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## Comments

@WebHubTel wrote:

This point is now being substantiated and amplified here:

Note the points about the significance of large droplet transmission, and nasal tissues as a key site for the infection.

`@WebHubTel wrote: > Why was the wearing of face masks by doctors in the USA not encouraged? Every doctor interviewed by the media said they were not that effective. How can they not be effective? Even if they reduced transmission of droplets by 30% that would be effective in reducing overall R0, and therefore growth in the logistic function. So why were they not recommended? Could it be the doctors realized people would start hoarding surgical masks in an emergency, thus reducing the amount available at hospitals? > > Why not tell people how to make their own mask? Even if was only 20% effective instead of 30% effective it would be good. Maybe this isn't done because western people don't understand stochastic thinking, but perhaps the people of the far east do? This point is now being substantiated and amplified here: * Sui Huang, Institute for Systems Biology, [COVID-19: Why we should all wear masks - there is new scientific rationale](https://medium.com/@Cancerwarrior/covid-19-why-we-should-all-wear-masks-there-is-new-scientific-rationale-280e08ceee71), medium.com. Note the points about the significance of large droplet transmission, and nasal tissues as a key site for the infection.`

@WebHubTel

In comment #43 on this thread you cited a web page on how to make a DIY mask out of paper towel and rubber bands.

I did a quick Google search on making DIY masks, and found this survey.

Not sure how authoritative this is, but it's something, and presents some concepts to consider. Cotton T-shirt was one of them. Paper towel not mentioned. A good advantage of paper towel of course it it's disposability.

Perhaps paper towel -- two layers? -- could get upvoted, on account of the significance of large droplet size?

`@WebHubTel In [comment #43 on this thread](https://forum.azimuthproject.org/discussion/comment/21974/#Comment_21974) you cited a web page on how to make a DIY mask out of paper towel and rubber bands. I did a quick Google search on making DIY masks, and found [this survey](https://smartairfilters.com/en/blog/best-materials-make-diy-face-mask-virus/). Not sure how authoritative this is, but it's something, and presents some concepts to consider. Cotton T-shirt was one of them. Paper towel not mentioned. A good advantage of paper towel of course it it's disposability. Perhaps paper towel -- two layers? -- could get upvoted, on account of the significance of large droplet size?`

There are even some elements of game theory here. Will a mask give a false sense of security, allowing more people to venture out?

`There are even some elements of game theory here. Will a mask give a false sense of security, allowing more people to venture out? <blockquote class="twitter-tweet"><p lang="en" dir="ltr">This is what <a href="https://twitter.com/nntaleb?ref_src=twsrc%5Etfw">@nntaleb</a> calls the Ludic Fallacy. These guys either did experiments in the class room or in their minds. What they are saying are FAR from the reality. At least from my experience in China, NOBODY had this false feeling of protection while wearing the masks. <a href="https://twitter.com/hashtag/RWRI?src=hash&ref_src=twsrc%5Etfw">#RWRI</a> <a href="https://t.co/c0uQCdksPv">pic.twitter.com/c0uQCdksPv</a></p>— Zhuo Xi (@birdxi1988) <a href="https://twitter.com/birdxi1988/status/1244452039498911746?ref_src=twsrc%5Etfw">March 30, 2020</a></blockquote> <script async src="https://platform.twitter.com/widgets.js" charset="utf-8"></script>`

@WebHubTel In comment #49, you talk about a "huge hole" in the technique of Hubbert Linearization. Does that consideration also apply to Hubbert Linearization for the pandemic?

Can you elaborate more on the ideas in that comment. There's terminology, context and ideas that I'm not familiar with.

`@WebHubTel In comment #49, you talk about a "huge hole" in the technique of Hubbert Linearization. Does that consideration also apply to Hubbert Linearization for the pandemic? Can you elaborate more on the ideas in that comment. There's terminology, context and ideas that I'm not familiar with.`

Cross posting three comments that I just made on the Azimuth blog:

`Cross posting [three comments](https://johncarlosbaez.wordpress.com/2020/03/31/how-scientists-can-help-fight-covid-19/) that I just made on the Azimuth blog: > Modeling each country separately leaves holes in the overall model for a pandemic. E.g. if the curve goes down, travel restrictions are lifted, and then it goes back up due to what’s happening in other countries. Compartmental models use ODEs and assume a well-mixed population. What about a multi-level approach, where each country or well-mixed region has a compartmental model with its own parameters. Then there could be transitions between the compartments in different countries, reflecting flows due to travel. This looks like a potential application of composition of open networks. Perhaps a good composition rule could produce an aggregated, abstracted compartmental model for the whole globe. Or help us in other ways to understand the dynamics of the whole. What do you think?`

