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## Comments

Made multiple updates.

For anyone who wants to lend a hand, there are many other references in our Corona discussion which deserve to be added to this syllabus / bibliography.

`Made multiple updates. For anyone who wants to lend a hand, there are many other references in our [Corona discussion](https://forum.azimuthproject.org/discussion/377/logistic-equation-with-recent-commentary-on-corona-virus) which deserve to be added to this syllabus / bibliography.`

via Twitter @johnurbanik is setting up a project and said it is OK to advertise it here

His goal is to develop better models and tooling, and implement an expanded compartmental (as compared to SEIR) model.

https://github.com/understand-covid/proposal

`via Twitter @johnurbanik is setting up a project and said it is OK to advertise it here His goal is to develop better models and tooling, and implement an expanded compartmental (as compared to SEIR) model. https://github.com/understand-covid/proposal`

Some of these requests for help are almost desperate

https://epcced.github.io/ramp/

The bolded part is the important part, in so far as applying concepts from other disciplines as I commented elsewhere.

`Some of these requests for help are almost desperate https://epcced.github.io/ramp/ > RAPID ASSISTANCE IN MODELLING THE PANDEMIC: RAMP >This is a nationwide call for rapid assistance in modelling the pandemic (RAMP), addressed to specialists in any or all of the above areas, and indeed to the scientific modelling community more widely. **Possible assistance could include advice on importing modelling elements from other research domains** ; undertaking the software engineering needed to port vastly enlarged datasets into existing pandemic models; data analytics to create predictive empirical models from real-world data; offering new perspectives on existing modelling strategies; and adding to human and computing resources more generally. Another role could be to review and filter the numerous COVID-relating modelling efforts from scientists in other fields that are already starting to appear online, feeding through to SPI-M and/or other bodies, contributions that might have substantial impact on planning. > This call for assistance is addressed to the wider modelling community (including data analytics) in academia and industry. Our initial focus is on the UK community but this is an international emergency and we welcome contributions from non-UK based scientists, while realizing that they might wish to prioritize any similar initiative in their own countries. The bolded part is the important part, in so far as applying concepts from other disciplines as [I commented elsewhere](https://forum.azimuthproject.org/discussion/comment/22013/#Comment_22013).`

Hi David and Paul, I wonder if you know of a wiki or site online where I might share a simple model that I've developed for understanding how the coronavirus will play out in various countries.

The basic idea is that the ratio "New Cases (in the last day) / Total Cases" is very helpful for understanding the nature of the exponential growth in cases. On March 23, I wrote a letter, which you can find at that page, explaining how dire the situation is. The main point is that only China and South Korea have managed to get the virus under control, and they did it by chasing it down, which means finding absolutely every sick person and all of their contacts, and isolating them.

It's simply not enough to have a quarantine. In the best case, Italy has achieved a reduction down to a growth rate 6%. But that still means doubling every 12 days. Which means that just about everybody in the country will get sick. The US has brought its growth rate down from 25% to 13%, but it is still means doubling every 6 days. How low can these rates go? The list of countries lets us study how the actions taken match up with the lower growth rate achieved.

I'm surprised that this number is not referenced as it seems to be the most informative number to look at, I think. So I'm wondering where to share that. I thought there was a wiki for coronavirus volunteers but I can't find it now.

`<img src="http://www.ms.lt/derlius/20200331-NewCasesOverTotalCases-Coronavirus.png"> Hi David and Paul, I wonder if you know of a wiki or site online where I might share a <a href="http://www.ms.lt/sodas/Book/Coronavirus">simple model</a> that I've developed for understanding how the coronavirus will play out in various countries. The basic idea is that the ratio "New Cases (in the last day) / Total Cases" is very helpful for understanding the nature of the exponential growth in cases. On March 23, I wrote a letter, which you can find at that page, explaining how dire the situation is. The main point is that only China and South Korea have managed to get the virus under control, and they did it by chasing it down, which means finding absolutely every sick person and all of their contacts, and isolating them. It's simply not enough to have a quarantine. In the best case, Italy has achieved a reduction down to a growth rate 6%. But that still means doubling every 12 days. Which means that just about everybody in the country will get sick. The US has brought its growth rate down from 25% to 13%, but it is still means doubling every 6 days. How low can these rates go? The list of countries lets us study how the actions taken match up with the lower growth rate achieved. I'm surprised that this number is not referenced as it seems to be the most informative number to look at, I think. So I'm wondering where to share that. I thought there was a wiki for coronavirus volunteers but I can't find it now.`

