humanfist:

nostalgebraist:

I’ve seen this post going around, so I’ll repost here what I wrote about it in a Facebook comment.

This article simply does not make sense.  Here are some of its flaws:

- It assumes the time course of the epidemic will have a Gaussian functional form.  This is not what exponential growth looks like, even approximately.  Exponential growth is y ~ e^x, while a Gaussian’s tail grows like y ~ e^(-x^2), with a slower onset – the famous “light tails” of the normal distribution – and a narrow, sudden peak.  I don’t know why you’d model something that infamously looks like y ~ e^x as though it were y ~ e^(-x^2), even as an approximation, and the author provides no justification.

- Relative to a form that actually grows exponentially, most of the mass of a Gaussian is concentrated right around the peak.  So the top of the peak is higher, to compensate for the mass that’s absent from the light tails.  Since his conclusions depend entirely on how high the peak goes, the Gaussian assumption is doing a lot of work.

- No citation is provided for 40%-to-70% figure, just the names and affiliations of two researchers.  As far as I can tell, the figure comes from Marc Lipsitch (I can’t find anything linking it to Christian Drosten).  Lipsitch derived this estimate originally in mid-February using some back-of-the-envelope math using R0, and has since revised it downward as lower R0 estimates have emerged – see here for details.

- In that Lipsitch thread, he starts out by saying “Simple math models with oversimple assumptions would predict far more than that given the R0 estimates in the 2-3 range (80-90%),” and goes on to justify a somewhat lower number.

The “simple math” he refers to here would be something like the SIR model, a textbook model under which the fraction S_inf of people never infected during an epidemic obeys the equation R_0 * (S_inf - 1) - ln(S_inf) = 0.  (Cf. page 6 of this.)

Indeed, with R_0=2 we get S_inf=0.2 (80% infected), and with R_0=3 we get S_inf=0.06 (94% infected).  So I’m pretty sure Lipsitch’s estimate takes the SIR model as a point of departure, and goes on to postulate some extra factors driving the number down.

But the SIR model, like any textbook model of an epidemic, produces solutions with actual exponential growth, not Gaussians!  There is no justification for taking a number like this and finding a Gaussian that matches it.  If you believe the assumptions behind the number, you don’t actually believe in the Gaussian; if you believe in the Gaussian (for some reason), you ought to ignore the number and compute your own, under whatever non-standard assumptions you used to derive the Gaussian.

- What’s more, he doesn’t say how his plotted Gaussian curves were derived from his other numbers.  Apparently he used the 40%-70% figure together with a point estimate of how long people spend in the ICU.  How do these numbers lead to the curves he plotted?  What does ICU duration determine about the parameters of a Gaussian?  Ordinarily we’d have some (simplified) dynamic model like SIR with a natural place for such a number, and the curve would be a solution to the model.  Here we appear to have a curve with no dynamics, somehow estimated from dynamical facts like ICU duration.

- Marc Lipsitch, on his twitter, is still pushing for social distancing and retweeting those “flatten the curve” infographics.  I suppose it’s conceivable that he doesn’t recognize the implications of his own estimate.  But that is a strong claim and requries a careful argument.

I don’t know if Lipsitch has read this article, but if he has, I imagine he experienced that special kind of discomfort that happens when someone takes a few of your words out of context and uses them to argue against your actual position, citing your own reputation and credibility as though it were a point against you.

I dislike that this sloppiness is present in the main anti-flattening article, but at the same time I have yet to hear a single flattening proponent give any sort of model based estimate for how long social distancing would have to last despite this being one of the main factors determining if flattening is a viable strategy.  And this is despite having read more flattening related articles than is probably healthy and asked this question directly on several occasions (though the shear firehose of information does mean I could have missed something).

I’ve probably read less of this stuff than you, but personally I get the sense that epidemiologists are being cautious about quoting concrete numbers because they tend to get misunderstood, misused, or just fixated on to an inappropriate degree.

The 40%-to-70% figure, for example, was a very rough estimate based on the reasoning “it should be somewhere below the number I get out of a simple SIR model, and somewhere above the numbers from 2 historical examples.”  It was based on an early estimate of R_0 that’s higher than more recent estimates, and it doesn’t capture how the outcome varies with the interventions you perform (because those change R).  But it’s still being widely quoted and used in other people’s back-of-the-envelope calculations.

