealogy as that is the section of Ancestry.com that provides the facilities for my web that consists of over 4000 files that occupy over 500 MB of space on a disk somewhere. It could be argued that "How does Covid-19 relate to genealogy?" and my reply is the cause of death for many people worldwide will be shown as Covid-19 in future databases of genealogies of a family or area.
Purpose
This webpage is being developed to help myself and others to get a better insight into how the Covid-19 virus spreads and what factors may be of importance in the control of how it spreads.
When I started this project, my thoughts were along the route of what we are taught by others, However, it appears that some of the things that we are taught may not be correct if we don't fully understand the complete situation. We need to look at the situation that we are confronted with and from that determine what the rules should be. You will see many recommendations that are opposite to standard practice at the start of the summary but end with a different set of recommendations. You must start from the beginning and go to the end to find out why and how these recommendations have been developed. If I were to watch a TV program of what is being presented and miss the part that has been shown up to here, I would flip to another channel or turn the TV off when in fact I should study this in more detail
This project started out looking at the Covid-19 virus but has
now evolved into looking at the policies of how society should look at the
control of any disease or epidemic/pandemic. An epidemic is confined to one
area or country while a pandemic covers the whole world. It is not a factor of
will you get the virus it is only about when as you will get it. The rules for
each kind of event differ and this is what will shock you like it has shocked
me.
The rules for the control of a disease are different than the rules that should
be enacted for an epidemic. Think of this like the case of burning a pile of brush
where people try to put the fire out using fire control when in fact you are
trying to start a file to consume the pile of rubbish. You would say that is
genocide if people were equivalent to the rubbish. Now that the case where you
are trying to remove the rubbish in a forested area so that a forest fire cannot
start. If you manage to get the fire started that burns only the ground cover
then you don't want any firefighters putting that fire out but to keep it
burning this is a controlled burn. When the fire is sufficient to keep burning
the ground cover but not burning the trees then it will advance until all of the
ground cover is consumed and the forest remains. If it takes as long to light
the fire as it takes to consume the ground cover then you have spent twice as
much time as you should have. Anything that slows down the initial starting of
the fire is counter productive. Wet kindling, damp fire starter material, or
someone with a pail of water are not what you would use.
When going through the discussion shown below, the fact that I
had to use three different graphs with three different day scales makes it
difficult to follow. I could only use the tools available and apologize for the
confusion.
This is the first time that I have ever published a document while it was being
created. Anyone doing research knows that you set your goals, do the research
and use a research journal to record your findings, filter out the sidetracks,
produce the final results, then publish. In this case what I have done is
compress all of these into one operation (this project was started on Apr 4,
2020). Multiple rough drafts have been published and for anyone trying to see
the latest version, I have included the date and time at which the latest
document was published. A major deficiency in this operation has been the graphs
that started on day 1 and have been updated many times when they should have
been discarded and a final graph used in the final report. Since this has not
been done then I find it difficult to keep things straight using the graphs and
you as the reader will find it even more difficult. At least the info is in the
public domain so if for some reason, I can't produce a final report then all is
not lost. I leave it up to someone else to carry on. All of you have my research
journal, so take it and run, do more research and find the truth or get closer to
the truth.
Other examples of Mathematical Modeling and/or display of information acquired can be found at Other examples Some may ask why would other examples have anything to do with this analysis of Covid-19 data for Canada. The mathematical Model that is shown for the Covid-19 data is the same mathematical model that can be used to show World Population. When a real-time world Population display is viewed, it could just as well be a real-time display of World cases of the Covid-19 virus. The eleven sub-regions for the population display could be eleven different regions for the Covid-19 virus. The only difference between the two displays would be the critical values as determined in the summary as shown below.
Definitions.
Note:: Anyplace where the word Asymptomatic is found. may have the wrong conclusions because my understanding was that the person showed no symptoms but was infectious for life.
The term asymptomatic means a person who has not had the virus
but tests positive for the
virus but shows no symptoms and is thus a carrier. They do not recover but can
infect anyone who does not have immunity.
Np = number of people who make up the total segment that get the virus in the
study. These would include the following: Health Care worker, Emergency Medical
Service worker, essential services workers (essential workers include many
different groups that one would not normally think of but all people in the food
industry are in this group if their group provides an essential service like
take-out meals, grocery store workers, etc, plus any others that are required to
maintain public involvement. If anyone leaves their self isolation and goes out
into public to purchase groceries, medications, medical services, or any exposure where
they cannot maintain social distancing then they are part of this group.
Tau0 = time in Phase 1 for the number of virus cases to increase
by a factor of 10
Tau1 = time in Phase 2 for the number of virus cases to increase by a factor 10.
Found to be 4.56 days
Tau2 = time in Phase 2 for the number of virus cases to double. Found to be 3.16
days
Tau3 = time from when a person is exposed to the virus to when they can actively
transmit it to others.
Np= total number of cases during a wave of the epidemic.
The summary is being shown here so that anyone reading this will be aware of the suggestions for Public Policy and what practices should be followed during the different phases of the virus epidemic.
The summary items shown below are based on the analysis of the number of recorded cases of the virus in Canada. The last item is based on the mathematical model and testing to try to modify spread of the epidemic. It is assumed that the mass testing would only result in 5% success rate which is the value found in Saskatchewan. Different scenarios for outcome are shown for a) no testing, b)5% , and 10% success rates.
First of all, no county can escape the virus, all it can do is to try and minimize the number of deaths and have the least amount of financial impact on the economy by minimizing the duration of any shutdown.