Andrius. The New Cases/Total Case plotted against Total Cases linearizes the logistic curve as shown on the other Azimuth thread

https://forum.azimuthproject.org/discussion/comment/21989/#Comment_21989

`Andrius. The New Cases/Total Case plotted against Total Cases linearizes the logistic curve as shown on the other Azimuth thread https://forum.azimuthproject.org/discussion/comment/21989/#Comment_21989`

https://johncarlosbaez.wordpress.com/2020/03/31/how-scientists-can-help-fight-covid-19

`https://johncarlosbaez.wordpress.com/2020/03/31/how-scientists-can-help-fight-covid-19`

`* Jeff Stern, [How open-source software is fighting COVID-19](https://opensource.com/article/20/3/open-source-software-covid19).`

Paul and David, Thank you for the links to John and Jeff's posts about how volunteers are helping!

`Paul and David, Thank you for the links to John and Jeff's posts about how volunteers are helping!`

Paul, thank you for pointing me to your comments about Hubbert Linearization. Do you think that model is valid here? How would you argue validity?

My personal viewpoint is that I'm skeptical that such linearity will hold here. I see two cases: either we can and do chase down each and every case, or we can't and don't. If we don't chase down every case, then we are simply slowing down the exponential spread, but it is still exponential. In the chart I made, only South Korea and China seem to have a chance of having snuffed out the virus. In the case of Singapore, Taiwan, and Japan, the growth rates continue at 6% or higher. Italy has brought it down to 6%. But what is the reason for thinking that it can be brought down even lower? That is still doubling every 12 days or so.

Dr.Birk showed a graph that the New Cases per day in Italy has dropped. But that may be simply a dip in the rate of growth. If the rate of growth does not dip further, then the exponential growth is maintained. Basically, I am saying that the virus spreads perniciously and if you don't chase it down, it simply won't snuff itself out until it has infected most everybody.

`Paul, thank you for pointing me to your comments about Hubbert Linearization. Do you think that model is valid here? How would you argue validity? My personal viewpoint is that I'm skeptical that such linearity will hold here. I see two cases: either we can and do chase down each and every case, or we can't and don't. If we don't chase down every case, then we are simply slowing down the exponential spread, but it is still exponential. In the chart I made, only South Korea and China seem to have a chance of having snuffed out the virus. In the case of Singapore, Taiwan, and Japan, the growth rates continue at 6% or higher. Italy has brought it down to 6%. But what is the reason for thinking that it can be brought down even lower? That is still doubling every 12 days or so. Dr.Birk showed a graph that the New Cases per day in Italy has dropped. But that may be simply a dip in the rate of growth. If the rate of growth does not dip further, then the exponential growth is maintained. Basically, I am saying that the virus spreads perniciously and if you don't chase it down, it simply won't snuff itself out until it has infected most everybody.`

I made a spreadsheet with some calculations on when the peak might occur given a particular growth rate. Consider countries currently ranging from a 2000 cases per million (Spain) to 200 cases per million (Turkey). If a 6% rate is achieved, then the peak will be reached in 13 weeks (Spain) to 18 weeks (Turkey).

My main point is that I think that the various quarantine policies may slow the exponential growth but it still remains exponential until it reaches the peak. I am curious if I'm wrong. If I'm not wrong, then it would be good to see more realistic projections for what will happen. But, in any event, the projections do seem to be more realistic nowadays.

Is there any point to making such calculations?

