Friday, March 13, 2020

Exponential functions and hysterical epiphanies

The ongoing insane hysteria about the new kind of cough has revealed a new bunch of hysterical ideologues who are dreaming about a nanny state and who previously pretended to be deep thinkers who understood why socialism or the constant martial law were wrong.

It seems that lots of people who should be smart and/or who think that they are smart have had this epiphany:
Wow, this graph must be the famous exponential function. It's really the exponential function and it will therefore diverge to infinity, and very quickly so. Why didn't they tell me that there are exponential functions in Nature? If I had been told earlier, I would have been on the side of the likes of CNN for years!
So deep. But not so much. ;-)

Judging by these reactions, it really looks like these people just encountered the concept of exponential functions for the first time in their lives. And they are greatly moved by this encounter. Well, the rest of us have known exponential functions for a longer time – and, like in my case, for a decade before we became tangible conservatives. I think that I was 3 when my grandfather impressed me with the chessboard thought experiment for the first time:

The Shah (sorry, Ayatollah, but you didn't invent chess and the game is not named after you so far) has placed one little seed on the first square of a chessboard, twice as much (2) on the next one, twice as much (4) on the next one... and so on. How many seeds are on the last square of the chessboard?

The answer is 263, almost 1019, of course, and the total number of seeds on the chessboard is 264–1. Even with 0.01 gram seeds, those seeds are some 1014 kilograms, exceeding the mass of Mount Everest. You can obviously obtain higher numbers surpassing the mass of the visible Universe is you pick a greater multiplier than 2 or (and that's easier) a larger chessboard.

Exponential functions sort of grow very quickly rather soon – although, as you may figure out soon, there are even "faster growing" functions, parameterically speaking. When we return to the hysterical epiphany, it really doesn't seem to be affected by the numerical values of any numbers. Instead, it's the qualitative trait of exponential functions, their ability to quickly become "incomparable" to the starting point, that impresses all those "thinkers" who become hysterical. And the word "qualitative" means "non-quantitative" and in this sense, it's not far from "religious" or "humanities-based". Paying too much attention to such qualitative properties – and ignoring the actual numbers – is a sign that someone is simply not thinking as a rational, quantitative person who realizes that mathematics is absolutely essential and who gets "lost in it" is an intellectually worthless pseudointellectual.

Let me share some secrets with you now:

* Exponential functions indeed exist mathematically.
* However, they are never a permanently valid description of objects and processes in Nature because
* some other complications soon or later ruin the simple-minded exponential growth.
* The longer and more accurate exponential growth you want, the harder and less likely it becomes.
* Despite (and maybe especially because!) the existence of approximate exponential functions in Nature, a system supporting human freedom is better than a nanny state.

Let's talk about these points a bit more slowly. First, in the Platonic world of mathematical structures, exponential functions indeed exist as elements, members, citizens, comrades, or animals, or whatever is the ecological or social system introduced by Plato (one of the first comrades in the world). However, pure mathematics – and the Platonic world is pure mathematics – has by definition no direct connection with the real world.

Second, exponential functions aren't ever perfect in Nature. In Nature, they may follow from differential equations. The simplest one that produces exponential functions is\[

\frac{dy}{dt} = ky

\] which is solved by \(y(t)=C\exp(kt)\). It is increasing or decreasing for positive or negative \(k\), respectively. Another important differential equation is\[

\frac{d^2y}{dt^2} = Ay

\] which is solved by \(y(t)=C\exp(+\sqrt{A}t)+D\exp(-\sqrt{A}t)\) for a positive \(A\) and, for a negative \(A\), by \(y(t)=C\cos(\sqrt{-A}t)+D\sin(\sqrt{-A}t)\). Nice. Fundamentally, differential equations like that arise from the Heisenberg equations of motion for some operators (observables) but in this text, we are focusing on the classical limit in which the operators are replaced by some mean values i.e. \(c\)-numbers.

It's very clear that no observable in the world obeys simple equations like those two equations above precisely. If an observable \(y\) obeyed such a simple equation, which doesn't include other observables, then the Lagrangian would segregate the terms with \(y\) and without \(y\). It would mostly follow (also from the principle of reaction) that the other observables (like the brightness on our retina) are not affected by \(y\), and therefore we couldn't observe \(y\). That variable would be detached from the real world of objects that may be connected to our perceptions.

That means that in reality, those equations are just never perfect. Exponentially growing things and populations ultimately stop growing. What are the longest examples of a nice and precise exponential function in Nature?

A population of viruses may indeed grow by a factor of a trillion. But we get bigger factors in cosmology. In cosmic inflation, the Universe must have undergone at least 55 e-foldings for cosmology to work. The Universe must have increased by more than 23 orders of magnitude (in the decadic system). If the dark energy in the present Universe is close to the cosmological constant (by its behavior) that will last, our Universe is getting very empty (it is already 70% empty) and will be growing exponentially forever, with the doubling time around 11 billion years (counted in the cosmic time, a proper time measured by clocks in the CMB frame).

