指数增长与流行病：英语精读讲义

The phrase exponential growth is familiar to most people.

大多数人都熟悉“指数增长”这个词。

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

然而，人类的直觉有时很难真正认识到它的含义。

We can anchor on a sequence of small-seeming numbers 習語

我们的思维可能会锚定在一连串看似很小的数字上，

and then become surprised when suddenly those numbers look big,

然后当这些数字突然变大时感到惊讶，

even if the overall trend follows an exponential perfectly consistently.

即使整体趋势完美而一致地遵循着指数规律。

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

这正是记录在案的 COVID-19 病例数据，

aka the coronavirus,

也就是冠状病毒，

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

至少是在我写这篇文章（录制此视频）时的数据。

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

作为一个从不错过任何数学课机会的人，

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

我觉得现在对我们所有人来说，可能是一个回归基础的好时机，

on what exponential growth really is,

去了解什么是真正的指数增长，

where it comes from,

它是怎么产生的，

what it implies,

它意味着什么，

and maybe most pressingly,

也许最紧迫的一点是，

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

如何知道它何时会结束。

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

指数增长意味着，从一天到下一天，

it involves multiplying by some constant.

它涉及乘以某个常数。

In our data, the number of cases in each day

在我们的数据中，每天的病例数

tends to be a multiple of about 1.15 to 1.25 用詞

往往是前一天病例数的

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 短語搭配

病毒是此类增长的教科书级例子，

because what causes new cases are the existing cases.

因为引起新增病例的正是现有病例。

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

所以，如果某天的病例数是 n，

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

我们假设每个携带病毒的人在特定的一天里，

on average, e people on a given day,

平均会接触 e 个人，

and each one of those exposures has a probability p

并且每次接触都有概率 p

of becoming a new infection,

变成新的感染，

then the number of new cases on a given day

那么在特定的一天里，新增病例的数量

is e * p * n.

就是 e * p * n。

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

而 n 本身就是其自身变化的一个因数，

is what really makes things go fast

这正是让事情发展得如此之快的原因，

because if n gets big,

因为如果 n 变大了，

it means the rate of growth itself is getting big.

就意味着增长的速度本身也在变大。

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

一种思考方式是，当你把新增病例

to get the next day's count,

加上去得出第二天的总数时，

you can factor out the n. 短語動詞

你可以提取出公因式 n。

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

所以，这就等同于乘以某个

that's bigger than one.

大于 1 的常数。

This is sometimes easier to see

有时这会更容易看出来，

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

如果我们把图表的 y 轴设为对数刻度，

which means that each step of a fixed distance

这意味着固定距离的每一步

corresponds to multiplying by a certain factor.

都对应于乘以一个特定的系数。

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

在这种情况下，每一步就是另一个 10 的次方。

On this scale,

在这个刻度上，

exponential growth should look like a straight line.

指数增长看起来应该是一条直线。

Looking at our data,

看看我们的数据，

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

似乎花了 20 天时间才从 100 涨到 1000，

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

然后又花了 13 天从 1000 涨到 10000。

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

如果你做一个简单的线性回归来寻找最佳拟合线，

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

你可以通过看那条线的斜率得出结论，

like we tend to multiply by 10 every 16 days,

比如我们倾向于每 16 天乘以 10，

on average.

平均而言。

This regression also lets us be a little more quantitative

这种回归也能让我们更加定量地

about exactly how close the exponential fit really is.

准确了解这种指数拟合到底有多接近。

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

套用一下这里的统计学专业术语，

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

答案是它真的超级接近。

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

但如果这是真的，人们可能很难消化这到底意味着什么。

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

当你看到一个国家有，比如说 6000 个病例，

and another with 60,

而另一个国家有 60 个，

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

很容易让人觉得第二个国家的情况好 100 倍，

and hence fine.

因此没问题。

But if you're actually in a situation

但如果你真的身处在这样一个环境中：

where numbers multiply by 10 every 16 days,

数字每 16 天乘以 10，

another way to view the same fact 表達脈絡

看待同一个事实的另一种方式是，

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

第二个国家的情况大约落后第一个国家一个月。

Now, this is, of course, rather worrying

如果顺着这条线画下去，

if you draw out the line.

这当然是相当令人担忧的。

I'm recording this on March 6th,

我录制这个视频的时间是 3 月 6 日，

and if the present trend continues,

如果目前的趋势继续下去，

it would mean hitting a million cases in 30 days,

这将意味着 30 天内达到 100 万病例，

hitting 10 million in 47 days,

47 天内达到 1000 万，

100 million in 64 days,

64 天内达到 1 亿，

and 1 billion in 81 days.

81 天内达到 10 亿。

Needless to say, though, 表達脈絡

不用说，尽管如此，

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

你不可能永远这样画下去。

It clearly has to start slowing down at some point.

很明显，它肯定会在某个时刻开始放缓。

But the crucial question is when?

但关键问题是何时？

Is it like the SARS outbreak of 2002,

是像 2002 年的 SARS 爆发那样，

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

上限停留在 8000 个病例左右，

or the Spanish flu of 1918,

还是像 1918 年的西班牙流感那样，

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

最终感染了世界人口的大约 27%？

In general, with no context,

通常，在没有背景信息的情况下，

just drawing a line through your data

仅仅在你的数据上画一条线

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

并不是做预测的好方法。

But remember,

但请记住，

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

在这里，有一个现实的理由让我们预期这是指数级的。

If the number of new cases each day

如果每天新增的病例数

is proportional to the number of existing cases, 用詞

与现有病例数成正比，

it necessarily means each day you multiply by some constant.