`> What I just described just involves composition of deterministic networks. But to what is to extent is stochasticity fundamental to the evolution of a pandemic. E.g. a super-spreader went to a funeral and started a huge wave of disease in one region. To that extent, perhaps it makes sense to retain a stochastic Petri net framework. This could also on a regional basis, and the local networks composed into a global network. Or one could imagine hybrid models that use deterministic nets for the well mixed populations, but have stochastic regions as well. Certain parts of the network may be more critical and sensitive, and may deserve to be modeled at a more fine-grained, stochastic level. I’m thinking of individuals who have many connections. Whether or not a key politician practices social distancing could have a big ripple effect, due to the large number of social connections – and that a stochastic, individual consideration.`

`> A general underlying question here is to what extent stochastic, individual differences will actually “wash out” in the evolution of a pandemic – or whether there may still be large variances in overall development due to chance factors. To the extent that a deterministic compartmental model actually produces good predictions – then that argues that the stochastic factors do wash out.`

Huge hole in that a strict linearization only works for a single formulation, that of a logistic. In other words, its like saying that a quadratic formula only works for a quadratic equation. But not everything is a quadratic, leaving a hole for every other order.

`Huge hole in that a strict linearization only works for a single formulation, that of a logistic. In other words, its like saying that a quadratic formula only works for a quadratic equation. But not everything is a quadratic, leaving a hole for every other order.`

I pointed out that there is a way to derive precisely the logistic function via a stochastic spread in characteristics here: https://forum.azimuthproject.org/discussion/comment/21938/#Comment_21938

This essentially says that an aggegration of sub-populations will generate a logistics sigmoid if the individual populations follow a Max Entropy spread in characteristics.

I have never found anything like this elsewhere in the research literature -- trying "stochastic logistic function" on Google Scholar: https://scholar.google.com/scholar?q="stochastic+logistic+function"

`I pointed out that there is a way to derive precisely the logistic function via a stochastic spread in characteristics here: https://forum.azimuthproject.org/discussion/comment/21938/#Comment_21938 This essentially says that an aggegration of sub-populations will generate a logistics sigmoid if the individual populations follow a Max Entropy spread in characteristics. I have never found anything like this elsewhere in the research literature -- trying "stochastic logistic function" on Google Scholar: https://scholar.google.com/scholar?q=%22stochastic+logistic+function%22`

From the Azimuth blog noticed that Graham Jones has started a group on ResearchGate

If it's the same Graham Jones, he was active on this forum early on. Understandable that so many projects would start up or split off during a crisis.

`From the Azimuth blog noticed that Graham Jones has started a group on ResearchGate > "Goal: Investigating methods which use a combination of genetic and epidemiological data to make inferences about the way that SARS-CoV-2 spreads and evolves. This could include statistical inference based on mathematical models of evolution and/or machine learning approaches. The project is exploratory, aimed at bringing together researchers with different areas of expertise, and figuring out what the ‘real’ projects should be." > https://www.researchgate.net/project/Phylodynamic-methods-for-SARS-CoV-2 If it's the same [Graham Jones](https://forum.azimuthproject.org/profile/164/Graham%20Jones), he was active on this forum early on. Understandable that so many projects would start up or split off during a crisis.`

Haven't posted too much more as it really does appear that much of the predictive modeling involves aspects of game theory. In other words, high morbidity projections are used to convince the public to participate in mitigation procedures such as social distancing and partial quarantines. This then has the effect of making the projections wrong when put into practice. Then the second-guessing comes in that the original analysts failed with respect to predictive modeling. No-win on this one.

Leave with these posts over at http://peakoilbarrel.com/the-oil-shock-model-and-compartmental-models/ or https://geoenergymath.com/2020/04/01/the-oil-shock-model-and-compartmental-models/

A similar game-theoretic behavior underlies resource depletion but it is in slow-motion compared to pandemic modeling.