But cosmology is a fundamental scientific field and other fields like biology are more dirty and messy which also means that their realization of exponential functions is less precise and less long-lasting. Quite generally, things in biology smell, decay, rot away, putrefy, go mouldy etc. As you could have learned from Sheldon Cooper or similar giants (and their real-world role models), those are obvious reasons why biology is inferior relatively to physics (and OLED is inferior relatively to QLED).

Let's pick Covid-19 as the ludicrous topic that is being marketed as the main problem of Planet Earth by the fake news media now. Some people are impressed by the exponential-like growth of the number of cases or infections up to one thousand or few thousands per country. In those fits, the daily multiplier seems to be as large as 1.3. That's so scary for them. Equivalently, the doubling time is some 2.65 days. After 63 of such 2.65-day periods, i.e. after 170 days, you get the "Mount Everest" multiplier from the chessboard. The cataclysm is clearly half a year away! Well, most of the people don't even calculate the time that is left but they still believe that they "know" that the Armageddon is a matter of months.

Because these unhinged hysterical people think that we should look at least "half a year into the future", they are already there. They are acting as if the dying of all people (or at least those 0.5% of mankind from the fatality rate) were a fact. A fact now. Instead, if they looked a bit closer at the graphs that already exist, it would become very clear to them that the coefficient 1.3 per day simply never lasts. It can never last. It always drops.

Media have admitted (but all the brainwashed hysterical sheep were guaranteed to overlook) that China's new cases already peaked one long month ago. Surprising, hours ago, Reuters admitted that the simplest indicator suggests that South Korean cases peaked, too: 110 new cases appeared (less than 114 a day earlier) which is fewer than the 177 people released on that day. People are being released more quickly than new people are being infected, the population of active cases in South Korea is shrinking. Incidentally, the main global statistical page added some national sub-pages and it is clear that Korea's new daily cases peaked on March 5th or so.

Yes, the slowdown requires the institutionalized fight against the infections. But by far the most important task is simply to look for, identify, and quarantine the infected people or the people who are likely to be infected. Everything else is secondary in comparison – and only a nearly complete curfew and/or mandatory face masks can make a similar difference.

As Edwin suggested, it's almost guaranteed that the fast exponential growth of the "infected cases" at the beginning is due to the tree-like discovery. In reality, these hot spots that had no regulation to start with developed at least hundreds of cases almost simultaneously. Suddenly, one or two cases became obvious because the people were sick enough. So authorities started to look at the ill people and their contacts and their contacts... and the exponential growth indicated how quickly they could test the people and expand the trees of suspects.

That's a correct description for hot spots where the infections spreads via "community spread", i.e. internally. In Czechia, this sector is still negligible. Czechia has 117 infections now, see the nice new statistical page about Czechia and Covid-19. Two men are in a serious condition (including the Uber driver), the 84-year-old mother-of-a-skiing-daughter surprisingly got pretty well again. We have zero fatalities now. The "places where the people got infected" show a nice histogram:
  1. Val Gardena 11
  2. Prague 10
  3. Auronzo di Carore 8
  4. Passo Tonale 7
  5. Vigo di Fasso 6
It's followed by Milan, Děčín, Boston, Bratislava, San Cassiano, Selva di val gardena, Val di Fassa at 2, plus 12 places with 1 infection. I find this toplist comical because it's clearly nothing else than an unpaid advertisement for the Italian ski resorts in the Dolomites. In fact, almost all the infections "acquired from Czechia" are traced to the Dolomites, too.

In Czechia, Covid-19 is a mild cold mostly suffered by the upper middle class that goes to the Dolomites for skiing. And some of their family members. Forget about the idea that the infected people are "representative" of the Czech society. These people are managers from good enough companies, dentists with good enough patients, and similar rich enough folks. Next to them, you have just a few infections traced to the BioGen conference in Boston (a place I know rather well); and a few linked to Austria or Germany.

Czechia also shows some rather fast exponential growth of cases from our first three cases on March 1st. They may be mostly indicative of the infection rate in our main source, Northern Italy. But in reality, even our accelerating growth is mostly due to the "gradual kickstarting" of the testing and research. But the tree-like organization of the tracing isn't too important for us; instead, the sheer volume of testing is the key explanation. Around March 1st, we tested around 10 people a day. Today, we test about 500 people a day. Of course we are bound to get a greater number of new cases now.

On the other hand, it's rather likely that we've already found most of the infected people on the Czech territory. The suspects are nicely chosen and the percentage of infections (from the number of tests) goes down since March 1st (from 20+ percent to below 10 percent). If we're overlooking (or anyone is overlooking) some hot spot, well, this hot spot becomes visible after less than 100 cases or so because someone will probably feel visibly ill. And that hot spot is then tamed by the standard tracing.