I imagine that concrete numbers about social distancing, from a similarly reputable researcher or group, would likewise undergo “community spread” and acquire an aura of being “the estimate” – which could actually be a downgrade in public knowledge, insofar as the conclusion “social distancing is helpful” can be drawn much more confidently than any particular quantitative version of it.

I am not an epidemiologist myself and only know what I’ve read in the last few weeks, so take everything I say (including OP) with a correspondingly sized grain of salt, but … my impression is that model-based quantitative estimates are hard, because everything is sensitive to the details of numbers like R which interventions will change to some extent but not to an extent we can know with any quantitative precision.  Meanwhile, we have some compelling case studies – comparing US cities in 1918, or Hubei vs. the rest of China in 2020 – suggesting that social distancing works extremely well.

If we use a mathematical model, we have enough degrees of freedom (especially if it is even remotely realistic), and enough uncertainty associated with numeric inputs like R_0/R, that we can probably generate a whole range of estimates that make social distancing look relatively good/bad, short/long, etc.

Because it and other interventions will push R downward to some extent, they will not just “flatten” a constant-mass curve but actually lower the total number of people that are ever infected (yet another problem with the OP is that it ignores this!).  So very optimistic estimates about this effect could yield very optimistic conclusions, e.g. the extreme case where R gets close to 1 and the thing just fizzles out.  That extreme may feel unrealistic, but rejecting it on the grounds of “feeling unrealistic” is not a model-driven conclusion, it’s guesswork based (at best) on case studies that kind of passes through a mathematical model, superfluously, on its way to becoming a conclusion.  Might as well just skip the model and say “the case studies show you should do social distancing fast and hard,” which is what the experts are doing.  See e.g. the paper on Wuhan vs. Guangzhou by Li, Lipsitch and others, which basically says “fast and hard interventions saved Guangzhou, so they should be done in the US” without explicitly modeling what the latter might look like.

It’s reminiscent of the persistent situation in some parts of economics, where it’s easy to make memorable and memetic qualitative arguments that something is good or bad – stuff like the broad idea of gains from trade, analogous here to “flatten the curve!” – and it’s also easy to produce compelling case studies in which something appeared to succeed or fail.  But if you try to bridge the two with a more quantitative, “crunchy” math model, you have enough degrees of freedom that you can paint in virtually whatever details you want between the lines given by the other available information, or even stray outside those lines if you aren’t careful.  The tail is wagging the dog: at best you get out what you already knew, but you have to do a lot of work to even achieve that, and even then you’ll end up with the false precision of the sci-fi character who reports “the ship has a 98.7594738% chance of blowing up in the next 60 seconds.”

(Final disclaimer: again, I am not an epidemiologist!!)

invertedporcupine:

The simulations are fairly mesmerizing to watch.

These are illustrations of the SIR model I was talking about earlier, BTW.  The “S,” “I” and “R” in the name are the three different colors.

[EDIT: hello SSC readers!  This is a post I wrote quickly and with the expectation that the reader would fill in some of the unstated consequences of my argument.  So it’s less clear than I’d like.  My comment here should hopefully clarify things somewhat.]

———————–

[EDIT2: people seem really interested in my critique of the Gaussian curve specifically.

To be clear, Bach’s use of a Gaussian is not the core problem here, it’s just a symptom of the core problem.

The core problem is that his curves do not come from a model of how disease is acquired, transmitted, etc.  Instead they are a convenient functional form fitted to some parameters, with Bach making the call about which parameters should change – and how much – across different hypothetical scenarios.

Having a model is crucial when comparing one scenario to another, because it “keeps your accounting honest”: if you change one thing, everything causally downstream from that thing should also change.

Without a model, it’s possible to “forget” and not update a value after you change one of the inputs to that value.

That is what Bach does here: He assumes the number of total cases over the course of the epidemic will stay the same, whether or not we do what he calls “mild mitigation measures.”  But the estimate he uses for this total – like most if not all such estimates out there – was computed directly from a specific value of the replication rate of the disease.  Yet, all of the “mild mitigation measures” on the table right now would lower the replication rate of the disease – that’s what “slowing it down” means – and thus would lower the total.

I am not saying this necessarily means Bach is wrong, either in his pessimism about the degree to which slowing measures can decrease hospital overloading, or in his preference for containment over mitigation.  What I am saying is this: Bach does not provide a valid argument for his conclusions.