`Haven't posted too much more as it really does appear that much of the predictive modeling involves aspects of game theory. In other words, high morbidity projections are used to convince the public to participate in mitigation procedures such as social distancing and partial quarantines. This then has the effect of making the projections wrong when put into practice. Then the second-guessing comes in that the original analysts failed with respect to predictive modeling. No-win on this one. Leave with these posts over at http://peakoilbarrel.com/the-oil-shock-model-and-compartmental-models/ or https://geoenergymath.com/2020/04/01/the-oil-shock-model-and-compartmental-models/ A similar game-theoretic behavior underlies resource depletion but it is in slow-motion compared to pandemic modeling.`

Here's the stochastic Monte Carlo simulation of comment #11. Samples of a sequence of ensemble runs.

`Here's the stochastic Monte Carlo simulation of comment #11. Samples of a sequence of ensemble runs. ![](https://imagizer.imageshack.com/img130/8448/ddsim.gif)`

Recent paper making the rounds:

This is non-logistic growth. Power-law is essentially an acceleration over time -- a non-autonomous differential equation -- in contrast to the classical autonomous differential equation to describe exponential contagion growth followed by a logistic asymptote.

https://ltcconline.net/greenl/courses/204/firstOrder/autonomous.htm

I mentioned autonomous vs non-autonomous growth in comment above and compared them in Mathematical Geoenergy

This was an early graph of wiki page (i.e. wiki word) growth

It would be interesting to find out how many pages are added per day now, many years later

English articles: 6,067,626

https://en.wikipedia.org/wiki/Wikipedia:Size_of_Wikipedia#Graphs_of_size_and_growth_rate

https://stats.wikimedia.org/#/en.wikipedia.org/content/pages-to-date/normal|table|all|page_type~content|monthly

The power is still roughly 2 even though it appears to be slowing down. Note that this is plotted on a log-log scale -- contagion growth is plotted on a semi-log scale, which would have shown a more apparent flattened bend in the curve and earlier in the timeline.

There is an equivalent stochastic power-law "logistic" which may explain an asymptotic flattening -- as there are only so many words available in the english language to draw from (or human wikipedia editors to contribute). This is straightforward to derive in the same way that a stochastic logistic can be derived, mentioned in comment #11.

Example stochastic model fit

.. on semilog plot

I don't really know what implications this has for contagion growth modeling, as the exponential growth mechanism is well established.

`Recent paper making the rounds: > ["Strong correlations between power-law growth of COVID-19 in four continents and the inefficiency of soft quarantine strategies"](https://aip.scitation.org/doi/10.1063/5.0009454), *Chaos* This is non-logistic growth. Power-law is essentially an acceleration over time -- a non-autonomous differential equation -- in contrast to the classical autonomous differential equation to describe exponential contagion growth followed by a logistic asymptote. https://ltcconline.net/greenl/courses/204/firstOrder/autonomous.htm I mentioned autonomous vs non-autonomous growth in [comment above](https://forum.azimuthproject.org/discussion/comment/22022/#Comment_22022) and compared them in Mathematical Geoenergy > " This kind of growth has an underlying mechanism of a constant acceleration term; in other words, the rate of growth itself increases linearly with time. To first order, this explains scenarios that involve a rapidly increasing uptake of resources and particularly those that spread by word of mouth. The growth of wiki words in Wikipedia provides the best current‐day example of quadratic growth." This was an early graph of wiki page (i.e. wiki word) growth ![](https://imagizer.imageshack.com/img924/1907/onAKOm.png) It would be interesting to find out how many pages are added per day now, many years later English articles: 6,067,626 https://en.wikipedia.org/wiki/Wikipedia:Size_of_Wikipedia#Graphs_of_size_and_growth_rate [https://stats.wikimedia.org/#/en.wikipedia.org/content/pages-to-date/normal|table|all|page_type~content|monthly ](https://stats.wikimedia.org/#/en.wikipedia.org/content/pages-to-date/normal|table|all|page_type~content|monthly) ![](https://imagizer.imageshack.com/img923/6089/Hau3PR.png) The power is still roughly 2 even though it appears to be slowing down. Note that this is plotted on a log-log scale -- contagion growth is plotted on a semi-log scale, which would have shown a more apparent flattened bend in the curve and earlier in the timeline. There is an equivalent stochastic power-law "logistic" which may explain an asymptotic flattening -- as there are only so many words available in the english language to draw from (or human wikipedia editors to contribute). This is straightforward to derive in the same way that a stochastic logistic can be derived, mentioned in [comment #11](https://forum.azimuthproject.org/discussion/comment/21938/#Comment_21938). [Example stochastic model fit](https://imagizer.imageshack.com/img923/7500/1v8Ukb.png) [.. on semilog plot](https://imagizer.imageshack.com/img923/6352/5c1XS0.png) I don't really know what implications this has for contagion growth modeling, as the exponential growth mechanism is well established.`

Paul, thank you very much for posting the paper from Chaos. I would have to study the paper to understand the conditions in which exponential growth becomes power law growth. Regardless, the main conclusions are basically the same as I have argued. It's remarkable how quickly they were able to get and publish their results. I am glad that different voices are converging on similar perspectives. And that all manner of people around the world are getting involved and speaking up.

