The Roman Philosopher Lucius Anneaus Seneca (4 BCE-65 CE) was perhaps the first to note the universal trend that growth is slow but ruin is rapid. I call this tendency the "Seneca Effect."

Sunday, March 26, 2023

The Worst Model in History: How the Curve was not Flattened

"Flattening the Curve" was an incredibly successful meme during the early stages of the Covid epidemic. Unfortunately, it was based on a model that we can describe as the worst ever proposed in history (or maybe the second worst, after the one that assured Napoleon that invading Russia in Winter was a good idea). Here, I explain why the model was so bad, and I also include a discussion on whether climate change models might suffer from the same problems.  

You may have heard the quote, "all models are wrong, but some can be useful." It is true. But it is also true that wrong models can be misleading, and some can be lethal. In history, some of these lethal models were fully believed ("let's invade Russia, what could go wrong?), while the lethal consequences of following some current models are still not understood by everyone ("economic growth can continue forever, why not?"). Other models are telling us of the lethal consequences of not following them; it is the case with climate models. There are many kinds of models, but you can't deny that they are important in determining human actions. 

In this post, I'll discuss the model that gave rise to the concept of "Flattening the Curve" at the beginning of the Covid-19 epidemic. It was based on the idea that "non-pharmaceutical measures" (NPIs) would slow down the diffusion of the virus and avoid overloading the healthcare system. It was one of those models that looked good at the beginning, but that turned out to be a disaster. Among other things, it gives us a chance for a critical examination of climate models: could they suffer from the same problems? 

About the "Flattening the Curve" story, this idea of slowing down the diffusion of a viral infection was not wrong in itself. For millennia, people had noted that many illnesses were transmitted from person to person and that staying away from sick people could reduce the chances of infection. But country-wide lockdowns, universal masking, and the like had never been tried before. So how would you know that they could have a significant effect? 

Indeed, before the great Covid scare, the general opinion among practitioners and experts was that quarantines and other drastic measures were counter-productive, if not completely useless. Then, in early 2020, a new concept burst into the scene and took the memesphere by storm: "Flattening the Curve." It was expressed in the form of a graph that appeared over and over on the media in slightly different forms, but always showing the same concept. Here is an example among the many.

Image from "The New York Times," 2020,

Let's start by noting that the model is based on the typical shape of the curves describing an epidemic cycle. It occurs when something grows (e.g., a virus) by exploiting a resource (e.g., human beings). If the resource is limited, as is the case for the number of people that can be infected, then the growth of the infection will start slowing down, reach a maximum, and then decline. The result will be a "bell-shaped" curve, a behavior that has been known from the time of the Great Plague of London in the mid-17th century. (note, incidentally, that epidemic curves do not normally show the "Seneca Effect," that a faster decline in comparison to growth. It is because the system is relatively simple, and viruses are not affected by "pollution"). 

So, the "Flattening the Curve" model was based on something real; nevertheless, it had enormous problems. Take a careful look at the figure above. The model implies no less than two separate miracles. The first is that the "zero" of the x-axis is supposed to coincide with the "first case." It implies that, miraculously, the government would be so farsighted to decide to lock down a whole country on the basis of a single observed case or just a few. Such a government never existed, and you may argue that it cannot exist in the real world. In practice, NPIs were mandated only when the epidemic was well on its way and fast growing. Note also how the "Protective Measures" curve touches exactly the limit of the healthcare system's capacity without overcoming it. How the measures could be calibrated so precisely is another miracle. 

The need for two miracles is bad enough for a single model, but there is a much worse problem with it: the model shows two curves with the same shape; they differ only in scale, a parameter that cannot be reliably determined in the early phases of an epidemic cycle. Then, of course, in the real world, the epidemic will follow only one of the two curves, and how do you know which one? In other words, how do you know if the measures are having any effect? Remarkably, the question was almost never publicly asked during the epidemic. The "flattening the curve" model soon became a political issue and, in politics, there are questions that you are not allowed to ask. 

So, let me try to step out of politics and use science to ask a forbidden question: how would the curve react to the "measures" applied while the curve has already started to grow? Everybody expected an effect, of course, and, obviously, a strong effect if it had to be worth the effort. Tomas Pueyo correctly used the term "the hammer" to describe the expected effects of NPIs (one of the very few correct observations he ever made). And if you hit something with a hammer,  you do expect some immediate effect. But what kind of effect, exactly? 