The first thing that a country should do is to identify different populations within their borders according to risk of death if they contact the virus. Then place those with the higher risk into a population which is cared for by workers that have little to no chance of getting the virus and spreading it in this secured group.
Testing should be done to prove that a person is virus free and has no chance of spreading the virus to any person who has a higher probability of death if they become infected.
All persons entering Canada from abroad must self-isolate for 14 days and at the end of this period, they must be tested to prove they are Virus free and certified as so. If they have been in contact with anyone else during the isolation period and they test positive the all of the contacts must also remain in isolation for a further 14 days until they are proven virus free.
Testing during the epidemic should be divided into four different
phases, beginning(Phase 1), mid(Phase 2 and 3), and
end(Phase 4). There are
separate policies in each phase that either shorten the time of each
phase, ensure that a person is not an asymptomatic carrier, or is a
certified person who has had the virus. The policy in Phase 1 can also take
two different approaches; a) try to prevent the spread of the epidemic, or
b) acknowledge the fact that this is an epidemic and not try to prevent it. In Phase 1a, the start, testing would
be to find those infected and remove them to isolation/quarantine, In Phase
1b only test those that have had the virus and when they are recovered
test them to prove they are not an asymptomatic carrier and identify them as
a certified person who has had the virus. Phase 2 and 3:
minimal testing other than to protect the most vulnerable, and testing of
all people who have recovered from the virus and certify them as in Phase
1a. In Phase 4, the testing is the opposite and would be used to find those who
have
not had the virus and isolate them. If they are from the essential
services sector then they should be replaced by a person who has had the
virus and has recovered. Testing would also be for those who get the virus and
certify them as in the other phases.
This is just like fighting a fire,
initially the purpose is to save the property(Phase 1), Phase 2: try to
contain the fire but just let it burn, Phase 3 try to remove any material
that the fire would consume so that it dies out. If Mother Nature comes
along with three inches of rain while the fire is raging then you will have
the situation like what Australia just went through with their wild fire.
Trying to fight the fire or testing to try and slow or stop the epidemic
during Phase 2 and 3 is a waste of money and resources and is only for show that
something is being done. Testing during this phase is only to verify that
none of those who have had the virus have remained as carriers.
At or near the end of the time, during Phase 3 and 4, when the virus is still active then testing should be to determine those in the active population who have not had the virus and remove them from the active population and place them in the inactive population. If they are from the essential services sector then they should be replaced by a person who has had the virus and has recovered.
As the epidemic progresses, those who have had the virus and fully recovered and certified should now be the workers that take care of the most vulnerable, if qualified.
The time from the first case to the end of an epidemic is found to be 5-6 months using the data for Canada where the provinces of Quebec, Ontario, Alberta, and BC had the dominant number of cases. These five provinces had a population of 30.7 million out of a total of approximately 37 million. The number, Np, of essential workers and others in the population that was infected, was found to be 30,000 or 0.08% of the total population.
The peak number of cases hospitalized is shown to be determined as the percentage of cases hospitalized times Np/(2 times Tau1) From the values shown, this works out to Percent Hospitalized(Ph) * 30,000/(2*4.5). This gives a value of approx. 34 /percent hospitalized/day
The return to normal or controlled return of the economy should only be after no cases of the virus for FOUR weeks. The county could expect to see another wave start within those not effected by the virus if the economy is opened prematurely. A second wave will most likely occur if the total population has not been infected during the first wave. The second wave then starts from an external source.
When an area tries to return to normal in a slow and controlled fashion then those who go out into public may be only a reduced part of the population that are not at high risk. If anyone is at a high risk of death if they contact the virus then they should remain in isolation.
Traceback: This is an important factor only at the beginning when testing is trying to locate all contacts of the person who could possibly infect others. All of the contacts must be located and isolated and tested every day to determine if and when they tested positive for the virus. As each person tests positive, they are remove to quarantine and the time until they show symptoms of the virus must also be collected. Anything less than this approach is a waste of time and resources. This information must be used in planning and policy or again is a waste of time and resources. Traceback during Phase 2 and Phase 3 is of little or no benefit and is just like fighting a ranging fire like the wild fires in California in the fall or the wildfire in Australia in the fall of 2019. Where the fire came from or trying to put it out using techniques for a small fire are only for show and do not have any effect on how the wild fire spreads.
A mathematical model was developed plus additional mathematical equations that allowed for exponential decay. The development of this mathematical model is shown in more detail in Major Contribution shown below.
The mathematical model and actual data fit within 1% over the range where calculations are made to determine the parameters required for this analysis. See Figure 8 for plots of both data sets vs. Time. Two additional curves were added to those shown in Figure 8. The resulting graph is shown how the spreadsheet analysis can be used to predict when the peak will occur and the total population that will be infected. These values are Tp and Np and are determined where the two curves called "Find Np Tp" and "Exponential" intersect each other, Tp is the value day(46) when they intersect, and Np is determined by calculating e to the power of the y-axis at Tp. This value was found to be 25084 or 25000 and not 30000 as was originally determined. From this figure, it can be seen the actual cases are greater than this value for Np as the actual cases are now being effected by a new wave that was just starting and not noticeable for several days later. The total Cases for Canada are now at 48,500 as of 28 April 2020
A number of graphs that show the number of cases/day vs. time with different values of Np are included as it appears that people are using raw data to determine when the peak load on the hospital system will occur and what the magnitude will be. The curves shown are based on the Mathematical Model so they don't have the noise that actual data would include. The value of Np differs for each graph. Note: the values are plotted on a Ln scale: Np=20000, Np=30000, Np=40000, and Np=60000 .
