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### KeqingMains CN Translation: The Truth Behind Elemental Reaction/Application Cooldowns WITH PROOF - RELEVANT TO MELT/VAPORISE COMPOSITIONS

Translation link here

Translated by: iDreamz梦

Reviewed/Edited by: IonFox, Doug

Hello everyone. Today's translation is HUGE because it sheds light on a topic that many people have been debating for the past month: Do elemental reactions have a cooldown? The short answer is NO.

Key findings / TLDR

- Vaporise and Melt have special mechanics that cannot be explained in a few words. You MUST understand how they work to optimise teambuilding and DPS (eg setting up Mona ult, how to maximise vaporise procs in a rotation with Diluc, etc.)
- Elemental Reactions do not have a CD
- Skills and attacks have their own unique Elemental Aura application cooldowns (applying an elemental gauge onto an enemy). Repeating the same attack/skill onto an enemy within this cooldown period will result in the element NOT being applied to the enemy (and thus, rendering you unable to trigger another reaction)
- All element-applying skills/attacks have a "strength" (Strong or weak or weaker)
- The strength of both the aura element and triggering element will determine whether or not all elements are wiped from the enemy after a reaction, or if a single element persists.
- Certain characters having either "Strong" or "weak" elemental skills/attacks can make or break your team composition

### Where is the promised exponential growth in COVID-19?

*R₀*of 3 to 4 at the beginning of the epidemic, we should have seen exponential growth even after half of the population has been infected.

If at the beginning of the epidemic

*R₀ = 3*(one person infects 3 other people), we expect

*R₀*to be 1.5 when half of the population has been infected or been immunized, because there is 50% fewer people that can be infected. Similarly,

*R₀ = 2.7*if 10% of population is immune: 2.7 is 10% less than 3.

*Note about R₀: some people define R₀ as a constant for the whole epidemic, but the definition I'm using is more common and also more useful.*

The exponential growth is expected to slow down in the classic SIR models, but it should still be noticable well into the epidemic. And there should be almost no noticable difference in the exponential growth before the first 10% of population has been infected. For a detailed mathematical proof, see section 3 of Boďová and Kollár.

However, the graphs of total confirmed cases for the top countries at the start of the epidemic don't look exponential. Exponential growth is a straight line on a semi-log graph -- the dotted lines in the graph show different exponential functions doubling every day, every two days, etc. And the plotted numbers of total confirmed cases are nothing but straight lines. Where is the promised exponential growth?

If you instead look at the graphs on a log-log plot, where a polynomial forms a straight line, you see that a polynomial is a better fit. In this case a cubic polynomial for total confirmed cases:

- Italy growing as 2.5 t
^{3}or slower - Spain growing as 4.5 t
^{3}or slower - USA growing as 18 t
^{3}or slower

*Technical note: Each exponential function eventually beats a polynomial, but in the beginning a polynomial might grow faster.*)

And this doesn't seem to be the case only for these three countries. Mathematicians have analyzed data from many countries and have found polynomial growth almost everywhere. By now the pile of papers noticing (and explaining) polynomial growth in COVID-19 is quite big.

- Maier and Brockmann
- Ziff and Ziff
- Manechein et al.
- Brandenburg
- Komarova and Wodarz
- Li, Chen and Deng
- Radiom and Berret
- Szabó
- Bianconi and Krapivsky
- Merrin
- Boďová and Kollár

#### A toy example of polynomial growth

How could we get polynomial growth of infected people? Let me illustrate this with an (exaggerated) example.Imagine 100,000 people attending a concert on a football field. At the beginning of the concert, a person in the middle eats an undercooked bat and gets infected from it. The infection spreads through air, infecting everyone within a short radius and these people immediately become contagious. The infection travels roughly one meter each minute.

After about 100 minutes, people within 100 meters have been infected. In general, after

*t*minutes, about

*π t*square meters have been infected, so the number of infected people grows quadratically in this case. The cubic rate of growth mentioned above suggests that the disease spreads as in a 3-dimensional space.

^{2}The crucial detail in this example is that people do not move around. You can only infect few people closest to you and that's why we don't see exponential growth.

#### Modeling the number of active cases

We've seen the number of total confirmed cases, but often it's more helpful to know the current number of*active cases*. How does this number grow?

There's an interesting sequence of papers claiming that the growth of active cases in countries implementing control measures follows a polynomial function scaled by exponential decay.

- Polynomial growth in age-dependent branching processes with diverging reproductive number by Alexei Vazquez. [Gives a theoretical basis for the subsequent papers.]
- Fractal kinetics of COVID-19 pandemic by Ziff and Ziff. [Analyzing data from China.]
- Emerging Polynomial Growth Trends in COVID-19 Pandemic Data and Their Reconciliation with Compartment Based Models by Katarína Boďová and Richard Kollár. [Analyzing data from the whole world and making predictions.]