This is my general model. As long as it is possible to look at the contacts of all the positive people (and I think it's more important for taming the virus than the availability of hospital beds), we generally start to see the "hot spots" when they have about 100 cases started by a local infection. At that moment, the infections bringing you up to a thousand or a few thousand may be unavoidable because some people will be unavoidably found after they were able to infect someone. But once you start to see one thousand or few thousand cases, the intense enough policies and monitoring has already stopped the actual further growth.

To summarize, you always see hot spots going from "one hundred to one thousand or few thousands" and then the growth just turns off. (The slowdown is also going to be helped very soon by the coming semi-global warming, the so-called spring.) It is purely irrational hysteria to think or say that the growth from 1,000 to 10,000 per hot spot is as easy as the growth from 100 to 1,000. It's simply not. This virus spreads just like a damn cold, but a cold that is treated as if it were the plague. BTW it's somewhat more likely, I think, that some copies of the virus will survive to 2021. I hope that people will treat it more rationally in 2021.

If a country with a program to fight the contagion has X thousand active cases, you may imagine it's something in between X hot spots that are already peaking, and 10X hot spots that are starting. From the observation to the peak, you will rarely get a bigger piece of the exponential function's graph than the increase by one order of magnitude, or 1.5 orders of magnitude. To imagine that many, many orders of magnitude of growth are a "fact" – assuming the data from the present – is just stupid. If this were possible, mankind would have been extinct for millions of years.

China is already in the state where most of the new infections may be classified as imports from some other intensely infected places (Italy has a vastly higher number of active cases per million citizens than China has ever had, of course). When it's so, the current Czech description works well enough. They're just personal stories of some people who understandably didn't want to sacrifice a prepaid trip to the Dolomites. And the places that are dominated by local community spread simply won't grow too much over several thousand cases because most of the infected people are found and even if it's not a majority, the personal contacts in such a community "known to be at risk" are heavily suppressed.

These places with not-yet-tamed community spread will be marked, avoided by tourists, and perhaps harassed by travel bans which is why the export of the virus to other places will be tiny (or people who return from there may be mandatorily quarantined). I find it obvious that something basically equivalent is taking place in all the relevant countries which is why the disease will be tamed. It could be brutally tamed within a month with dictatorial draconian, China-style policies. Without them, it may take a few months. But the down trend will be obvious enough, anyway. I find it likely that with no policies to fight the contagion at all, the virus could ultimately spread to something like 1/2 of mankind. But it couldn't do it at the same moment.

So I was discussing the glorified virus for a long time as a fashionable example of the fact that the exponential functions aren't ever exact in Nature. New terms arise. And the precise mimicking of many (M) e-foldings becomes increasingly unlikely with an increasing M. It's just absolutely stupid – a mistake equivalent to the conflation of mathematics and physics (or even mathematics and biology) – to assume that the exponential function that created the increase by one order of magnitude must continue for 2, 3, or even 7 additional orders of magnitude which would be needed to spread the virus to most of mankind.

It just won't fudging happen, hysterical crackpots aligned with MSDNC and CNN (although many of you shouldn't be, having pretended to be right-wing). The disease's spreading isn't too much different from that of flu except that flu has an easier life because no one really fights it at the level of nations (and during "epidemics" in Bohemia, something like 2-3 percent of people may be infected with it – so it's still extremely far from 100%). Unlike flu, Covid-19 is fought like a plague so the virus is driven to fast near-extinction within the sources of community spread. The increase from 100 (which is needed for a community spread hot spot to be observable through some severe cases) to 3,000 may look dramatic but it's just a result of the unavoidable delay between the infection and detection (or counter-action such as quarantines).

To be honestly hysterical about Covid-19 means to be mathematically or scientifically illiterate or to completely confuse the real world with the idealized world of mathematics.

Approximations of exponential functions are relevant for many places in the real world but they are never perfect and they are never lethal. The exponential growth of many things – GDP and the number of products – is actually a great virtue of a nicely operating free and/or capitalist society. This clearly impressive, quasi-exponential growth is something that the Soviet bloc could have only been jealous of (because its economy has always resembled stagnation of a sort, and if there were growth, the linear description of that growth was always a good enough approximation). And when some bad things exponentially grow, like viruses considered very bad by the "holders of power" which are really humans in this case, these bad things will rather soon experience a tough backlash and the exponential growth stops. This is what happens to hot spots of Covid-19 with community spread when the number of infections exceeds several thousand. As long as someone tries to find and isolate most of the clear enough infected people or the people who are at risk due to their visits to the hot spots with community spread, the growth quickly reverses to a decline. It just can't fudging be otherwise.

The free and capitalist society without unnecessarily martial laws is great for many reasons – and its ability to pick and exploit the exponential growth of good things while the growth of bad things is ultimately tamed is one of the most admirable traits of capitalism!

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