`Paul, thank you very much for posting the paper from Chaos. I would have to study the paper to understand the conditions in which exponential growth becomes power law growth. Regardless, the main conclusions are basically the same as I have argued. It's remarkable how quickly they were able to get and publish their results. I am glad that different voices are converging on similar perspectives. And that all manner of people around the world are getting involved and speaking up.`

Power-law growth of log-slope=2 is essentially a constant acceleration over time. For resource depletion, it's a fairly intuitive approach as one thinks of economic output in terms of a first-order linearly growing technology or a gradually growing population over time. When these are multiplicative the power can be greater than 2.

That's why I had originally considered this growth mode for resource depletion, as oil molecules don't act as exponential contagions -- they may provide some

autonomousfeedback in terms of self-accelerating technology growth but it's not as strong as the initially unconstrained positive autonomous feedback of virus contagion growth during an epidemic.`Power-law growth of log-slope=2 is essentially a constant acceleration over time. For resource depletion, it's a fairly intuitive approach as one thinks of economic output in terms of a first-order linearly growing technology or a gradually growing population over time. When these are multiplicative the power can be greater than 2. That's why I had originally considered this growth mode for resource depletion, as oil molecules don't act as exponential contagions -- they may provide some *autonomous* feedback in terms of self-accelerating technology growth but it's not as strong as the initially unconstrained positive autonomous feedback of virus contagion growth during an epidemic.`

The role of dispersion as described in comment #11. A single set of equations without a stochastic spread is inadequate for projection.

"

"

A tweet by Levitt, a structural biologist from Stanford

"

"

`The role of dispersion as described in [comment #11](https://forum.azimuthproject.org/discussion/comment/21938/#Comment_21938). A single set of equations without a stochastic spread is inadequate for projection. " <blockquote ><p lang="en" dir="ltr">One extra tidbit tonight:<br><br>Look at national & subnational daily deaths side-by-side.<br><br>NY daily deaths descending, but US plateau continues.<br><br>Why?<br><br>US is compound of multiple peaks like NY. As one state’s daily toll falls, another rises, keeping daily deaths high at national level <a href="https://t.co/Idz2vMcHRc">pic.twitter.com/Idz2vMcHRc</a></p>— John Burn-Murdoch (@jburnmurdoch) <a href="https://twitter.com/jburnmurdoch/status/1256717914973057024?ref_src=twsrc%5Etfw">May 2, 2020</a></blockquote> " A tweet by Levitt, a structural biologist from Stanford " <blockquote class="twitter-tweet"><p lang="en" dir="ltr">Team analysis of 17 peaked locations world-wide<a href="https://t.co/gdwfzjE0TQ">https://t.co/gdwfzjE0TQ</a><br><br>Comprehensive analysis 14Mar20 <a href="https://t.co/ufMaJRJPFg">https://t.co/ufMaJRJPFg</a><br><br>Analysis of COVID in China from 2/2 to 3/3 shows best & worst of real-time science <a href="https://t.co/Pe3k6XK28M">https://t.co/Pe3k6XK28M</a></p>— Michael Levitt (@MLevitt_NP2013) <a href="https://twitter.com/MLevitt_NP2013/status/1250009834574970880?ref_src=twsrc%5Etfw">April 14, 2020</a></blockquote> <script async src="https://platform.twitter.com/widgets.js" charset="utf-8"></script> "`

“Logistic map”: an analytical solution - fractional iteration of the logistic equation. https://www.math.tamu.edu/~berko/papers/pdf/paRBBSH95.pdf

`“Logistic map”: an analytical solution - fractional iteration of the logistic equation. https://www.math.tamu.edu/~berko/papers/pdf/paRBBSH95.pdf`

Among the armchair epidemiologists there seems to be a misguided belief that the factor (1 – 1/R0) will set the asymptote for "herd immunity" of a population. Any value of R0 above unity is exponential contagion growth and below unity it is damped. So they apparently think that keeping R0 just above unity can keep the level of infection to a fraction of the population. See this blog post (by the well-known AGW skeptic Nic Lewis hosted on the well-known AGW skeptic Judith Curry's blog)

https://judithcurry.com/2020/05/10/why-herd-immunity-to-covid-19-is-reached-much-earlier-than-thought/

So if R0 is reduced to 1.1 instead of a highly contagious 5, they think that the fractional level of effective immunity can be reduced to 1-1/1.1 ~ 0.09. Apparently they believe that this provides a rationale for society to go back to BAU.