*N*(

*t*) = (

*A*/

*T_G*) ⋅ (

*t*/

*T_G*)

^{α}/ e

^{t/T_G}

Where:

*t*is time in days counted from a country-specific "day one"*N(t)*the number of active cases (cumulative positively tested minus recovered and deceased)*A*,*T_G*and*α*are country-specific parameters

## How does the model fit the data?

The model fits the data very well for countries whose first wave is mostly over. Some examples:An example of a country that doesn't fit is Chile (plotted prediction uses data available on May 2) which seems to be catching a very strong second wave. For a survey of more countries, see Boďová and Kollár.

Unfortunately, the exact assumptions of the model haven't been formulated. Even the obvious candidates like

*social distancing*or

*contact tracing*need to be first better understood and quantified before we can formulate exact assumptions of the model, so it's hard to say whether the bad fit for Chile is because of a flawed model or unfulfilled model assumptions (i.e. model does not apply there).

Regarding the countries that fit well, could it be that with so many parameters we could fit almost any curve? The formula

*N*(

*t*) = (

*A*/

*T_G*) ⋅ (

*t*/

*T_G*)

^{α}/ e

^{t/T_G}has three free parameters

*α*,

*A*and

*T_G*. A simple analysis shows that

*A*and

*T_G*only scale the graph of the function vertically and horizontally. The observation is left as an exercise to the reader. In the end the only really useful parameter for "data mining" is

*α*, which gives the curve different shapes.

This picture shows different curves with

*α*equal to 1, 2 and 3, with

*A*and

*T_G*chosen in such a way that the maximum is at

*t=1*with the value of 1. Changing

*α*doesn't allow

*that many*shapes.

## Predictions

Above we showed how the model fits existing data, but can it be used to make predictions? Me and my friends made a tool that calculates the best fitting curve every morning for 45 different countries. We show them on different dashboards:- Single country with a slider -- you can use the slider to pick the date of the prediction. See an example for Italy.
- Single country with multiple predictions -- the plotted predictions are from yesterday, a week ago, two weeks ago, etc. See an example for Czechia.
- Many countries on one page.

## Bayesian modelling

I have always been a fan of anything Bayesian, so I wanted to use this opportunity to learn Bayesian modelling. The uncertainty of predictions seemed like a great candidate. Please note that this is outside of our area of expertise, it was a fun hobby project of myself and my friends.The results look good for some countries but worse for others. For example, this is a visualization of Switzerland's plausible predictions about 7 weeks ago. The real data in the 7 weeks since then is well within the range of plausible predictions. However, plotting the same graph for Czechia didn't go so well for predictions made 5 weeks ago. The real data was worse than the range of predictions.

Summary of what we did:

- If you use cumulative data, the errors of consecutive days correlate. Instead, we fitted the difference/derivative (daily change of active cases), to get rid of the correlation of errors.
- The distribution of daily errors of the best Boďová-Kollár fit was long-tailed, so we tried Laplace and Cauchy distribution. Laplace was the best (corresponding to the L1 metric).
- The code is written in PyMC3 and you can have a look.

One important issue with the data is that each country has a different error distribution, because of different reporting styles. If anyone has any ideas on how to improve this, feel free to contact me. Even better, you can install our Python package and run the covid_graphs.calculate_posterior utility yourself.

## Discussion

The classic SIR models with exponential growth have a key assumption that infected and uninfected people are*randomly mixing*: every day, you go to the train station or grocery store where you happily exchange germs with other random people. This assumption is not true now that most countries implemented control measures such as

*social distancing*or

*contact tracing*with quarantine.

You might have heard of the term

*six degrees of separation*, that any two people in the world are connected to each other via at most 6 social connections. In a highly connected world, germs need also very short human-to-human transmission chains before infecting a high proportion of the population. The average length of transmission chains is inversely proportional to the parameter

*R₀*.

When strict measures are implemented, the random mixing of infected with uninfected crucial for exponential growth is almost non-existent. For example with social distancing, the average length of human-to-human transmission chains needed to infect high proportion of the population is now orders of magnitude bigger. It seems like the value of

*R₀*is decreasing rapidly with time, since you are meeting the same people over and over instead of random strangers. The few social contacts are most likely the ones who infected you, so there's almost no one new that you can infect. Similarly for contact tracing and quarantine -- it's really hard to meet an infected person when these are quickly quarantined.

The updated SIR model of Boďová and Kollár uses

*R₀*that is inversely proportional to time, so

*R₀ ~ T_M/t*, where

*t*is time in days and

*T_M*is the time of the peak. This small change in the differential equations leads to polynomial growth with exponential decay. Read more about it in section 5 of their paper.

#### FAQ

*But we aren't testing everyone!*-- Yes, we aren't, but it seems that the model applies fairly well even to countries that aren't doing such a good job testing people. What matters is the shape of the curve and not the absolute value at the peak. This is still useful for predictions.*What if the lack of exponential growth is caused by our inability to scale testing exponentially?*-- If the growth of cases was exponential, we would see a lot of other evidence, for example rapidly increasing number of deaths or rapidly increasing positive test rate.*What about the number of deaths?*-- According to the authors this model could be modified to also predict deaths, but this hasn't been done.

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