These graphs are for Np=30000 and differing values of tau1
which is determined by tau1=1/(rate for virus spreading- testing success
rate) Figure 8a is for 5%,
Figure 9 is for no testing,
characteristics of virus only, Figure
9a is for 10% mass testing success rate. These figures show that
changing the value of mass testing successes does not change the peak of the
curve but only changes the time at which the peak is reached. At any given
time prior to reaching the peak, the longer the peak has been delayed, gives
a lower number of cases which could be called "Lowering the Curve" but from
the diagrams shown, it only changed the time scale by stretching out the
time for the growth and decay of the epidemic, If the purpose of delaying
the time of the peak was to allow the purchase or arrival of critical
supplies and/or Services then the long time is of major benefit. Of all the discussions in all
the media, this has not been explained and people are blaming others for not
either doing or providing the desired/required amount of mass testing that
they think is required.
On 22 Apr 2020, the Saskatoon Star Phoenix newspaper had an article that
showed daily cases , Fig. 10,
reported for nine different provinces of Canada. This diagram has a
different starting time for each province as the time scale is Days since 30
Cumulative cases. The Daily reported cases are shown on a log scale but they
are for a 17 Day Rolling Average. The rolling average is used to reduce the
noise present in the daily data. To do this requires a considerable amount
of forethought as it is a filtering process and the method shown by doing a
Google search on the internet showed a Bright Hub Project Management report
that explained their version of Rolling Average. If one is looking for only
the values then what they show would be satisfactory but if both the value
shown and time then one has to the averaging differently. You must use a
method that I call a Boxcar Averaging. The number of samples to be filtered
must be an odd number so that the averaged value will for the value at time
T. To do this, (N-1)/2 values prior to T are summed as well as (N-1)/2
values ahead of T as well as the value at T are all summed up then the two
values at the extreme ends I'll call first and last are determined and 1/2
of this sum is subtracted from the previous sum then that final sum is
divided by (N-1) not N. This process is really digital integration. If the
values are divided by N that is rectangular integration while the method
shown above is trapezoidal integration which is more accurate. People who
use this method may jump on me as what I show requires information to
predict the future values. This is not true as we are filtering data that
has already been gathered unlike in business where the method is used to
smooth the data up to the present time. This is done by determining when the
next smoothed point would require data that has not been gathered yet, so
the width of the box is reduced by 2 and the process described above is used
then the next point is smoothed by reducing N by 2 again and this is
repeated until N equals 3. The last data point could be smoothed/averaged by
some method but the time at which it should be located would not be at the
time it was gathered, This is a trade-off of incorrectly located
smoothed data or raw data at the correct location.
Since the method used by the authors of Fig 10, daily cases, is unknown then
any further analysis or statements would not be of much help. However, they
can still be looked at in terms of when did the individual provincial data
start to be smoothed, From Figure 2, it can be seen that 30 cumulative cases
is the start of the straight line at day 16 of the mathematical model
straight line smoothing curve. At the upper end of the straight line on
/figure 2, the actual data starts to deviate away beginning at day 33. The
difference between these two values gives 17 days which is exactly the width
of the smoothing window used to generate the data in Fig 10. Since this time
is very close to the total time of Phase 2 and Phase 3 has this same time of
decay, the large values from day 20 onward in Figure 10 are caused not by
cases per day for that day but by the peak number of cases 17 days
previously. More information can be found in the Platt
report
Important information that everyone needs to follow.
Many will find this information boring or beyond their limit. However, what is most important is that all of us do our part to reduce the spread of this epidemic and have as few deaths as possible. To achieve this you/we MUST follow the directions of the Health Authorities in your area and limit the probability of you either receiving the virus or unknowingly transmitting it to others be they either young or old. The main items stressed are:
Social Distancing; If two people are more than 6 feet apart then the probability of getting the virus from the other person, if they have the virus, is very small. The probability should be inversely proportional to the distance squared . If they are miles away then it is zero but within a closer distance it would be finite but small. I may be stepping out of my area of competence but my feeling is that the human body may be able to handle small amounts of the virus if the immune system is strong. However, if the amount of virus is above a certain threshold then the immune system can not kill the virus before it multiplies beyond some limit. This could mean that if you are out in public and are more than 6 feet away from anyone you meet and the amount of virus that you pick-up is below your threshold then you are ok, but if you meet N people, who have the virus, then the sum from each person may exceed your threshold and you become ill. There are people in the medical field that may be able to indicate if this is true or not.
Facemasks must be worn when in public if you cannot maintain a Social Distancing of 6 to 10 feet.
Wash your hands with soap and water to wash away any virus that you may have picked up from touching some object that has been contaminated.
If you are sick or feel you might be coming down with some illness, then stay at home and isolate yourself from others so you don't infect them. Have a test to see if you have the virus, if so, then you must take extra precautions to quarantine yourself. This rule is the opposite to what is required in Phase 1 as it reduces the number of people who start the growth in Phase 2. The shortest time spent in Phase 1 would be for more than one person to start the expansion and not to try to isolate any contacts of the single person who started the epidemic. Sounds wrong but needs to be tested and proven either true or false.
During Phase 2, remain in self-isolation and only venture out into public when absolutely necessary. This keeps the population who are eligible to be infected smaller then if you are adding to this population. When you return, then you must isolate and be tested, or your partner becomes the canary to test if you have gotten the virus.
In Phase 4, only when no new cases of people being infected with virus in your area for more than 28 days will your health advisors declare that the virus has been eliminated. If any case is reported in your area then they could have infected others who will be getting the virus and spreading it to others before they become isolated and that will start a new out-break.