`I made a <a href="http://www.ms.lt/derlius/20200331-CoronavirusData.ods">spreadsheet</a> with some calculations on when the peak might occur given a particular growth rate. Consider countries currently ranging from a 2000 cases per million (Spain) to 200 cases per million (Turkey). If a 6% rate is achieved, then the peak will be reached in 13 weeks (Spain) to 18 weeks (Turkey). <ul><li>3%: from 26 to 36 weeks with 9% of population sick at peak <li>4%: from 20 to 27 weeks with 12% <li>5%: 16 - 22 with 14% <li>6%: 13 - 18 with 17% <li>7%: 12 - 16 with 19% <li>8%: 10 - 14 with 21% <li>9%: 9 - 12 with 23% <li>10%: 8 - 11 with 24% <li>11%: 8 - 10 with 26% <li>12%-13%: 7 - 9 with 28% <li>14%-16%: 6 - 7 with 31% <li>17%-20%: 5 - 6 with 35% <li>21%-24%: 4 - 5 with 38% <li>25%-30%: 4 with 41% </ul> My main point is that I think that the various quarantine policies may slow the exponential growth but it still remains exponential until it reaches the peak. I am curious if I'm wrong. If I'm not wrong, then it would be good to see more realistic projections for what will happen. But, in any event, the projections do seem to be more realistic nowadays. Is there any point to making such calculations?`

In summary, I have seen a lot of wishful thinking, dubious models and misinterpretations of the S-curve model. What happened in China and South Korea had nothing to do with the logistic curve, but there was this notion that other countries will automatically have the same results without the effort of chasing down the virus. A logistic curve only kicks in when so many people have gotten sick that there are few left to get infected. Basically, a logistic curve only tells you how things die down after peak. Before the peak you basically have exponential growth unless you can chase down the virus and snuff it out.

That is why at this stage it is all about exponential growth rates and how they express various possible quarantine policies. So the point would be to focus on that number in policy communication. And to realize that the lower the number, the longer the quarantine, which if we are lucky, would be four months to the peak, and perhaps another four months after that. And that the world will be compartmentalized for quite a bit longer than that. And that the developing world will likely have higher growth rates and a huge catastrophe within four to six weeks. It would not be suprising to have death rates of 5% or whatever is the number of acute cases that need intensive care which won't be available. And to realize that there is no way to avoid most everybody on the planet getting sick unless the virus is completely chased down, which seems less and less possible.

I think this is simple math, ordinary mathematical thinking, which I suggest could be one main contribution of the Azimuth Project. Distilling ideas like a propagation rate, a carbon budget, a tipping point, etc.

`In summary, I have seen a lot of wishful thinking, dubious models and misinterpretations of the S-curve model. What happened in China and South Korea had nothing to do with the logistic curve, but there was this notion that other countries will automatically have the same results without the effort of chasing down the virus. A logistic curve only kicks in when so many people have gotten sick that there are few left to get infected. Basically, a logistic curve only tells you how things die down after peak. Before the peak you basically have exponential growth unless you can chase down the virus and snuff it out. That is why at this stage it is all about exponential growth rates and how they express various possible quarantine policies. So the point would be to focus on that number in policy communication. And to realize that the lower the number, the longer the quarantine, which if we are lucky, would be four months to the peak, and perhaps another four months after that. And that the world will be compartmentalized for quite a bit longer than that. And that the developing world will likely have higher growth rates and a huge catastrophe within four to six weeks. It would not be suprising to have death rates of 5% or whatever is the number of acute cases that need intensive care which won't be available. And to realize that there is no way to avoid most everybody on the planet getting sick unless the virus is completely chased down, which seems less and less possible. I think this is simple math, ordinary mathematical thinking, which I suggest could be one main contribution of the Azimuth Project. Distilling ideas like a propagation rate, a carbon budget, a tipping point, etc.`

Not sure how to do that. What I pointed to is simply a mathematical identity relating to the use of the logistic equation. If the data doesn't follow a logistic function, then the linear relationship no longer holds.

I have been applying very similar compartmental models to the discipline of modeling oil depletion for the past 15 years. The logistic function comes up quite often in both discussion of epidemiology and peak oil (where the logistic is known as the Hubbert Curve). This morning I finished a blog post comparing the two modeling disciplines -- https://geoenergymath.com/2020/04/01/the-oil-shock-model-and-compartmental-models/ .