The issue is that new carriers of disease and relaxation of community standards will almost guarantee that everyone will become exposed to coronavirus at some point. IOW, I think the skeptics are playing mathematical games with this factor and in reality, the only solution is to maintain some type of lock-down approach until a vaccine is found.

I could be wrong about this interpretation, but in dealing with compartmental models for resource depletion over the years, I have found all that matters is the available pool to draw from. By slowing down extraction rates, this does not mean that we will extract less. For contagions, the available pool is the human population, and so by the same token, slowing the infection rate down will not change the asymptotic level.

The following document is what the contrarians need to be reading, and not analyses from armchair skeptics experienced at climate war polemics:

https://wwwnc.cdc.gov/eid/article/25/1/17-1901_article

this another the key point made in the conclusions

`Among the armchair epidemiologists there seems to be a misguided belief that the factor (1 – 1/R0) will set the asymptote for "herd immunity" of a population. Any value of R0 above unity is exponential contagion growth and below unity it is damped. So they apparently think that keeping R0 just above unity can keep the level of infection to a fraction of the population. See this blog post (by the well-known AGW skeptic Nic Lewis hosted on the well-known AGW skeptic Judith Curry's blog) https://judithcurry.com/2020/05/10/why-herd-immunity-to-covid-19-is-reached-much-earlier-than-thought/ So if R0 is reduced to 1.1 instead of a highly contagious 5, they think that the fractional level of effective immunity can be reduced to 1-1/1.1 ~ 0.09. Apparently they believe that this provides a rationale for society to go back to BAU. The issue is that new carriers of disease and relaxation of community standards will almost guarantee that everyone will become exposed to coronavirus at some point. IOW, I think the skeptics are playing mathematical games with this factor and in reality, the only solution is to maintain some type of lock-down approach until a vaccine is found. I could be wrong about this interpretation, but in dealing with compartmental models for resource depletion over the years, I have found all that matters is the available pool to draw from. By slowing down extraction rates, this does not mean that we will extract less. For contagions, the available pool is the human population, and so by the same token, slowing the infection rate down will not change the asymptotic level. The following document is what the contrarians need to be reading, and not analyses from armchair skeptics experienced at climate war polemics: https://wwwnc.cdc.gov/eid/article/25/1/17-1901_article > *Complexity of the Basic Reproduction Number (R0)* > Abstract > "The basic reproduction number (R0), also called the basic reproduction ratio or rate or the basic reproductive rate, is an epidemiologic metric used to describe the contagiousness or transmissibility of infectious agents. R0 is affected by numerous biological, sociobehavioral, and environmental factors that govern pathogen transmission and, therefore, is usually estimated with various types of complex mathematical models, which make R0 easily misrepresented, misinterpreted, and misapplied. R0 is not a biological constant for a pathogen, a rate over time, or a measure of disease severity, and R0 cannot be modified through vaccination campaigns. R0 is rarely measured directly, and modeled R0 values are dependent on model structures and assumptions. Some R0 values reported in the scientific literature are likely obsolete. R0 must be estimated, reported, and applied with great caution because this basic metric is far from simple." this another the key point made in the conclusions > "R0 values are nearly always estimated from mathematical models, and the estimated values are dependent on numerous decisions made in the modeling process. "`

Interesting as the data is playing itself out, the model is changing from a logistic-based shape to a Gompertz-based bell-curve which can show a large asymmetry or skewness. The skewness can be short-tailed or long-tailed based on the growth. The skewed tail is short in the case of accelerating growth via maximal resource depletion, and is long or fat in the case of decelerated growth (compared to an exponential) via a contagion lock-down.

See this twitter thread: https://twitter.com/WHUT/status/1257338423851057159

The Gompertz is more fat-tail with decelerating growth, as citizens continue to quarantine and take preventative measures:

In contrast, here's an example of a severe resource extraction process (phosphates for fertilizer from Nauru island). Demand for phosphates shows an exponential increase on a finite supply and this is what happens -- the Gompertz short-tail in action. This is an acceleration not decel in the face of a hard constraint, which is the "herd immunity" level of completely exhausted resource levels.