Developing the Model.
First of all, one has to gather the data, process the data to remove the noise, then analyze it. On 12 April, I saw the data that I am using being shown as number of cases of the virus each day, that is the raw data that contains a large amount of noise or variability. If a case is not reported on one day but reported the next day that produces noise but the total for the two days is correct. To reduce the noise, the total number of cases from day 1 should be the format used for analysis. Then to reduce the noise even further, the data should be plotted on an appropriate graph that will allow further smoothing. This might be linear, semi-log, or log-log graph paper. Yes, graph paper, the old way, not on a computer because the next smoothing step is much easier and faster done by doing a best straight line fit to the data over a segment than to try and get a five figure accuracy on a computer result when the same can be done to the desired accuracy using a graph. If one were publishing a scientific paper and wanted better quality graphs then the computer would be used to produce the same results and graphs. From Figure 2, the straight line approximation to the data covers four decades that take 42 days or three decades in 31 days. This gives either 10.5 or 10.33 days per decade. If you look at the actual data points, you can see that in the mid portion of the straight line and along the line they vary on each side and this is the noise that is being reduced so that two of the most important parameters, tau1 and tau2, can be obtained as shown below.
While viewing the report on TV, on 12 April, I realized what others were talking about when they would refer to flattening the curve and how this was reducing the number of deaths so their policy was better than those who had not flattened the curve and had higher number of deaths. What these people don't realize is that in both cases, the epidemic is not over and the total number of deaths in each case would be the same for the two different policies, if they had been used on the same group of people, however, the time required to end the epidemic would be vastly different for each policy.
The mathematical model is developed using actual data for the number of cases in Canada. Once the mathematical model has been developed then it can be used to predict when certain critical events will occur. The model can be updated and changed as new data is received. Using the raw data to do the predictions is not the way to proceed as it is very noisy. This noise needs to be removed and that is where the mathematical model does the job.
The data used is shown in Figure 1 which is assumed to be correct as it was published in the local newspaper on 3 April 2020. Figure 1 opens in a new window or tab which allows the user to view this text as well as the figure. The data shows the totals for each day which is made up as the sums from each province or territory. The next diagram is Figure 2 which is a plot of the number of cases vs. day of the month(also day from 0 to 50). the first data point is for Feb. 21 when the number of cases were only 9 to the last point which is for 5 April at day 45(Note: this point is off the graph at the very top edge of the page.) The straight line curve only gives a close fit to the data over the range from day 13 to day 35. There is a small section from day 19 to day 25 where the data does not fit the straight line as well and this is assumed to have been caused by a change in method of reporting or some other unknown factor. Other equations and diagrams are shown in Figure 3. These will be referred to as Fig. 3a....n as required in the following sections.
New information as of 14 April 2020 is shown in Figure 1b
Analysis of graphic data.
The straight line approximation to the curve is shown in Fig 2, gives an intercept at day 0 of a value between 1.7 and 1.8 which is also the square root of PI(1.77245). The slope of the straight line is one decade on the graph for each Tau1 days of time. Since the natural log of 10 is 2.30258, then the value of tau1 is calculated over 4 decades and the time is 42 days which gives (42/(4*2.3) therefore Tau1= 4.57 days. For those who have been trained as Electrical Engineers they will see this as the value of Tau in the equation for the exponential equation e to the power of t/tau. Sorry for not being able to show this as a proper equation but my webpage editor does not allow me to use the proper symbols. Fig. 3a
The value of Tau2, time for population to double, is found in a similar way (42/4)*.301 = 3.16 days
The final resulting equation for the number of cases vs. time is given by 1.77 times e to the power of t/4.57 where t is the time in days. See Fig. 3a
As shown in Figure 2, the data deviates from the straight line at both ends just like a Probability curve that shows the accumulated probability for a Normal distribution. This distribution curve would be the slope of the actual data as one goes from day 0 to the final value which has not been shown in the data. This would mean that one should be showing a graph which is the difference at t +/- 1 day divided by 2. This is only true in a mathematical sense as delta t is small and not 1 day as shown by the data that has been gathered.
Development of Theoretical modeling equation
The normal approach used to develop a population growth is to say that the rate of growth is proportional to the amount present, Fig 3b. This leads to the result that when the total values is plotted on semi-log graph paper the equation becomes a straight line with a slope that is equal to the proportionality constant. The starting assumption is that any item that has already been generated is available for the total time of the equation, This is unlike the Covid-19 case, growth where the person is only infectious for a limited time until they recover. They may show no symptoms so can spread the virus to others. This fact yields another term in the basic equation which is the Probability of infection(Pi). This probability also depends on how far the virus has to travel to be able to infect a recipient. If the recipient is beyond a certain distance then the probability is zero. The actual infection rate also depends upon the availability if a person to be infected. When the virus first starts to spread, the total population of people in the area where the source is located are eligible to be infected so this probability can be thought of as one. However, as the total number of people who have had the virus becomes larger, this decreases the population that is available to be infected. This results in the spreading of the virus to decrease in time rather than increase as it did when it was just starting to spread. This probability of a person getting the virus will be called Po.
The equation of growth can now be shown as dN/dt= alpha * N*Pi*Po. See Figure 4a. Figure 5 shows plots for the data at both the lower and upper ends of the graph and cover the time t=0 to t<12 and t>35.
Where N is the number of people infected,
alpha= the proportionality constant, (shown as 1/tau1 in Fig 4a)
Pi= Probability of infection,
and Po= Probability of a person receiving the virus.