In a previous post, I described a simple SIR (sane, infected, removed) epidemic model, not a sophisticated model but several steps higher on the scientific scale than a purely qualitative two-curve diagram. The model can be easily tweaked to show the effects of a sudden reduction in the transmission factor (Rt) of the infection as a result of NPIs (note that it doesn't apply to vaccines, which can only be introduced gradually). Below, you see a typical result of my calculations. 

The vertical axis is the infected fraction of the population (the "prevalence"), which should be proportional to the number of measured positive cases. The horizontal scale is the time; a typical epidemic cycle lasts a few months. The graph is roughly modeled on the Italian case in early 2020, and it assumes that the "measures" are mandated on the 20th day of the start of an infection cycle that lasts a few months. The model assumes that the NPIs reduce the infectivity (Rt) of the virus by 50% (as it was commonly expected to happen). 

The result is that the slope of the prevalence curve changes when the NPIs are put in place. You can play with the parameters in different ways, but, for a significant decrease in the virus transmission rate, you will always see a discontinuity in the curves in correspondence to the start of the measures. NOte also that there is a certain latency time before a contact with an infected person will lead to a positive result to a PCR test, but for Covid this latency is estimated as of a few days, no more than five. The effect of the latency time will be to smooth the transition, but the change of slope should remain detectable. Overall, this is what the real "flattening the curve" should look like.

Of course, there exist much more sophisticated epidemiological models, but good modelers know (or should know) that complicated models are not necessarily better than simple ones. Here I don't want to enter into the academic debate on the effect of NPIs (it never reached policymakers and the public, anyway). Just as a quick note, you may wish to take a look at this 2020 paper. It was published by the group led by Neil Ferguson at the Imperial College in London, who was one of the main proponents of lockdowns. The authors argue that lockdowns were effective, but, if you examine the paper carefully, for instance, looking at fig. 2 of the extended results, you'll see that their own results do not support their conclusions. (and I am not the only one who noted the problem).

But rather than going into the details of complicated models, let's just use common sense. The NPIs are a sudden change in the parameters of the system. When the government orders people to stay locked at home, most of them do that immediately. So, you do expect an immediate effect on the shape of the epidemic curve. The problem is that you don't see anything like that in real-world data. Below, the case of Italy in 2020. NPIs were enacted on March 9th, when the curve had reached about 25% of the peak. The curve continued to grow along the same trajectory for 19 days more. 

Italy is just one case. Maybe, if you are a real first-class sleuth, you might find some cases where you can evidence a discontinuity in an epidemic curve in correspondence with the NPIs being enacted. But we have hundreds, probably thousands, of examples, and they are almost always smooth, except for the unavoidable random noise. The conclusion can only be that if the NPIs had an effect, it was very small. Incidentally, these observations are consistent with the recent Cochrane Review that used different methods to examine the effectiveness of face masks and other NPIs in slowing down the diffusion of viruses. No detectable effects were found. 

In the end, more than two years of "measures" were imposed on citizens on the basis of a model that implied miracles and didn't include methods to verify the effect of the recommended actions. The damage done to society was enormous in psychological, economic, and human terms, all for effects that turned out to be so small not to be measurable. We are still reeling from the disaster, and it may take several more years before we completely recover -- if we'll ever recover. 

The question, then, becomes how it could be that almost everyone in the world was completely overtaken by such a bad model -- possibly the worst one ever developed in history? It is a story related to the military implications of epidemics as bioweapons, but I'll tell it in a future post. 


  1. I'm reasonably close to the science of economics and I believe that you partially misunderstood it.

    Economic growth going forever is not a dogma, it's just a reasonable simplification of reality, just as other sciences sometimes assume lack of air resistance or unrealistic perfect sphere to simplify calculations.

    Obviously when the solar system is finally consumed and transformed into economically profitable matter, with most energy from the sun captured (in science fiction- Dyson Sphere) then the growth will stop, but economy and finance needs models for the next fifty years, not five hundred.

    My cornucopian disagreement with you is not about the growth going literally forever, I "just" think that might of the industrial civ can consume the solar system without collapsing, especially with future AI systems etc.

    This does not mean that collapse is utterly impossible, but I see more danger in nuclear wars and solar storms (1859, The Carrington Event) than in resource depletion; even you nowadays seem to recognize that there is plenty of solar power that our civ could build.

