指數增長與流行病：英語精讀講義

The phrase exponential growth is familiar to most people.

大多數人都熟悉「指數成長」這個詞。

And yet, human intuition has a hard time really recognizing what it means sometimes. 片語搭配

然而，人類的直覺有時很難真正意識到它的含義。

We can anchor on a sequence of small-seeming numbers 慣用語

我們的思維可能會錨定在一連串看似很小的數字上，

and then become surprised when suddenly those numbers look big,

然後當這些數字突然變大時感到驚訝，

even if the overall trend follows an exponential perfectly consistently.

即使整體趨勢完美且一致地遵循著指數規律。

This right here is the data for the recorded cases of COVID-19,

這正是記錄在案的 COVID-19 病例數據，

aka the coronavirus,

也就是冠狀病毒，

at least at the time that I'm writing this.

至少是在我寫這篇文章（錄製此影片）時的數據。

Never one to waste an opportunity for a math lesson, 文法

作為一個從不錯過任何數學課機會的人，

I thought this might be a good time for all of us to go back to the basics 文法

我覺得現在對我們所有人來說，可能是一個回歸基礎的好時機，

on what exponential growth really is,

去了解什麼是真正的指數成長，

where it comes from,

它是怎麼產生的，

what it implies,

它意味著什麼，

and maybe most pressingly,

也許最迫切的一點是，

how to know when it's coming to an end.

如何知道它何時會結束。

Exponential growth means that as you go from one day to the next, 文法

指數成長意味著，從一天到下一天，

it involves multiplying by some constant.

它涉及乘以某個常數。

In our data, the number of cases in each day

在我們的數據中，每天的病例數

tends to be a multiple of about 1.15 to 1.25 用詞

往往是前一天病例數的

of the number of cases the previous day.

1.15 到 1.25 倍左右。

Now, viruses are a textbook example of this kind of growth 片語搭配

病毒是此類成長的教科書級例子，

because what causes new cases are the existing cases.

因為引起新增病例的正是現有病例。

So, if the number of cases on a given day is n,

所以，如果某天的病例數是 n，

and we say that each individual with the virus is exposed to,

我們假設每個攜帶病毒的人在特定的一天裡，

on average, e people on a given day,

平均會接觸 e 個人，

and each one of those exposures has a probability p

並且每次接觸都有機率 p

of becoming a new infection,

變成新的感染，

then the number of new cases on a given day

那麼在特定的一天裡，新增病例的數量

is e * p * n.

就是 e * p * n。

And the fact that n itself is a factor in its own change

而 n 本身就是其自身變化的一個因數，

is what really makes things go fast

這正是讓事情發展得如此之快的原因，

because if n gets big,

因為如果 n 變大了，

it means the rate of growth itself is getting big.

就意味著成長率本身也在變大。

One way to think about this is that as you add the new cases 表達脈絡

一種思考方式是，當你把新增病例

to get the next day's count,

加上去得出第二天的總數時，

you can factor out the n. 片語動詞

你可以提取出公因式 n。

So, it's just the same as multiplying by some constant

所以，這就等同於乘以某個

that's bigger than one.

大於 1 的常數。

This is sometimes easier to see

有時這會更容易看出來，

if we put the y-axis of our graph on a logarithmic scale,

如果我們把圖表的 y 軸設為對數刻度，

which means that each step of a fixed distance

這意味著固定距離的每一步

corresponds to multiplying by a certain factor.

都對應於乘以一個特定的係數。

In this case, each step is another power of 10.

在這種情況下，每一步就是另一個 10 的次方。

On this scale,

在這個刻度上，

exponential growth should look like a straight line.

指數成長看起來應該是一條直線。

Looking at our data,

看看我們的數據，

it seems like it took 20 days to go from 100 to 1,000

似乎花了 20 天時間才從 100 漲到 1000，

and 13 days to go from that to 10,000.

然後又花了 13 天從 1000 漲到 10000。

And if you do a simple linear regression to find the best fit line, 用詞

如果你做一個簡單的線性迴歸來尋找最佳擬合線，

you can look at the slope of that line to draw a conclusion 片語搭配

你可以透過看那條線的斜率得出結論，

like we tend to multiply by 10 every 16 days,

比如我們傾向於每 16 天乘以 10，

on average.

平均而言。

This regression also lets us be a little more quantitative

這種迴歸也能讓我們更加定量地

about exactly how close the exponential fit really is.

準確了解這種指數擬合到底有多接近。

And to use the technical statistical jargon here, 表達脈絡

套用一下這裡的統計學專業術語，

the answer is that it's really freaking close. 用詞

答案是它真的超級接近。

But it can be hard to digest exactly what that means if true. 慣用語

但如果這是真的，人們可能很難消化這到底意味著什麼。

When you see one country with, say, 6,000 cases 語氣意圖

當你看到一個國家有，比如說 6000 個病例，

and another with 60,

而另一個國家有 60 個，

it's easy to think that the second is doing 100 times better,

很容易讓人覺得第二個國家的情況好 100 倍，

and hence fine.