Major contribution to the Theoretical modeling equation
The probability of a person receiving the virus has one major component that is the population still not infected which can be shown as Po=(Total Population available at the start - the number infected)/ Total Population.
This gives Po = (1-N/Np) where Np is the total number of people who were eligible to be infected. The value of Np is calculated to be 30,000 as shown in Fig. 5 This actual value is important as it appears to be the number of people shown as essential in this population. Any calculation that will give an approximation or prediction of this value is of use. The final number of cases will equal Np but we only know that value when the epidemic has ended.
The final equation, as shown in Fig 4b, would be: N= (No[Np+No] e to the power t/tau1)/(Np+No
*e to the power t/tau1) See Eq-1 in Fig 4b
Limits at t=0 N=No and t large N=Np
A more sophisticated version of the modeling equation is shown in Fig 4c given by Eq-3 This version takes into account the finite time between when a person is exposed to the virus and when they become active spreaders. The theoretical equation assumes that tau3 is zero. the value of tau3 is another unknown but is somewhere in the range of 5 to 13 days.
Determination of components in equation shown above.
Figure 2 and 5 are the main items used in this section.
When starting the analysis, the only data known is what has been gathered
to-date. When this data is plotted on semi-log paper the portion of the curve,
Fig 5a, will most likely look like from day 0 to any where in the region of day
30 to 40. A person can start to do analysis at this time but it would be better
to have more data up to about day 40 to 50 when the curve starts to bend down at
the upper section as shown in Fig 5b. Note: days are shown at the top, not
bottom of the page.
The value of Tau1 is calculated as the time for the population of cases to in
crease by a factor of 10. This allows tau1 to be calculated as shown above in
the section "Analysis of graphic data". Tau1 = 4.57 days
The growth rate can also be expressed as P= (2) to the power t/tau2 where tau2
is the time for the number of cases to increase by a factor of 2, tau2 is about
3.16 days.
The main items that need to be calculated are the values of: time for
Phase 1, Np, Tp, and Th as shown in Fig 5b.
Time for Phase 1 from Fig 5a lower left is shown as 17-(-15) = 32 days
Tau1 is the 1/slope of the straight line using Naperian log = 4.57 days
Tau2 is the slope of the straight line in Figure 1 using log to the base 10 = 3.16 days
The time at which the peak number of cases per day is determined by taking the second derivative of the equation Eq-1 in Fig 4b and determining at what time this equation is equal to zero. The results show this happens when the population is Np/2. This gives the resulting equation Np/2= e to the power Tp/tau1, using the exponential form or Np= (2 )to the power [(Tp/tau2)+1] using the doubling equation. The actual data compared to the straight line segments show these results. This gives Np= 30,000
Tp=61-17 as shown on fig 5b and Fig 5a = 44 days from the time when when there was a total of one case as shown in Fig 5a. This is the duration of time for Phase 1.
Th is the time when the projected theoretical line intersects the value of Nd/2 would be determined as Th= Tp-Tau2. Th=44-3.16= 41 days
The total time for the epidemic to grow would be the sum of Time for Phase 1 + time for Phase 2 which is (32+41) = 73 days. If one assumes that the decay during Phase 3 and Phase 4 are similar then the total time for a wave or cycle is given as double the sum of Phase 1+ Phase 2 or 146 days which is approx. 5 months. As a margin of safety, one month could be added making the total time 5-6 months. These values may not be exact but the this give the approximate time for all four phases to take place. Data has not been obtained for Phase 3 and 4 and they are being assumed the same as Phase 1 and 2 which may be incorrect. These values will be updated when all of the data has been analyzed after a full cycle/wave.
The time at which the number of new cases/day is at a peak value is when the number of cases is equal to Np/2. The time at which this peak occurred from the start of the epidemic would be the same amount of time for the epidemic to die out. For the data shown in Fig 1 and 2, this gives a total time of 2*41 days for the main portion of the epidemic but the two sections, one at the start and the other at the end of the curve require another total of 64 days. This give a gross day count from beginning to end of the epidemic of 82 plus 64 or total of 146 days. This is approximately 5 months for this population to die out. Since the population that is capable of getting the virus is not static but spreads with time then as a new "Hotspot" starts, it will go through the same process as the one being studied. If a second "Hot Spot"(HS) occurs, then the data being analyzed should not be the total cases to date but the cases to date from the assumed start of the new HS. If this is not done then the data and statistics for the second HS will be masked by the first one. If one considers the case where the second HS is the same then the compete curve would be compressed into a dataset that had a new value of Np2 equal to 2Np as shown in Fig 5b. Any analysis of this data would be completely false.
Predicting the value of Np before reaching the peak cases per day as shown in #4 above.
The question many will ask is have we reached the peak and when will it happen? With reference to Fig 5b there is a time shown as Th. The theoretical number of cases would be on the straight line approximation. Let's call this Nh, The actual number of cases is known and will be referred to as Na. Using Eq-1 from Fig 4b, it can be shown that Np(predicted) =Nh*Na/(Nh-Na). From Fig 5b, these values are: Np(predicted)=16,000*9700/(16000-7900) = approx. 19000. This is only a rough approximation but at least we know what decade the value of Np will be. The error is 33%. This prediction was four day before the time of the actual peak so we can do a total of five predictions before the time when Na is .5Nh which is the time of Tp when the peak is reached. People who use the doubling relationship to follow the growth in cases will be lead astray as what they are doing is taking the slope of the actual data and this keeps changing and becoming smaller as one approaches the peak at Tp. The number of cases at Tp will only double at time equal to infinity if growth in cases are the values being considered. If one were to look at the number of remaining cases and the time going in reverse where t=0 is when last case happened then the mathematical curve and plotted data would be the same as for the growth portion but now it is the decay portion of the cycle. This is fine for analysis after the events are all competed but what is required now is looking ahead and not behind in time.