So the distinction is in how society will speed head-first into natural resource depletion, but do whatever it takes to prevent a completely infected population when it comes to a medical crisis, and thus acting to suppress the rise of contagion growth and consequently prolonging the tail.

`Interesting as the data is playing itself out, the model is changing from a logistic-based shape to a Gompertz-based bell-curve which can show a large asymmetry or skewness. The skewness can be short-tailed or long-tailed based on the growth. The skewed tail is short in the case of accelerating growth via maximal resource depletion, and is long or fat in the case of decelerated growth (compared to an exponential) via a contagion lock-down. See this twitter thread: https://twitter.com/WHUT/status/1257338423851057159 The Gompertz is more fat-tail with decelerating growth, as citizens continue to quarantine and take preventative measures: > "Updated plot of NHS England and Swedish Health Service death by date (not by reporting) data. X axis is days since first death. Red - England, blue - Sweden. Lines are best fit Gompertz function. Note the growth was NEVER exponential." > ![](https://pbs.twimg.com/media/EY2bnUxWkAA3Fhv.jpg) In contrast, here's [an example of a severe resource extraction process (phosphates for fertilizer from Nauru island)](https://books.google.com/books?id=xb17DwAAQBAJ&pg=PA129&lpg=PA129&dq=phosphates+nauru+pukite+mathematical+geoenergy&source=bl&ots=oZ1ZGqVqMK&sig=ACfU3U2KwYM0hUuJrxlrlbZG3xyHOwrqWg&hl=en&sa=X&ved=2ahUKEwj03rPFv97pAhUzAp0JHRiuDzAQ6AEwAHoECAkQAQ#v=onepage&q=phosphates%20nauru%20pukite%20mathematical%20geoenergy&f=false). Demand for phosphates shows an exponential increase on a finite supply and this is what happens -- the Gompertz short-tail in action. This is an acceleration not decel in the face of a hard constraint, which is the "herd immunity" level of completely exhausted resource levels. ![](https://pbs.twimg.com/media/EXL4KDbXsAI18re.png) So the distinction is in how society will speed head-first into natural resource depletion, but do whatever it takes to prevent a completely infected population when it comes to a medical crisis, and thus acting to suppress the rise of contagion growth and consequently prolonging the tail.`

Finally, noticing some discussion on applying dispersion of rates in the analysis of COVID-19 growth

https://www.theatlantic.com/health/archive/2020/09/k-overlooked-variable-driving-pandemic/616548/

Go back to the 2nd entry in this thread describing dispersion on the logistic:

https://forum.azimuthproject.org/discussion/comment/2160/#Comment_2160

Discussion on Twitter:

`Finally, noticing some discussion on applying dispersion of rates in the analysis of COVID-19 growth https://www.theatlantic.com/health/archive/2020/09/k-overlooked-variable-driving-pandemic/616548/ > "By now many people have heard about R0—the basic reproductive number of a pathogen, a measure of its contagiousness on average. But unless you’ve been reading scientific journals, you’re less likely to have encountered k, the measure of its dispersion. The definition of k is a mouthful, but it’s simply a way of asking whether a virus spreads in a steady manner or in big bursts, whereby one person infects many, all at once. After nine months of collecting epidemiological data, we know that this is an overdispersed pathogen, meaning that it tends to spread in clusters, but this knowledge has not yet fully entered our way of thinking about the pandemic—or our preventive practices." Go back to the 2nd entry in this thread describing dispersion on the logistic: https://forum.azimuthproject.org/discussion/comment/2160/#Comment_2160 Discussion on Twitter: <blockquote class="twitter-tweet"><p lang="en" dir="ltr">When <a href="https://twitter.com/jburnmurdoch?ref_src=twsrc%5Etfw">@jburnmurdoch</a> said this was "probably the best article on Covid I've read," I had to read it immediately. It's really good, as everything from <a href="https://twitter.com/zeynep?ref_src=twsrc%5Etfw">@zeynep</a> seems to be.<a href="https://t.co/BHVQA2WNX3">https://t.co/BHVQA2WNX3</a></p>— Paul Graham (@paulg) <a href="https://twitter.com/paulg/status/1314990334682763264?ref_src=twsrc%5Etfw">October 10, 2020</a></blockquote> <script async src="https://platform.twitter.com/widgets.js" charset="utf-8"></script>`