I have learned quite a bit from that experience and so can pick up the SIR contagion modeling approach handily, yet by the same token, I understand where the weaknesses are and the deliberative process that is required to do a comprehensive analysis. For example, I don;t know what is harder to do: estimate the peak date in peak oil, or estimate what the final herd immunity level or peak infected population level is in a pandemic. Both are very sensitive to corrective actions and stochastic side-effects.

`> "Do you think that model is valid here? How would you argue validity?" Not sure how to do that. What I pointed to is simply a mathematical identity relating to the use of the logistic equation. If the data doesn't follow a logistic function, then the linear relationship no longer holds. I have been applying very similar compartmental models to the discipline of modeling oil depletion for the past 15 years. The logistic function comes up quite often in both discussion of epidemiology and peak oil (where the logistic is known as the Hubbert Curve). This morning I finished a blog post comparing the two modeling disciplines -- https://geoenergymath.com/2020/04/01/the-oil-shock-model-and-compartmental-models/ . I have learned quite a bit from that experience and so can pick up the SIR contagion modeling approach handily, yet by the same token, I understand where the weaknesses are and the deliberative process that is required to do a comprehensive analysis. For example, I don;t know what is harder to do: estimate the peak date in peak oil, or estimate what the final herd immunity level or peak infected population level is in a pandemic. Both are very sensitive to corrective actions and stochastic side-effects.`

This is a google doc spreadsheet listing compartmental models for epidemic simulation

https://docs.google.com/spreadsheets/d/1hUZlVDPfa5C8KgURoP_3dAiUQgI6rdb7A5e_g8NcPaY/

There are many ways to simulate, the spreadsheet has columns for languages, approaches, etc

`This is a google doc spreadsheet listing compartmental models for epidemic simulation https://docs.google.com/spreadsheets/d/1hUZlVDPfa5C8KgURoP_3dAiUQgI6rdb7A5e_g8NcPaY/ There are many ways to simulate, the spreadsheet has columns for languages, approaches, etc`

Hi David and Paul,

On April 9, the Lithuanian web portal Alkas.lt published my article, "Logic and reality indicator for determining how well the coronavirus is being contained".

I want to suggest here at the Azimuth Project that we might have a significant impact by helping develop and promote mathematical thinking in public policy. Ideas such as "tipping point", "carbon budget", "flattening the curve", can make all the difference. Of course, they can also lead us astray. The idea of "flattening the curve" and references to "the peak", in my mind, often suggest wishful thinking that we just need to slow the pandemic down and it will magically go away. The point of my article is to argue that it won't magically go away and we will have to do massive testing of the population before we can loosen the quarantine. A slogan could be "Wait for the Test".

I machine translated my article and created a page: Coronavirus Policy View 1. My thought is that the Azimuth Project could host many views, possibly contradictory, with the aim of encouraging, developing and promoting mathematical thinking. This would be material for people to use in writing opinions and editorials or simply communicating their own views.

What do you think?

`Hi David and Paul, On April 9, the Lithuanian web portal Alkas.lt published my article, "Logic and reality indicator for determining how well the coronavirus is being contained". I want to suggest here at the Azimuth Project that we might have a significant impact by helping develop and promote mathematical thinking in public policy. Ideas such as "tipping point", "carbon budget", "flattening the curve", can make all the difference. Of course, they can also lead us astray. The idea of "flattening the curve" and references to "the peak", in my mind, often suggest wishful thinking that we just need to slow the pandemic down and it will magically go away. The point of my article is to argue that it won't magically go away and we will have to do massive testing of the population before we can loosen the quarantine. A slogan could be "Wait for the Test". I machine translated my article and created a page: <a href="https://www.azimuthproject.org/azimuth/show/Coronavirus+Policy+View+1">Coronavirus Policy View 1</a>. My thought is that the Azimuth Project could host many views, possibly contradictory, with the aim of encouraging, developing and promoting mathematical thinking. This would be material for people to use in writing opinions and editorials or simply communicating their own views. What do you think?`

Placed the compartmental model mathematical analogy on the POB blog:

http://peakoilbarrel.com/the-oil-shock-model-and-compartmental-models/

received 360 comments so far, and many people get the idea.