Model equation near t=0 from Figure 2 or t=-15 to 33 from Fig 5a this is Phase 1
In Phase 1, from figure 2, the data deviates from the theoretical equation that fits from day 0 to 12. This deviation is best explained by the fact that a single person would be the one that started the virus to enter the population and it takes several days before the first person infected and acquired the virus to be capable of spreading it to others. Of all the people infected, by the first person, they would not show on any data acquired until they in turn were reported as sick with the virus. This initial period is more of discrete distribution rather than a continuous one that is described by a mathematical equation. The period of time required to become ill from the virus is some finite time period and if it is assumed to be 13 days then anyone in contact with this period during which they could spread the virus would be the people in the next generation. This process would keep repeating generation after generation until the generation rate is governed not by the time from contact to onset of the illness but by the spread as shown in the graph from t= -15 to 16 left graph in Fig 5b (Phase 1). The Public Policy during this time should not be to test contacts of the person who is ill, but only to quarantine them for 14 days if they are going to be in contact with people who have a higher death rate if they are infected.
From Figure 2 in the lower right corner is the symbol To which is the time when the straight line intersects the line on the graph for 1 person or when the value of the log is zero. This value is 3 days which mean that all people who got the virus in this first section can be placed into a discrete model where the number increases according to which generation. The equation for this time can be thought of as P=(1+x) to the power n where n is the number of generations, and x is the number infected by one person. Using this relationship, the value of x is found to be 15. This means that the spreading rate is that each person infects 15. The value of n is calculated as (t+3)/13, where t is the time in days and 13 is the assumed number of days from contact until a person becomes capable of infecting others.
From Fig 5a, straight line approximation shown in the lower left portion, the value of tau0 is found to be 17/2.3 or 7.39 days. A very rough approximation would be to assume that this growth and decay is similar to the main growth and decay but now, the this new total population would be somewhere in the range of 8-10 would be used in an equation like Eq-1 from Fig 4b. The one datapoint shown at day 0 with a value of 2 but at day 1 the value is 3 looks like it should have been 3 at day 0 which would have made that point fall right on the straight line approximation.
Model equation near t>12 - t<38 in Figure 2 or t>35 in Fig 5a or T>52 in Fig 5b. Phase 2 and 3
This is the mid-portion of the pandemic where the spread is like a wildfire that is out of control and any attempts to stop it are futile. Think of a wild fire and dropping water on it or fighting it in any way does not stop the spreading. The only factor that will stop the spread of the virus, is to reduce the available population so that it is limited by the time when the number infected is limited by the total population that is still available to be infected. If this population is more widely dispersed then that will lower the probability of spreading. For example, if one million people who are dispersed over a large area vs. the same million people in a higher density location. These people are not isolated but must work and provide services to the rest of society. Testing the general population during this time period is of very little benefit as it does not slow down the spread by any significant number and is a waste of time and resources. Only test those who have recovered from the virus to prove they are not carriers.
Model equation near t>35 from Fig 2 or t> 52 from Fig 5b
This is the region where the effect of the finite population limits the uncontrolled growth of the virus as shown above. The population that is still not infected consists of at least two main groups, one is the group that were available during the uncontrolled growth and the second group are those who are the Health Care Workers and others who take special care to prevent themselves from getting the virus. This group is much smaller than the general population. It is very difficult to model this group without knowing which population a person if from when they acquire the virus as the growth rates for each group are different. However, one can assume a stable situation exists and now the decease in growth process can be thought of the spreading situation but in reverse to the explanation for t=0 to t<12. The more testing that one can do should be to remove those who have not been infected from the population and replace them by people who have already recovered. Any person who is tested and found to be positive should remain in this population and not be isolated which is exactly opposite to the present Public Policy.
How long in the section t>35 or when the growth is less than the uncontrolled section. Phase 4
In the case of an exponential decay, the time for the population to decease depends upon the value of tau for this section. As the number of active cases become smaller, then the statistics will be governed by the same factors as described for t<12 in Fig 2. The amount of testing during this time period is one of the major factors and cannot be over stated. The testing in this Phase 4 is the opposite to the testing in Phase 1. Again, it is test, test, test and test again but you have to be testing the correct population. The population that must be tested are:
Any person who has not been in complete isolation for more than 2 weeks and tested at the end of that time to prove they are virus free. If they are living with another person then the isolation time could be 4 weeks only if the other person does not become infected. The test at the end of 4 weeks would not be required. These people are the ones that should be removed into isolation so their possibility of being infected is zero
If a person is in self isolation and completely by themselves for more than 2 weeks or even the complete duration of the epidemic then it is possible that they could be an asymptomatic carrier so must be tested to prove they are virus free.
Other factors that may be important.
Comparison of the Model to actual data
The only portion of the model data vs. actual data that can really be compared is for the portion of curve shown in Fig 1 and Fig 5a at the top end when the curve deviates from the straight line. As time goes to infinity, the actual data will asymptotically approach the value of Nap. This is the most difficult portion to compare as the model equation should be the one shown as Eq-3 in Fig 4c. the only way to test if the values reported in this report are correct is to obtain the actual data that has not been generated at the current time. There will be a new Figure 6 required to show the plot from t>30 for fig 2 or t > 52 in Fig 5b, and see if the real data points approach the line shown a Ned or some other value. This is where more work needs to be done.
