Friday, February 28, 2020

Applying The Mathematics Of Traveling Waves & Diffusion To The Spread Of Pandemics - What's The Takeaway Applied To COVID-19?

No photo description available.
Graph for the wave of advance for the Bubonic Plague of 1347, after Noble, J.V.: 'Geographical and Temporal Development of Plagues', Nature, 250, pp. 726-728.

As the COVID-19 virus continues its march toward being officially designated a pandemic, it is instructive to examine how pandemics spread- based on previous models.    Almost all such models are mathematically based on the concept of semi-linear diffusion, and diffusion itself (as a physics concept) is very useful in analyzing everything the spread of molecules of gas released in a container, to the behavior of plagues, to the spread of galactic civilizations, e.g.

How Difficult Would It Be For Advanced Extraterrre...

But in this post I want to focus on the spread of plagues, in diffusive "waves"  such as set out by Noble (1974) wherein he devised a model for the Bubonic Plague of 1347.  For reference his starting equations were:

1)   ¶ I /  t  =  KIS  -    m I   +  D Ñ 2  

2)  ¶ S /  t  =  - KIS  +   Ñ 2  S

The problem posed by these two partial differential equations entails predicting the (diffusive) wave advance of the given plague or pandemic over time and in one linear dimension (x) - which could be km or m across E or W terrestrial longitude.  Here,  S and I denote the densities of the susceptible and infected populations (per sq. km or per sq. m).  D is the diffusion coefficient and   is the mortality rate for the given disease. The coefficient K defines the rate at which the disease is transmitted locally.

Newman and Sagan (1981) referenced in the preceding blog link, in discussing Noble's plague model, note that "there is in fact a third species" implicitly included" which is 'B' - or the density of individuals who've contracted the disease and either i) died, or ii) recovered.   The mortality rate  then describes the rate at which B grows, or writing the applicable partial:

¶ B /  t  =    B

The  total population for the Noble model can then be expressed:

ò  (I + S + B)  d 2 x

which remains constant (given B includes all those killed by the disease).   As far as we know right now, the  m -value is 2 %  which rivals the lower estimate for the 1918 Spanish flu pandemic.  To get a grounding here, if the worst seasonal flu - say H3N2 - has    = 0.5%  and has 80,000 killed in a year in the U.S. (as in 2014), then COVID-19 on the same basis of infected population potential would see 4 times that many killed, or 4 x 80,000 =  320,000.  

But it could be even worse for COVID-19  if one considers a global population of 7.9 billion, none of which have immunity - or would have any useful vaccine in time. (Estimated time to that now 12- 18 months). Hence, if the entire global population were to be infected with this virus and assuming the m -value of 2 %  doesn't change - say by mutation - then we could be looking at:

(0.02) x  7.9 x  10 9

  Or, 158 million dead across the planet.   For perspective, that compares with 50-75 million from the 1918 Spanish flu and 1 million from the Hong Kong flu in 1968.

 In the end,  all of this is still hypothetical and depends on how the equations actually work out.  Hence, zooming in on equations (1) and (2) we first would need to know the rate of change in the number of the infected within a small area (say the city limits of San Francisco or Colorado Springs) and as the rate of transitions (K I S) changes.  The latter will be arising from the susceptible population  minus the "removal rate" (- m I) from mortality - but also dispersal (Ñ 2  S).    

In other words, controlling the scale of the incipient pandemic will depend on defined regions-areas controlling the dispersal- diffusion of the susceptible population.  Since S is essentially the population of the whole planet - since there is no immunity - then one can see the Herculean task.   Basically, halting all travel afar for any of the susceptible population.  Question: How do you keep 7.9 billion people from moving into other areas and infecting others?  I mean, it's evident in the most recent case in California that we have the first incident of "community spread".      

That is, a woman who suffered with severe pulmonary and respiratory symptoms for 4 days (at VacaValley Hospital) who was then transferred to UC Davis Medical Center because the smaller hospital staff felt unequipped to deal with her symptoms. It was at UC Davis she was finally tested and was diagnosed positive with coronavirus, and is now on a respirator.  The sobering aspect is there's been no clear source infection or chain of transmission  identified. This is coupled with unprotected health  workers- at both the community hospital and UC Davis coming in contact with her.  They are isolated and now being monitored for potentially being exposed. So the question of additional exposure of healthcare workers who treated her remains as well as the original source of infection for her. (The community hospital where she first presented is less than 10 miles from Travis AFB, so it is possible HHS workers tending to a group of returned Americans at Travis AFB may have somehow infected her.   Those workers, as Rachel Maddow pointed out last night, had no protective clothing when they attended the quarantined group, and then left the base to migrate to their homes, hotels, airplanes etc.)

We came to this critical information thanks to another patriotic whistleblower, e.g.