"Flattening the curve" has almost opposite contexts -- in a pandemic the spike in the curve is related to out of control growth whereas immediate flattening in production is due to an unforeseen suppressive shock. The price of oil is down to $12 per barrel today, which no one could have predicted at the beginning of the year.

Oil Prices Have Dropped Below $12. Crude’s Problems May Get Worse Before They Get Better.

`Placed the compartmental model mathematical analogy on the POB blog: http://peakoilbarrel.com/the-oil-shock-model-and-compartmental-models/ received 360 comments so far, and many people get the idea. "Flattening the curve" has almost opposite contexts -- in a pandemic the spike in the curve is related to out of control growth whereas immediate flattening in production is due to an unforeseen suppressive shock. The price of oil is down to $12 per barrel today, which no one could have predicted at the beginning of the year. [Oil Prices Have Dropped Below $12. Crude’s Problems May Get Worse Before They Get Better.](https://www.barrons.com/articles/oil-futures-prices-have-dropped-51587387499)`

A review in the BMJ of SARS-CoV-2 diagnostic and outcome prognosis preprints and peer-reviewed studies in a collaboration between european researchers and the Cochrane Prognosis Group finds them all subject to mostly sampling bias and clinically unusable.

`A [review in the BMJ](https://bit.ly/34YyXfz) of SARS-CoV-2 diagnostic and outcome prognosis preprints and peer-reviewed studies in a collaboration between european researchers and the Cochrane Prognosis Group finds them all subject to mostly sampling bias and clinically unusable.`

Solution to century-old math problem could predict transmission of infectious diseases.

`[Solution to century-old math problem could predict transmission of infectious diseases](https://phys.org/news/2020-05-solution-century-old-math-problem-transmission.html).`

Wow, that's great news!

`Wow, that's great news!`

That is a solution to a lattice random walk (LRW) in a confined space. It would likely find a better application to modeling epitaxial crystal growth for nanostructures. See this research I did years ago

The diffusional confinement is very evident here, as the step edges provide the confinement boundary conditions. I have a program in appendix F of my thesis to generate a diffusional concentration profile that can be compared to the closed-form solution. BCF stands for the Burton-Cabrera-Frank model. One issue I see is that these are absorbing boundary conditions and not reflecting boundary conditions (but then again isn't a contagion model absorbing?)

If someone wants to work on this at an applied level, I am up for it. I have other diffusion models that I was able to convert to an analytical closed-form solution, and it is a real benefit to do this because then the numerical iteration is not needed.

`That is a solution to a lattice random walk (LRW) in a confined space. It would likely find a better application to modeling epitaxial crystal growth for nanostructures. See [this research I did years ago](https://link.springer.com/chapter/10.1007/978-3-642-73632-2_2) ![](https://pbs.twimg.com/media/EZRsxVaWkAAnReW.png) The diffusional confinement is very evident here, as the step edges provide the confinement boundary conditions. I have [a program in appendix F of my thesis](https://books.google.com/books?id=QbKN59MGbrUC&dq=pukite+thesis+diffraction&source=gbs_navlinks_s) to generate a diffusional concentration profile that can be compared to the closed-form solution. BCF stands for the Burton-Cabrera-Frank model. One issue I see is that these are absorbing boundary conditions and not reflecting boundary conditions (but then again isn't a contagion model absorbing?) ![](https://pbs.twimg.com/media/EZRxAAMXkAYWXSZ.png) If someone wants to work on this at an applied level, I am up for it. I have other diffusion models that I was able to convert to an analytical closed-form solution, and it is a real benefit to do this because then the numerical iteration is not needed.`

Understanding epidemiology models from Ars Technica.

`[Understanding epidemiology models](https://arstechnica.com/science/2020/06/understanding-epidemiology-models/) from Ars Technica.`

Coronavirus in the U.S.: Latest Map and Case Count from The New York Times.

`[Coronavirus in the U.S.: Latest Map and Case Count](https://www.nytimes.com/interactive/2020/us/coronavirus-us-cases.html) from The New York Times.`