因此沒問題。

But if you're actually in a situation

但如果你真的身處在這樣一個環境中：

where numbers multiply by 10 every 16 days,

數字每 16 天乘以 10，

another way to view the same fact 表達脈絡

看待同一個事實的另一種方式是，

is that the second country is about a month behind the first.

第二個國家的情況大約落後第一個國家一個月。

Now, this is, of course, rather worrying

如果順著這條線畫下去，

if you draw out the line.

這當然是相當令人擔憂的。

I'm recording this on March 6th,

我錄製這支影片的時間是 3 月 6 日，

and if the present trend continues,

如果目前的趨勢繼續下去，

it would mean hitting a million cases in 30 days,

這將意味著 30 天內達到 100 萬病例，

hitting 10 million in 47 days,

47 天內達到 1000 萬，

100 million in 64 days,

64 天內達到 1 億，

and 1 billion in 81 days.

81 天內達到 10 億。

Needless to say, though, 表達脈絡

不用說，儘管如此，

you can't just draw out a line like this forever.

你不可能永遠這樣畫下去。

It clearly has to start slowing down at some point.

很明顯，它肯定會在某個時刻開始放緩。

But the crucial question is when?

但關鍵問題是何時？

Is it like the SARS outbreak of 2002,

是像 2002 年的 SARS 爆發那樣，

which capped out around 8,000 cases, 片語動詞

上限停留在 8000 個病例左右，

or the Spanish flu of 1918,

還是像 1918 年的西班牙流感那樣，

which ultimately infected about 27% of the world's population?

最終感染了世界人口的大約 27%？

In general, with no context,

通常，在沒有背景資訊的情況下，

just drawing a line through your data

僅僅在你的數據上畫一條線

is not a great way to make predictions. 語氣意圖

並不是做預測的好方法。

But remember,

但請記住，

there's an actual reason to expect an exponential here.

在這裡，有一個現實的理由讓我們預期這是指數級的。

If the number of new cases each day

如果每天新增的病例數

is proportional to the number of existing cases, 用詞

與現有病例數成正比，

it necessarily means each day you multiply by some constant.

那就必然意味著你每天都在乘以某個常數。

So, moving forward d days

因此，向前推進 d 天

is the same as multiplying by that constant d times.

就等於乘以那個常數 d 次。

The only way that stops 文法

這種情況停止的唯一途徑，

is if either the number e or p goes down.

是數字 e 或 p 下降。

It's inevitable that this will eventually happen,

這是不可避免最終會發生的，

even in the most perfectly pernicious model for a virus,

即使在一個最完美地具有破壞力的病毒模型中，

which would be where every day each person with the infection

也就是每天每個感染者

is exposed to a random subset of the world's population,

都會接觸世界人口的一個隨機子集，

at some point most of the people they're exposed to

在某個時刻，他們接觸到的大多數人

would already be sick,

都已經病了，

and so they couldn't become new cases.

所以這些人無法成為新病例。

In our equation,

在我們的等式中，

that would mean that the probability of an exposure becoming a new infection

這將意味著一次接觸變成新感染的機率，

would have to include some kind of factor

必須包括某種係數

to account for the probability that someone you're exposed to 片語動詞

來解釋你所接觸的人

is already infected.

已經被感染的機率。

For a random shuffling model like this,

對於像這樣隨機洗牌的模型，

that could mean including a factor like 1 -

那可能意味著要包含一個像 1 減去

the proportion of people in the world who are already infected.

世界上已感染人口比例的係數。

Including that factor,

把這個係數算進去，

and then solving for how n grows,

然後求解 n 是如何成長的，

you get what's known in the business as a logistic curve, 片語搭配

你就會得到業內所稱的「羅吉斯曲線」（logistic curve），

which is essentially indistinguishable from an exponential at the beginning, 用詞

這種曲線在初期基本上與指數曲線無法區分，

but ultimately it levels out 片語動詞

但最終它會趨於平緩，

once you're approaching the total population size,

一旦你接近總人口規模，

which is what you would expect.

這也是你所預料的那樣。

True exponentials essentially never exist in the real world.

真正的指數成長在現實世界中基本上是不存在的。

Every one of them is really the start of a logistic curve.

每一個指數成長其實都是羅吉斯曲線的開端。

Now, this point right here,

現在的這個點，

where that logistic curve goes from curving upward 文法

也就是羅吉斯曲線從向上彎曲

to instead curving downward,

變成向下彎曲的點，

is known as the inflection point.

被稱為反曲點（inflection point）。

There, the number of new cases each day,

在那裡，每天新增的病例數，

represented by the slope of this curve, 文法

即這條曲線的斜率所代表的數值，

stops increasing and instead stays roughly constant

停止增加，轉而保持大致不變，

before it starts decreasing.