When is the Beast/virus beaten????
The major problem that society has when this model reaches the upper limit when the total number of people with the virus approaches Nap is that the population is not stationary or fixed in time. People move from isolation to the public to buy groceries, go to the Pharmacy to get medications, and any other time that they leave their location of isolation. This means that they now become part of the population that has not been infected and could be the person that starts the spread again, just like the first person. Only when there has been absolutely nobody infected for four weeks, can it be stated that the advance of the virus has been halted. If people come out of isolation in violation of the law or somebody decrees that all is over, when it is not, then the whole process can be started again if there is still any segment of society that has not acquired the virus previously.
When and how long should the Economy be shut down?
The main factor that should be considered for the time at which the economy should be shut down, schools closed, and other public gatherings, is dependant upon the capability of the Hospital system to handle the number of new cases of the virus each day that may require hospitalization and the availability of resources to take care of these new patients. The closer this number is to the capacity of the system, will reduce the amount of time the economy needs to be shut down. This is a very delicate balance between new cases/overloading the Health Care System and time to virus free. Not an easy projection. An approximation for the equation that gives the number hospitalized can be shown as:
Nh = percent hospitalized times Np/(2*tau1), this is the incremental hospital cases/day at the peak. If one assumes the rate of hospitalizations as 2% then Nh= Np/(100*4.57). Since Np is 30,000 then Nh=66/day at the peak. This is a very important number as it is the one that the Hospital System needs to know so that they have sufficient capabilities to handle this incremental load. The actual number hospitalized would be approx 2% times Np/2 minus those who died or have recovered.
Report on "Premiers get ready to reopen economies" by Brian Platt, published in Saskatoon Star Phoenix, 22 Apr. 2020
This report starts out with the headline as shown
above and develops the case showing the smaller provinces are past their peak
but cases mount in ON and QC which have not reached any peak. However, based on
the analysis that has been presented clearly shows that when all the provinces
are analyzed at once and ON, QC, BC, and AB are the dominant contributors to the
data, there is a definite peak as shown in Fig 8a.
Figure 11 shows both Fig 8a, on the
left, and Fig 10, on the right, which was from the Platt report. Fig 8a is based
on the theoretical Model and shows a peak between 45-50 days from the start of
Phase 2. The time that the single provinces would have each had 30 cases can be
estimated as about day 21 days from the start of Phase 2. Both graphs use a
logarithmic scale for cases/day so a visual comparison is possible but one has to
add 21 days to the graphs in Fig 10 as the day shown as 0 is actually day 30.
The curve for BC appears to have a peak in this time period. The summary and
other explanations relevant to this discussion are found in the
last section of the Summary
shown above
The raw data that was used to determine the mathematical model was smoothed by
using a best straight line fit as previously explained. Once the mathematical
model was developed and tested then the graph in 8a was produced that showed the
peak was prior to day 67 when the second wave or another "Hot Spot" was
starting. The graphs in Fig 11 for QC, ON, AB, and BC do not show they have
reached a peak because the method of data smoothing has removed the true signal
and replaced it by noise/unwanted data. From the start of Phase 1 to the peak,
the number of cumulative number of cases is determined by adding number of
cases/day for today to yesterdays sum. When the peak is reached then the
processing of cases/day becomes the reverse of what is done for Phase 1. In
Phase 2 the ending value of the cumulative cases is the starting value and the
cases/day are subtracted from the previous total. This is the decay phase and
now instead of looking for the time required to double the number of cumulative
cases, it is the time required to halve the number of cumulative cases. If a
graph was prepared to both of these at the same time, it would be like viewing
Figure 2 and a new Figure 2a that looks like the page flipped over making a left
and right side to this figure with exponential growth on the left and
exponential decay on the right. To use any data from the left graph to smooth
data in the right graph is totally wrong as they are two totally separate
events/phases.
Societal requirements are to minimize the effect on the total system
The only way that society will be able to be virus free is to either have the total society that must work and carry on the daily functions to have been infected by the virus or to have a vaccine that protects them from getting the virus. The main thing is that in the time before the vaccine is developed and tested, there will be waves of the virus in the total population. Now the critical factor is to reduce the total economic effect while a large proportion of the total population becomes infected and recovers. This relationship can be shown to give an equation of the form K*(2/x) *Ln(x/2). The value of x to give a minimum is when d/db(1/b*ln(b) is equal to zero. The resulting equation gives a value of b=e or the optimum value for the number of waves would be 2 times e or 4.6. Since a fractional wave is not possible then 5 waves would be the best. This assumes that the Hospital System could handle the number of cases without being overloaded.
To Flatten the Curve or not?
The main criteria should be to handle the number of virus cases in the minimum time and not overload the Health Care System while doing so. As explained earlier, the total number of cases will be the same no matter how high or flat the curve is. What lowering the curve requires is that the time required to have the number of cases double or increase by a factor of ten is made longer as the curve is lowered. Since the number of cases are the same in each situation, the total number of decades on the semi-log plot required to go from the first case to the final number of cases/2 is equal to half the duration time of the main part of the epidemic. The more flattened your curve is the longer this time will be.