His conclusions could be right.  Since I wrote this, he has updated his post with a link to the recent paper from Imperial College London, whose authors are relatively pessimistic on mitigation.

I had seen this study yesterday, because an acquaintance in public health research linked it to me along with this other recent paper from the EPIcx lab in France, which is more optimistic on mitigation.  My acquaintance commented that the former seemed too pessimistic in its modeling assumptions and the latter too optimistic.  I am not an epidemiologist, but I get the impression that the research community has not converged to any clear conclusion here, and that the range of plausible assumptions is wide enough to drive a wide range of projected outcomes.  In any case, both these papers provide arguments that would justify their conclusions if their premises were true – something Bach does not do.

P. S. if you’re still curious what I was on about w/r/t the Gaussian, I recommend reading about thin-/heavy-/exponential-tailed distributions, and the logistic distribution as a nice example of the latter.]

———————–

I’ve seen this medium post going around, so I’ll repost here what I wrote about it in a Facebook comment.

This article simply does not make sense.  Here are some of its flaws:

- It assumes the time course of the epidemic will have a Gaussian functional form.  This is not what exponential growth looks like, even approximately.  Exponential growth is y ~ e^x, while a Gaussian’s tail grows like y ~ e^(-x^2), with a slower onset – the famous “light tails” of the normal distribution – and a narrow, sudden peak.  I don’t know why you’d model something that infamously looks like y ~ e^x as though it were y ~ e^(-x^2), even as an approximation, and the author provides no justification.

- Relative to a form that actually grows exponentially, most of the mass of a Gaussian is concentrated right around the peak.  So the top of the peak is higher, to compensate for the mass that’s absent from the light tails.  Since his conclusions depend entirely on how high the peak goes, the Gaussian assumption is doing a lot of work. [EDIT: I no longer think Bach would have drawn a different qualitative conclusion if he had used a different functional form.  See the step function argument from ermsta here.]

- No citation is provided for 40%-to-70% figure, just the names and affiliations of two researchers.  As far as I can tell, the figure comes from Marc Lipsitch (I can’t find anything linking it to Christian Drosten).  Lipsitch derived this estimate originally in mid-February using some back-of-the-envelope math using R0, and has since revised it downward as lower R0 estimates have emerged – see here for details.

- In that Lipsitch thread, he starts out by saying “Simple math models with oversimple assumptions would predict far more than that given the R0 estimates in the 2-3 range (80-90%),” and goes on to justify a somewhat lower number.

The “simple math” he refers to here would be something like the SIR model, a textbook model under which the fraction S_inf of people never infected during an epidemic obeys the equation R_0 * (S_inf - 1) - ln(S_inf) = 0.  (Cf. page 6 of this.)

Indeed, with R_0=2 we get S_inf=0.2 (80% infected), and with R_0=3 we get S_inf=0.06 (94% infected).  So I’m pretty sure Lipsitch’s estimate takes the SIR model as a point of departure, and goes on to postulate some extra factors driving the number down.

But the SIR model, like any textbook model of an epidemic, produces solutions with actual exponential growth, not Gaussians!  There is no justification for taking a number like this and finding a Gaussian that matches it.  If you believe the assumptions behind the number, you don’t actually believe in the Gaussian; if you believe in the Gaussian (for some reason), you ought to ignore the number and compute your own, under whatever non-standard assumptions you used to derive the Gaussian.

- What’s more, he doesn’t say how his plotted Gaussian curves were derived from his other numbers.  Apparently he used the 40%-70% figure together with a point estimate of how long people spend in the ICU.  How do these numbers lead to the curves he plotted?  What does ICU duration determine about the parameters of a Gaussian?  Ordinarily we’d have some (simplified) dynamic model like SIR with a natural place for such a number, and the curve would be a solution to the model.  Here we appear to have a curve with no dynamics, somehow estimated from dynamical facts like ICU duration.

- Marc Lipsitch, on his twitter, is still pushing for social distancing and retweeting those “flatten the curve” infographics.  I suppose it’s conceivable that he doesn’t recognize the implications of his own estimate.  But that is a strong claim and requries a careful argument.

I don’t know if Lipsitch has read this article, but if he has, I imagine he experienced that special kind of discomfort that happens when someone takes a few of your words out of context and uses them to argue against your actual position, citing your own reputation and credibility as though it were a point against you.