    1. Indeed, there is no proof that we are going to collapse. But collapses are a normal feature of the history of civilizations, so it seems to be at least likely that ours will not be an exception. On the other hand, I noted in several posts that collapse is often followed by a rebound (the "Seneca Rebound"). So, as I use to say, collapse is not an event, it is a process. As such it can be steered, mitigated, acted upon, and even exploited. It is the cycle of life, as we know from the Lion King :-)

  2. There is one point that puzzles me. The club of rome has been telling us for 50 years that resources are not infinite and that we must expect shortages. The published model predicted the future shortage pretty accurately for today. So we are in the exciting phase, and what happens? The topic is changed. Suddenly, it's no longer about the range of resources, but about the climate. And what is the solution? Use fewer resources. Haha.
    So I suspect that climate change is just a placeholder for the end of the oil age, simply because you can't say the latter without running the risk of exacerbating the problems.
    You can't leave complex problems to politics? Where politics is only the space left to it by the economy?
    Let's assume that there is no solution for a mass society if the supply with central resources does not work out. Then, of course, chaos is inevitable, and the only thing the decision-makers can do is to organize the shortage before its time. Only then will they have enough resources to manage the event and ensure their power. They certainly want to avoid the chaos, and the danger of perishing in it.

    But something completely different? Has anyone read the Pfizerfiles?

    Mit besten Grüßen aus Berlin von Marcus

    1. I see no real resource issues yet, at least taking global view instead of looking only at Italy/other-western-nation. Try to take more international view. Steel output of India, as an example. 68.3 million tonnes in 2010, 124.45 million in 2022. This is no doubling of output, but close.

  3. I think that climate change is more or less a cover story for the real problem, resource depletion.

    King Hubbert modeled US conventional oil production in the 1950's. He predicted a peak for maybe 1970 and was correct by within a year or two. Then came the North Slope of Alaska, a shadow of its peak. Fracking as a gimmick is kind of played out, I guess.

    Industrial Civilization, like every other system in the universe, I guess, is subject to the law of the minimum, ie, a shortfall in any one critical component of a system, results in system failure. I don't know if anyone can accurately model its implications, as they might apply to our present situation. But I do think that a Seneca Cliff type event is a real possibilty.

    1. Hi Dave,
      Seems to be a standard comment here.

  4. Hello kind philosopher. I see you still tend the fire. From one come out from the cold and forlorn waste its warmth is a comfort and its light is a beacon.
    Bertrand Russel
    “I should like to say two things, one intellectual and one moral. The intellectual thing I should want to say is this: When you are studying any matter, or considering any philosophy, ask yourself only what are the facts and what is the truth that the facts bear out. Never let yourself be diverted either by what you wish to believe, or by what you think would have beneficent social effects if it were believed. But look only, and solely, at what are the facts. That is the intellectual thing that I should wish to say.

    The moral thing I should wish to say… I should say love is wise, hatred is foolish. In this world which is getting more closely and closely interconnected we have to learn to tolerate each other, we have to learn to put up with the fact that some people say things that we don’t like. We can only live together in that way and if we are to live together and not die together we must learn a kind of charity and a kind of tolerance which is absolutely vital to the continuation of human life on this planet.”
    Bob Marley
    "Emancipate yourselves from mental slavery
    None but ourselves can free our minds
    Have no fear for atomic energy
    'Cause none of them can stop the time
    How long shall they kill our prophets
    While we stand aside and look? Ooh
    Some say it's just a part of it
    We've got to fullfil the book
    Won't you help to sing
    These songs of freedom?
    'Cause all I ever have
    Redemption songs
    Redemption songs
    Redemption songs"

  5. The new practice of War, in tanks or syringes - now fossil fuel reserves are depleting to the ground - seems about - one doesn't know who is fighting who, what is fought against and what is really going on...

    ""It’s all lies!" | Was Putin, Or His "Body Double", Heckled During Visit To Occupied Ukraine City?"

    Could it be that what's going on is simply a global Energy Musical Chairs Game?

    "In any system of energy, Control is what consumes energy the most.
    No energy store holds enough energy to extract an amount of energy equal to the total energy it stores.
    No system of energy can deliver sum useful energy in excess of the total energy put into constructing it.
    This universal truth applies to all systems.

    Energy, like time, flows from past to future".


  6. Is Zelensky wanting to meet Xi Jinping the next flatting of the curve?

  7. Some of the flattened curve diagrams look like the total area under the curve is greater than the original curve.

    1. They were drawn by hand by people who had no idea of what they were doing. If you use the model, though, the area of the flattened curve should be somewhat smaller than that of the "original" curve