Total time for Epidemic to grow and die out
The time spent in the initial and ending portions of the epidemic are shown in Fig 5a. The additional time at the beginning is 27 days and it is being assumed that it would also be 27 days at the end. Anything that will help to reduce these times would be of major benefit in reducing the total time. Anything that extends these times is counter productive and should be avoided. Once the curve for your data starts to bend away from the straight line theoretical equation as shown in fig 5B, you know that the value of Np/2 is near. This value is not essential as you can see that it is somewhere in the decade above the last points of actual data. From Fig 5b, this time is in the range of 40-50 days. Double this value is the duration of the main epidemic which would be 80-100 days. Add the two end of 27 days each for a total of 134 to 154 days plus add another month to prove that nothing was missed. The total time is 4-5 months plus one additional month as a safety factor. Opening the economy or releasing restraints prior to this could just be asking for the epidemic to start spreading again. However, this may also be incorrect according to recent information on "Herd Immunity" where the policy should be to let the people acquire the immunity by letting the virus spread without trying to stop it.
Public Policy that closed down Society, Schools, and Social Gatherings
The current discussion is about the timing of how Society was closed down and it effected the outcome during the epidemic. When the model equation is looked at, it depends upon the value of Nd but the growth of the epidemic does not depend upon the value of Nd, only the decay portion which started when the number of people infected was equal to Nd/2. This means that the policy that reduced the value of Nd had 58 days to be applied for Canada. This means that Government Policy had lots of time to react and the speed at which they did, had no effect, provided they did it within slightly less than two months from the first case.
What effect did Public Policy have on the spread of Covid-19
When did the Policy to shutdown the school and work happen? Was it around March 11? In Fig 2 the actual data differs from the straight line for four (4) days then the actual data is right back on the straight line. If this disturbance was cause by the shutdown then it had no overall effect. However, the value of Np, the total number of cases, could be a function of this policy, however, it might also be from the total population of Canada. If Np id determined by the population then the shutdown had no effect on the outcome but has caused great economic problems. This may really shake the boat but needs to be looked into.
Is the Social Policy of maintaining a distance of six feet between contacts sufficient?
To understand if this policy is sufficient let's propose a little experiment. Consider the following: We have an empty building like a garage that is 24x24 feet available to our experiment. We place a total of nine stools in the space each with a distance of twelve feet between them, i.e. twice the recommended distance. Now we get nine volunteers to sit on the stools, each one has a cloth face mask, blindfold all of them, and give all of them a hand-held unit with two buttons on it so that they can push one of the buttons when they can sense a smell. The responses to the buttons being pushed is recorded is a time-stamped record that shows their location in the garage. Now for the source of our smell test, One person in a corner position is allowed to remove their blindfold and face mask and the are given a vaping device(s) that will produce smoke for the duration of the testing. The other person will be in the center of the room and they also remove their face mask and blindfold and they will chew one or more whole garlic cloves. If they only chew one then a garlic press could be used to crush the remaining cloves. The two people with the sources of odour start at time T0 which is the beginning of the experiment when the recorded responses come in from the remaining persons. If all of those who are wearing a mask report two smells then the experiment could end. At the end of 50 minutes masks should be removed for the last ten minutes. At the end of one hour the experiment ends and the air in the room is viewed to se how clear it is. I'll bet it is so thick with smoke that you could cut it with a knife. This shows that when each one in the room breaths, they are inhaling what is in the air around them and that includes what others have exhaled. If the face masks worked at 100% efficiency then nobody would have recorded smelling either the vape or garlic odour. The time-stamped record would tell how long it took for the odour to move from the source to the location where the recipient could smell it.
What results if the conditions are lessened before there are no reported cases for 30 days?
If social gatherings or workplace gatherings take place when there is a possibility that a person with the virus could be present and infect others then all people at the gathering must self isolate for two weeks and be tested to prove they have not been infected. This would be repeated for all gatherings until the conditions are met of zero cases in the area for one month. The people who make the decision to open prematurely bear a great responsibility in this case as it was their decision that caused the new outbreak if one happens as a result of the gathering. If people have gatherings that are unapproved then they are the ones responsible.
Can anyone be blamed for this epidemic and it's arrival in your area, province/state, or country?
The only way that it is possible to prevent the spread is to be isolated from contact with anyone who has the virus, that is the first and simple fact. That means the person responsible for getting the virus is you at the local level, or anything that allows movement of people or goods and services at the province/state level and this can be extended to the country level. This means that all international travel would have to be banned and all importation or export of goods and services would have to be banned. This would mean that each city, province, country, and continent would have to be isolated which is completely impossible. In all of these cases it is not someone else's fault but yours. Any who try a point a finger at someone else as the cause only need to bend their elbow and now it is pointing at themselves and that is the person who is the hardest to blame but is the one responsible.
Using this approach with other data sets
Since it is known that the spread of the virus is exponential growth and decay, any other data that is derived from the cases should analyzed using the same approach. Number of persons hospitalized or number of deaths should not be plotted as number per day but the total number at a given date. This is the smoothing process as those who are hospitalized or die per day has more noise or variation than the number of cases to date. When the peak hospitalizations or deaths is reached would be determined in the same way as shown for the number of cases. An example of noisy data is shown in Fig-07 that shows the number of cases/day. It is impossible to see where or if any peak has been reached as the noise masks the information.
This page is under development so please be patient.
This page was last modified on 07 Jan 2022 03:23:07 PM
Arnie Krause
I am showing three different ways that you can contact me by email. This is being done this way right now as I am in the process going to the method shown in C.
A) Email: [email protected] The first word in your subject should be "Covid-19"
B) Email using the normal method. Send me an email (notice the #Covid-19 after my email address. That is automatically placed into the Subject area.
C) Go to the following link: email-me Please add "Covid-19" to the Subject area in your email. Thank You.
Last updated: 07 January 2022
