Since R0 is not an invariant, it’s meaning changes from onset to offset. When the S curve is nearly level, R0 is essentially determined by the shape parameter, and the I curve behaves exponentially. There seems to be a law of diminishing returns going on. You can change parameters to extend the tail, hoping to fit data better, but you can not change the fundamental shape of the curve. The tail will always be strongly exponential. With data from US, Italy, UK, and others, it looks like the tail is more linear. I think I’ve heard Fauci use the word “plateau”, which sounds apt to me. If the decrease in daily cases does indeed follow a roughly linear decrease, then R0 no longer really applies. Something else is happening. Per the other idea of using PDEs (for which we have not seen any real calculations), it is possible that local outbreaks roughly described by SIR are superimposed into a less recognizable curve. I am still interested to do more data analysis, and plan to compare outbreak shapes and durations in the near future, but will wait until it’s mostly quelled. Then we can say more surely how and where SIR does/does not apply. Currently, I’m skeptical. Thanks for the article, avoid singing if possible! —Brad
On May 20, 2020, at 2:06 PM, Michael Kleber <michael.kleber@gmail.com> wrote:
This article has some good reporting on COVID researcher attention to modeling spreading events using a gamma distribution with mean R0 and dispersion k, including empirical estimates of the parameters and discussion of policy implications due to the importance of outliers on overall spread.
https://www.sciencemag.org/news/2020/05/why-do-some-covid-19-patients-infect...
--Michael
On Wed, May 20, 2020 at 2:41 PM Cris Moore via math-fun < math-fun@mailman.xmission.com> wrote:
short version: even for e.g. Poisson distributions the outbreak size has a heavy tail when R0 is close to 1. And with superspreaders the tail gets heavier.
Here is a piece where I did some simulations, and some generating function calculations, on various branching processes, showing that outbreaks are often much larger than the average 1/(1-R0). For my sins this got me quoted today in the Wall Street Journal…
https://www.santafe.edu/news-center/news/transmission-t-024-cristopher-moore... < https://www.santafe.edu/news-center/news/transmission-t-024-cristopher-moore...
- Cris
On May 20, 2020, at 12:35 PM, Henry Baker <hbaker1@pipeline.com <mailto: hbaker1@pipeline.com>> wrote:
Hi Cris:
Thanks very much for the link to this paper, which has a much more sophisticated model than the usual Ross/Kermack-McKendrick differential equation models.
I was just listening to a discussion on YouTube between Nassim Taleb (Mr. "fat tail" himself), and Yaneer Bar-Yam of the New England Complex Systems Institute.
Taleb's intuition is that with a 'fat tail' for R0, the larger values completely dominate the smaller values, so that instead of using mean(R0), we should use max(R0). In essence, the "super-spreaders" completely dominate the behavior, and neutralizing them can dramatically change the dynamics. Simultaneously, neutralize the most vulnerable members of the population, so that they don't get sick & die (although due to their early demise, their own R0's tend to be quite small).
Yaneer Bar-Yam says that it is more complicated than that, depending upon the % of super-spreaders.
Since Cris recommends the use of generating functions, I'm going to try to come up with a simple Maxima experiment to see how some of these "wide variance" models actually behave.
At 01:59 PM 4/11/2020, Cris Moore via math-fun wrote:
I hsve a little experience in these network models of epidemics, and friends of mine have much more.
The problem is that we don't know the network, since we don't know what types of links are relevant to the disease.
Prolonged exposure (as in the meetings you name) is clearly very effective, and many organizations are using 30 minutes at 6 feet or less; but this is just a guess.
SARS also spread a lot from touching surfaces.
That said, I totally agree with you that focusing on R_0, which is just a mean, is misguided.
Here's a paper that shows that the variance is often more important than the mean (and the tail, if heavy, even more so):
https://arxiv.org/abs/2002.04004 <https://arxiv.org/abs/2002.04004>
That also said, some of the more sophisticated models do try to take this kind of thing into account, along with demographics, regional variation, etc.
Cris
On Apr 11, 2020, at 12:44 PM, Henry Baker <hbaker1@pipeline.com> wrote: Perhaps the first order of business is to strangle the old differential equation model(s) in the cradle, and start using *Monte Carlo* simulations of network-type models.
Given the ubiquity of computing power these days, there's no excuse for restricting the search for the "car keys" only near the differential equation "lamp-post".
At 11:18 AM 4/11/2020, Fred Lunnon wrote:
No doubt you're right about all this --- but any sharpening requires further assumptions about the distribution of R_e among the population, for which any credible model is presumably currently unavailable. WFL
On 4/11/20, Henry Baker <hbaker1@pipeline.com> wrote: > I've been reading about Covid19 and listening to a number > of professional podcasts about Covid9 -- e.g., "This Week > in Virology", which amazingly enough, is entertaining > enough not to be a soporific. > > Reading between the lines, I'm coming to the conclusion > that the concept of "R_e" ("effective" R), may be fatally > flawed, as it tries to capture some sort of "mean" or > "average" R in an inherently exponential setting. > > Thus, if one person has an R_e of 0.9 and another person > has an R_e of 40.0, there isn't a good way to average the > two R_e's to compute a composite R_e. What if a vanishingly > small fraction of infected people induce the vast majority > of cases? We know that spreading is a *network* phenomenon > which means that it is highly likely to have a *fat tail* > distribution. > > If I'm correct about this, then the standard epidemic > model is also fatally flawed, and hence unreliable for > making trillion-dollar decisions. What won a Nobel > prize a century ago is today an undergraduate homework > exercise. We desperately need a better class of models > for today's pandemics. > > For example, in Massachusetts in the early going, almost > 100% of the confirmed cases stemmed from a single meeting > of a single company in a single downtown Boston hotel. > > As another example, in Chicago, most of the early cases > stemmed from a single funeral and a single birthday > party. > > So if it is true that there are "super spreaders", both > in terms of individual people and/or individual events, > there must be a better way to quickly identify and > isolate these people and events other than putting the > entire world on lockdown. >>>> I would imagine that some of the work on fractals should > be useful to look at this superspreader phenomenon. >>>> Perhaps fractures in glass or metal may have relevance, > as it may only take a single flaw in a crystal structure > to destroy the entire structure.
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