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Macromolecular Theory and Simulations ( IF 1.4 ) Pub Date : 2021-05-24 , DOI: 10.1002/mats.202100020
Hidetaka Tobita 1
Affiliation  

1 COVID-19 and Distribution

The year 2020 will be memorized as the year of pandemic. On April 11, 2020, under the state of emergency in Japan, I was staring abstractedly at the website of the Center for Systems Science and Engineering at Johns Hopkins University,[1] listing a large number of the COVID-19 cases by country. Simply because there were so many numbers, I tried the ranking plot, graphically showing the relationship between rank and size. When plotted on a double logarithmic paper, the top countries appear to follow the power law. Since then, my student and I collected the data every week, and Figure 1 shows some examples of the ranking plot.

image
Figure 1
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Ranking plots on the designated dates. The α-value shows the relationship, Rank ∼ xα for large x's.
Because the rank is proportional to the number fraction of the countries whose size is larger than the corresponding size x, the ranking plot shows the upper probability distribution given as follows.
R a n k x N x d x (1)
where N(x) shows the number-based probability density function (pdf).
With the relationship, Rank ∼ xα for large x's, the weight-based pdf, W(x) is given by:
W x x α , for large x s (2)
Assuming a continuous distribution, the weight average is given by:
x ¯ w = 0 x W x d x x α + 1 d x (3)

Equation (3) shows that when the exponent (−α+1) is greater than or equal to −1, the integration goes to infinity, which corresponds to the onset of gelation. The critical value of α is 2. The obtained values shown in Figure 1 are all smaller than 2, which might be interpreted as pandemic.

We found that a simple discrete-time stochastic model[2] that employs the weekly reproduction rate can reproduce the important characteristics of the obtained distribution.[3] On the basis of the model, the necessary conditions to stop the pandemic can also be deduced.

On the other hand, however, the opposite is not always true. Even when a certain mechanistic model can describe the phenomena successfully, this fact does not totally rule out the possibility of other mechanisms. The modeling study is a kind of guessing game, always trying to find out the true culprit. The distribution is a shadow of the stochastic processes.



中文翻译:

发行版

1 COVID-19和发行

2020年将被视为大流行之年。2020年4月11日,在日本处于紧急状态下,我抽象地凝视着约翰·霍普金斯大学系统科学与工程中心的网站,[ 1 ]按国家列出了大量COVID-19案例。仅仅因为有这么多的数字,我尝试了排名图,以图形方式显示了排名和规模之间的关系。当在双对数纸上作图时,排名靠前的国家似乎遵循幂律。从那时起,我和我的学生每周都会收集数据, 1显示了排名图的一些示例。

图像
图1
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在指定日期对地块进行排名。的α值示出的关系, 〜X - α为大X的。
由于排名与规模大于相应规模x的国家/地区的分数比例成正比,因此排名图显示了如下所示的较高概率分布。
[R 一种 ñ ķ X ñ X d X (1)
其中Nx)显示基于数字的概率密度函数(pdf)。
配合关系, 〜X - α为大X “s时,基于重量的PDF格式,w ^X)由下式给出:
w ^ X X - α 为了 X s (2)
假设分布是连续的,则权重平均值为:
X ¯ w = 0 X w ^ X d X X - α + 1个 d X (3)

等式(3)表明,当指数(-α+ 1)大于或等于-1时,积分达到无穷大,这对应于胶凝的开始。α的临界值为2。图1中显示的所得值均小于2,这可能被解释为大流行。

我们发现,采用每周复制率的简单离散时间随机模型[ 2 ]可以复制获得的分布的重要特征。[ 3 ]在该模型的基础上,还可以推断出停止大流行的必要条件。

但是,另一方面,并​​非总是如此。即使某个机制模型可以成功地描述现象,该事实也不能完全排除其他机制的可能性。建模研究是一种猜测游戏,总是试图找出真正的罪魁祸首。分布是随机过程的阴影。

更新日期:2021-05-25
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