U.S. workers lacking protective gear met evacuees, HHS whistleblower says

That is a pathway to disaster.   As Maddow put things in perspective last night:

"This president is telling people things that are not true about the coronavirus and specifically how it is being handled in the United States. And this is a problem. Again tonight the president said there were only fifteen cases of coronavirus in the United States.  It is not fifteen cases but at lest sixty cases in the U.S.  He also insists the number of cases is going to drop and become near zero... He even said the virus has a death rate like the common flu which is quite dramatically, mathematically incorrect. Insisting on that is causing confusion for people trying to understand the scale of this problem."

Last night even blathering "it's going to disappear, like a miracle"

 Making it worse, Trump in his presser two days ago insisted "it's not the right time to limit travel between countries with major outbreaks".  Given that attitude -  consistent with the disaggregation and liberal travel enabled in capitalist nations- it is a certainty the virus will spread and reach pandemic level soon.  After all, we like to believe in a democratic capitalist society we let businesses make their own decisions on whether to hold major conferences or allow international travel. The problem is that such bows to profit mongering enhance the dispersal term, Ñ 2  S.     Trump's comments effectively seek to increase that contribution to dispersal, while displaying little evidence of the ability to contain it if a major outbreak hits the U.S.  With minimal testing kits we have an invitation to disaster, and as Dr. David Agus put it on CBS this morning, "In California you have 8400 to be tested and only 200 testing kits, that doesn't equal."

As Newman and Sagan further note, and which is basically common sense - looking a the equations and variables- the rate of change of S within a small area is effectively a net loss owing to its transition over time to an infected population (I)  and a dispersed population.  The first as taken into account by the loss term (- KIS) in equation (2). and the second by the diffusive term, Ñ 2  S.     

Noble, in his paper (full citation below the graph) performed a numerical integration of the 1-dimensional analog to the equation (1) and (2).  He found this simulation quickly developed into a pair of traveling waves as shown in the graph.   Note particularly the change in the infected population I(x) and the susceptible population S(x).  The latter ultimately "plateaus" because the human "fuel" to feed it has effectively been exhausted.   The smaller wave I(x) dips to zero density (infected per square m) because those individuals have either recovered or perished from the disease.  

The main takeaway is that this kind of a semi-linear diffusion model based on a "multispecies" input delivers a set of traveling waves but differing in amplitude  and termination point.   There is no reason to believe that such analysis applied to the COVID-19 virus will be any different, after it fully plays out.  One can only hope that the infected density I(x) can be controlled, but again that will depend on the control of diffusion of the susceptible population.   If we are now seeing true instances of community spread the diffusion in the U.S. is already under way.  As this dispersion occurs we are also more likely to see a more accurate value of K, the rate of local transmission - related to R-nought- the generic transmission rate, but in a local sense.

How can we determine that?   Only 445 individuals have been tested nationwide thus far, and tens of thousands of kits are needed. California on the putative "front lines" has forty million people and a testing capacity to cover hundreds at the most.  And what are the Trumpies doing?  Lying and exaggerating about their readiness, and oh, now mandating the CDC no longer deliver any messages until they first go through Pence.  Not grasping that the COVID-19 virus has no regard or respect for puny Trumpy autocracy or bullshit.

 My best guess right now from looking at the diffusion wave model-  and reckoning in the scant existing data-  is that we could be looking at anywhere from 1 million dead worldwide to 20 million, depending on the response of different nations.  In the U.S. - by September- we could be looking at anywhere from 10,000 casualties to 500,000, depending on the preparations being laid now by the Trump government. In that estimate I have not included any deaths other than from the disease.  Hence, if ten thousand people perish in a quarantined city, say Santa Fe.  because adequate food (or water, or medical) supply lines aren't set up, that is not included.

Right now, the major uncertainties in applying the diffusion-wave model inhere in the virus' speed and the lack of preparation with testing kits - the only way to distinguish it from the flu. (Since the main symptoms include fever, sore throat, breathing problems, severe fatigue).   Hence, right now cases are falling through the cracks as the virus speeds around the world faster than screening capacity or measures can update and track it.  This will also likely affect the final value of K and  m  as well as I(x).  Prominent too, as I've noted, is how this so far incompetent Trump government plans to deal with it, especially in mustering an effective response.

What we now face - if we are to come through this with minimal damage - is to adopt a plan of action anathema to the core of rabid Trumpism.  In the words of Dr. David Agus: 

"This is a new era of caring not about yourself but your community. If you feel ill do not go to work.  Don't send your kid to school if he has a runny nose.  We should have d one that before but now it's a call for all of us to work together to be part of the solution."

See also:



"Imagine for a moment guys like Louie Gohmert or women like Joni Ernst in charge of getting food to your community when the whole interlocked web of transport is frozen by millions of sick and dying people, along with frightened people not yet infected huddling in their homes, and a nation-wide quarantine in place that has shut down most all vital supply lines.

And then imagine yourself huddled in front of your television set, desperate for news, and there's Trump, in a haz-mat suit, with tweets and reassurances that everything is under control, and Larry Kudlow, brilliant medico that he is, assuring us that the virus is actually going to turn out to be a good thing for investors and population control advocates."

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