Stochastics — and Quarantine Buddies

(S0) Not for everybody

You ~50% maths haters, please skip this text. You lack the very basics to understand it. (Probably not your fault, most school systems failed in this).

If you now reserve one hour per day to fix that mental incapacity, and work through one missed school year per week … in late summer you can graduate in maths :-) But dont’ worry for now — please you now simply go back to the main text, and continue reading reading in chapter 1.2.8. Thanks.

For the maths able people, this is an interesting real life example, and you’ll deepen your probability methods; or be able to better explain them to others.

What is a stochastic process? How to combine probabilities?

(S1.1) Outcome: Quarantine groups are problematic

We will see that the larger the quarantine group the more tragic any mishap.

“stochastic” def:having a random probability distribution or pattern that may be analysed statistically but may not be predicted precisely” (wikipedia)

Why use stochastic processes at all, if they never tell us a definite prediction?

(S1.1.1) Infection Processes can be seen as Random

Only in hindsight, we might see the Covid19 infection network: who infected whom. However, looking at the present, that is not yet possible. Simply due to these two infection channels:

  • pre-symptomatic spreaders (during their 1–14 days incubation period)
  • a-symptomatic spreaders (who never show symptoms but produce virus)

Often we do not know who is spreading the virus, until it is mostly “too late”.

On the contrary, being in one room with an infected means no guarantee to get infected; it can happen, but it does not need to happen. Countless factors (the air movements, your immune system, a sudden cough, etc) influence it.

So we must give up our causal way of thinking! From now on, we live with “less hard knowledge” about the world. That however does not mean we are blind. The situation is just too complex to model with causality — instead what maths is is successfully applying in such situations is … probability theory.

(S1.1.2) Mild - or severe — course of Covid19 ?

The above is about getting infected or not. Similar can be thought for the course of the illness itself, once you are infected. Whether mild or lethal depends on many factors. Age, lungs, heart, diabetes, immune deficiency, nutrition, hormonal cycles, etc — and yes, imagine that: it depends on your mood on the day when you acquire the virus; your immune system will perform better or worse in any moment.

The probability idea simply means: if you repeated the “experiment” of your life e.g. a hundred times — in how many cases would you get which outcome?

Now all of the above influences can be replaced by simple “probabilities”.

Disadvantage: You do not know the definite outcome anymore.
Advantage: We have learned to work well with such probabilities.

Since a “gambler’s dispute” in 1654 led to the creation of a mathematical theory of probability, by Blaise Pascal and Pierre de Fermat, the human species can now navigate uncertainty much better than before. (And uncertainty is one of the sources of fear, so good maths does help psychologically too).

Often a simple “random number generator” (like rolling a fair dice) is used to explore situations and their outcomes.

(S2) Why a stochastics detour? → Epidemiology!

It is worth it to invest some time now, because a lot in epidemiology can only be expressed as probabilities. Let’s learn how to to combine such probabilities of a very basic stochastic process — rolling a dice.

(S2.1.1) Stochastics: 1 person quarantine group

Example numbers, for illustration. Let’s use a six-sided DICE (a regular cube with six planes, and the six numbers 1 2 3 4 5 6 on them) to better understand this stochastic process. Six sides, all equally probable. No cheating.

A “six” in this game means something bad though: either dead, critical, or a severe course of Covid19 illness. Likely lasting lung damages, for example. Not nice weeks in a hospital; and perhaps the last weeks of a life, worst case.

In contrast (and for the “optimists” the only thing they ever want to hear about) … if the dice rolling results in a 1 2 3 4 or 5 then … the illness might not be nice — but you’ll survive it without heavy lasting damages. Congrats.

(Unlucky or reckless that you to had caught it — but lucky you to have a mild course. Hopefully you haven’t infected or even killed anyone, i.e. hopefully you didn’t pass it on. But for this reductionism here, let’s not think about any of those those secondary effects; instead) let’s for now focus on the egoistic perspective; to better understand the stochastic properties of this process).

Simplification:

5/6 = 83.333…% = mild course of illness. Lucky.
1/6 = 16.666…% = severe/critical/lethal cases. Sad.

Note that this & following chapters will change its basic “one dice probability” when we will have better numbers about who got infected and who didn’t. The above is near to the numbers in current computer simulations (see tweet below), but of course there is a yet unknown “gray area” of cases (that were so mild they were never even recorded). Long story short: the real dice will probably have more sides than six, i.e. the single person probability might look better than 5 out of 6.

This whole chapter is not about whether we must assume 59% or 84% per human experiment. Instead we are asking the question what happens when we combine such random processes.

Our useful example:
We live together in a small group, and we want to keep ANY tragedy out.

For now, and this teaching, the basic game will be (1 2 3 4 5) versus (6).

The dice rolling 83% versus 17% is sufficiently close to the risk for a 50y old:

(S2.1.2) Take a moment to reflect please.

Agreed with the above? Please immediately ask (someone) if you don’t understand some thought in this maths part here. Do not keep on reading until you grasped each step. You will not benefit, if you skip any of these steps.

Everyone else: Thank you for your attention. This will be interesting, and generally useful, for other statistical situations in your life. I promise, lol.