然後才開始減少。

So, one number that people often follow with epidemics

因此，人們在流行病期間經常關注的一個數字

is the growth factor,

就是成長係數，

which is defined as the ratio 片語搭配

它的定義是

between the number of new cases one day

某一天的新增病例數

and the number of new cases the previous day.

與前一天新增病例數的比值。

So, just to be clear, 表達脈絡

所以，為了明確起見，

if you were looking at all of the totals from one day to the next,

如果你關注的是一天到另一天的所有總數，

then tracking the changes between those totals,

然後追蹤這些總數之間的變化，

the growth factor is a ratio between two successive changes.

成長係數就是兩次連續變化之間的比率。

While you're on the exponential part, 表達脈絡

當你處於指數成長階段時，

this factor stays consistently above one.

這個係數始終保持在 1 以上。

Whereas as soon as your growth factor looks closer to one,

而一旦你的成長係數看起來接近 1，

it's a sign that you've hit the inflection.

這就是你已經達到反曲點的訊號。

This can make for another counterintuitive fact 片語動詞

這又會帶來一個違反直覺的事實，

while following the data.

當我們追蹤數據時。

Think about what it would feel like 文法

想想這會是什麼感覺：

for the number of new cases one day

如果某一天的新增病例數

to be about 15% more

多出大約15%，

than the number of new cases the previous day.

與前一天的新增病例數相比。

And contrast that with what it would feel like 表達脈絡

再把這種感覺拿來和另一種感覺相比：

for it to be about the same.

新增病例數與前一天大致相同時的感覺。

Just looking at the totals they result in,

如果僅僅看它們產生的總數，

they don't really feel that different.

其實感覺不到有什麼太大的不同。

But if the growth factor is one,

但是，如果成長係數為 1，

it could mean you're at the inflection point of a logistic,

這可能意味著你處於羅吉斯曲線的反曲點，

which would mean the total number of cases

這意味著病例總數

is going to max out at about two times 片語動詞

的最高峰大概是你現在所在位置的

wherever you are now.

兩倍左右。

But a growth factor bigger than one,

但一個大於 1 的成長係數，

subtle though that might seem, 文法

儘管看起來可能很微妙，

means you're on the exponential part,

卻意味著你正處於指數部分，

which could imply there are orders of magnitude of growth 用詞

這可能暗示著仍有幾個數量級的成長

still waiting ahead of you.

在前方等著你。

Now, while it's true that in the worst-case situation,

誠然，在最糟糕的情況下，

the saturation point is around the total population,

飽和點大概是總人口的數量，

it's of course not at all true

但這顯然根本不是事實，

that people with the virus are randomly shuffled

攜帶病毒的人並非像這樣在全球人口中

around the world's population like this.

被隨機打亂。

People are clustered in local communities.

人們是聚集在當地社區裡的。

However, if you run simulations

然而，如果你運行模擬，

where there's even a little bit of travel

即使這些集群之間

between clusters like this,

只有一點點旅行流動，

the growth is actually not that much different.

成長實際上也沒有太大不同。

What you end up with is a kind of fractal pattern, 文法

你最終得到的是一種碎形圖案，

where communities themselves function like individuals.

各個社區本身就像個體一樣運作。

Each one has some exposure to others

每一個都與其他社區有一定程度的接觸，

with some probability of spreading the infection,

且有一定的機率傳播感染，

so the same underlying and exponential inducing laws apply.

因此同樣適用引起指數級成長的底層規律。

Fortunately, saturating the whole population

幸運的是，讓全部人口達到飽和

is not the only thing that can cause 文法

並不是唯一能導致

the two factors we care about to go down.

我們關心的兩個因素下降的原因。

The amount of exposure can also go down

當人們停止聚會和旅行時，

when people stop gathering and traveling,

接觸量也可以隨之減少，

and the infection rate can go down

只要人們多洗手，

when people just wash their hands more.

感染率也會下降。

The other thing that's counterintuitive about exponential growth,

關於指數成長的另一個違反直覺之處在於，

this time in a more optimistic sense,

這一次是在比較樂觀的意義上，

is just how sensitive it is to this constant. 文法

它是多麼地對這個常數敏感。

For example, if it's 15%,

例如，如果它是 15%，

like it is as I'm recording this,

就像我錄製影片時這樣，

and we're at 21,000 cases now,

而我們現在的病例數是 21000，

that would mean that 61 days from now

那就意味著從現在起 61 天後

you hit over 100 million.

會突破 1 億。

But, if through a bit less exposure and infection,

但是，如果透過稍微減少一點接觸和感染，

that rate drops down to 5%,

這個比例降到了 5%，

it doesn't mean the projection also drops down

這並不意味著預測數字也隨之下降到

by a factor of three. 用詞

三分之一。

It actually drops down to around 400,000.

它實際上會降至 40 萬左右。

So, if people are sufficiently worried, 表達脈絡

所以，如果人們有足夠的憂患意識，

then there's a lot less to worry about.

那麼擔心的事就會少很多。

But if no one is worried,

但如果沒人擔心，

that's when you should worry.

那才是你該擔心的時候。