(S2.2.1) Stochastics: 2 people quarantine group

Now, imagine you are two people living together closely, and you even love each other. It would break two hearts, if (both or) one of you would die, or have a severe or critical course of #Covid19 illness, right?

What we want here is both people getting out of the pandemic alive & kicking, and no big suffering on the way. All good. All beautiful. Right?

So … how probable is it that two people do not roll a six, on their dice?

Here is the simple maths: For the probability of both outcomes combined, we must multiply their probabilities:

(5/6) * (5/6)
= 83.333…% * 83.333…%
= 69.444…%

Please try that on your calculator now. The 83.333…% percentage will look like this: 0.8333… because “per cent” means “per hundred” in Latin (see www.homeschoolmath.net).

Let’s put this percentage into words: In 69 out of 100 lives that you are living together through this pandemic, both of you would have the mild form, if you got infected. However, in 31 out of 100 lives, at least one of you would not be as lucky; or even both.

Right? Continue in (S2.3.1) now, for a slightly larger quarantine group of 3.

(S2.2.2) Advanced for 2

Advanced = You do not have to read this, and it doesn’t matter if you do not grasp it when you try — but for the maths-advanced people this is interesting:

The three other cases are
13.888…% = 5/6 * 1/6 = 1st mild 2nd bad
13.888…% = 1/6 * 5/6 = 1st bad 2nd mild
2.888…% = 1/6 * 1/6 = both bad double sad
— -
30.555% = that any of these 3 cases happens.

So here, if we don’t care which of the cases is happening (because we only want that not any of them happens) … we are simply adding the probabilities, for those 3 cases. 13.888… + 13.888… + 2.888… = 30.555…

Now look: That is another way how you can also get the “in 31 out of 100 lives” from the previous paragraph.

And of course 69.444…% +30.555 …% = 100.0 %

Magic? No. Probability maths.

Always good to get to the same outcome in 2 different ways. And once we see 100%, we know that we have not forgotten any of the cases. Whatever experiment you run, it will always have SOME outcome. That means: 100%.

(S2.3.1) Stochastics: 3 people quarantine group

Now what is the same calculation if 3 people live closely together?

Again assuming all three have identical probabilities of 5/6; and live so closely together that one infection would lead to three infections anyway; and it would break three hearts if one(or two or all three) of them had a severe/critical/dead outcome.

You can guess the formula now, right?

5/6 * 5/6 * 5/6
= 83.333…% * 83.333…% * 83.333…%
= 57.87037…%
That is the probability for three times the mild, non heart breaking, outcome, (i.e. that none of the 3 people rolls a “six” dice.)

And in contrast to that:
100% — 57.87037…% = 42.1296…%
for any of the outcomes in which one, two or all three are rolling a “six” with their dice, i.e. have a non-mild encounter with Covid19.

(S2.3.2) Advanced for 3

Now only for the advanced: If you want to, calculate all the remaining “probability channels” for one two or three tragedies, and then add up their respective probabilities.

Your sum should be 42.1296… percent. Then you have done it right, and understood basic probability theory. Congrats.

(If you get to a different number, please take a break, but then try again. Then take another break, and explain the problem and how you see it … to a rubber duck. In almost all cases, that then helps and you can refine your answer, and get 42%. If not, then pause again, and then talk to a human. Only if you still do not get the correct answer, play with the idea … to give up. Then solve it. Yes, that is how maths is for most of us. Doing it is frustrating. Until it’s not.)

(S2.4) Stochastics: 4 people quarantine group

Now what is the calculation if 4 people live closely together? Same assumptions, but this time the dice may not roll a six four times in a row.

Is the maths boring now? The outcome is not. Formula as before just longer:

5/6 * 5/6 * 5/6 * 5/6
= 83.333…% * 83.333…% * 83.333…% * 83.333…%
= 48.2253…%

With four people, it is already a bit more likely that at least one tragedy is happening (or two or three or four tragedies), than the case that all four would have a mild course of illness.

The (mild mild mild mild) outcome (is the only outcome that optimists ever want to talk about) happens in 48 out of 100 lives; then there’s no direct tragedy, and all 4 people survive the illness without much harm.

However, in 52 out of 100 lives, there would be sadness and grief among those 4 stochastic people, because 100 %— 48.2253 …% = 51.7747…%

(S3) Conclusions

  • multiply probabilities when you want both conditions to be fulfilled
  • add the probabilities for “any of the outcomes”
  • When a group grows, the “no tragedies outcome” becomes rather unlikely quickly. An improbable 83:17 case turns into a probable 48:52 case at only 4 repetitions.
  • The larger the group the better group consensus needed about personal levels of hygiene; and harder to “bridge” over different levels of hygiene
  • Maths is useful, especially when you know only probabilities, not more.

(S4) Todo: pics— let’s cooperate, you design genius

Are you a gifted designer? How would you draw these 4 experiments, so that the visual mind gets an additional “infection channel” for this information?

Thank you very much.

(S5) Main text is “Pandemic Tools for Minds”

→ continue reading at “practical conclusions” in chapter (1.2.8) now:

We need predictive politics. 我们需要预测性的政治. Wǒmen xūyào yùcè xìng de zhèngzhì. Wir brauchen prädiktive Politik. Necesitamos políticas predictivas.