那就必然意味着你每天都在乘以某个常数。

So, moving forward d days

因此，向前推进 d 天

is the same as multiplying by that constant d times.

就等于乘以那个常数 d 次。

The only way that stops 語法

这种情况停止的唯一途径，

is if either the number e or p goes down.

是数字 e 或 p 下降。

It's inevitable that this will eventually happen,

这是不可避免最终会发生的，

even in the most perfectly pernicious model for a virus,

即使在一个最完美地具有破坏力的病毒模型中，

which would be where every day each person with the infection

也就是每天每个感染者

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

都会接触世界人口的一个随机子集，

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

在某个时刻，他们接触到的大多数人

would already be sick,

都已经病了，

and so they couldn't become new cases.

所以这些人无法成为新病例。

In our equation,

在我们的等式中，

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

这将意味着一次接触变成新感染的概率，

would have to include some kind of factor

必须包括某种系数

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

来解释你所接触的人

is already infected.

已经被感染的概率。

For a random shuffling model like this,

对于像这样随机洗牌的模型，

that could mean including a factor like 1 -

那可能意味着要包含一个像 1 减去

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

世界上已感染人口比例的系数。

Including that factor,

把这个系数算进去，

and then solving for how n grows,

然后求解 n 是如何增长的，

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

你就会得到业内所称的“逻辑斯谛曲线”，

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

这种曲线在初期基本上与指数曲线无法区分，

but ultimately it levels out 短語動詞

但最终它会趋于平缓，

once you're approaching the total population size,

一旦你接近总人口规模，

which is what you would expect.

这也是你所预料的那样。

True exponentials essentially never exist in the real world.

真正的指数增长在现实世界中基本上是不存在的。

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

每一个指数增长其实都是逻辑斯谛曲线的开端。

Now, this point right here,

现在的这个点，

where that logistic curve goes from curving upward 語法

也就是逻辑斯谛曲线从向上弯曲

to instead curving downward,

变成向下弯曲的点，

is known as the inflection point.

被称为拐点。

There, the number of new cases each day,

在那里，每天新增的病例数，

represented by the slope of this curve, 語法

即这条曲线的斜率所代表的数值，

stops increasing and instead stays roughly constant

停止增加，转而保持大致不变，

before it starts decreasing.

然后才开始减少。

So, one number that people often follow with epidemics

因此，人们在流行病期间经常关注的一个数字

is the growth factor,

就是增长系数，

which is defined as the ratio 短語搭配

它的定义是

between the number of new cases one day

某一天的新增病例数

and the number of new cases the previous day.

与前一天新增病例数的比值。

So, just to be clear, 表達脈絡

所以，为了明确起见，

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

如果你关注的是一天到另一天的所有总数，

then tracking the changes between those totals,

然后追踪这些总数之间的变化，

the growth factor is a ratio between two successive changes.

增长系数就是两次连续变化之间的比率。

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

当你处于指数增长阶段时，

this factor stays consistently above one.

这个系数始终保持在 1 以上。

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

而一旦你的增长系数看起来接近 1，

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

这就是你已经达到拐点的信号。

This can make for another counterintuitive fact 短語動詞

这又会带来一个违反直觉的事实，

while following the data.

当我们追踪数据时。

Think about what it would feel like 語法

想想这会是什么感觉：

for the number of new cases one day

如果某一天的新增病例数

to be about 15% more

多出大约15%，

than the number of new cases the previous day.

与前一天的新增病例数相比。

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

再把这种感觉与另一种情况对比一下，

for it to be about the same.

也就是新增病例数大致相同时的感觉。

Just looking at the totals they result in,

如果仅仅看它们产生的总数，

they don't really feel that different.

其实感觉不到有什么太大的不同。

But if the growth factor is one,

但是，如果增长系数为 1，

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

这可能意味着你处于逻辑斯谛曲线的拐点，

which would mean the total number of cases

这意味着病例总数

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

的最高峰大概是你现在所在位置的

wherever you are now.

两倍左右。

But a growth factor bigger than one,

但一个大于 1 的增长系数，

subtle though that might seem, 語法

尽管看起来可能很微妙，

means you're on the exponential part,

却意味着你正处于指数部分，

which could imply there are orders of magnitude of growth 用詞

这可能暗示着仍有几个数量级的增长

still waiting ahead of you.

在前方等着你。

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

诚然，在最糟糕的情况下，

the saturation point is around the total population,

饱和点大概是总人口的数量，

it's of course not at all true

但这显然根本不是事实，

that people with the virus are randomly shuffled

携带病毒的人并非像这样在全球人口中

around the world's population like this.

被随机打乱。

People are clustered in local communities.

人们是聚集在当地社区里的。

However, if you run simulations

然而，如果你运行模拟，

where there's even a little bit of travel

即使这些集群之间

between clusters like this,

只有一点点旅行流动，

the growth is actually not that much different.

增长实际上也没有太大不同。

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

你最终得到的是一种分形图案，

where communities themselves function like individuals.

各个社区本身就像个体一样运作。

Each one has some exposure to others

每一个都与其他社区有一定程度的接触，

with some probability of spreading the infection,

且有一定的概率传播感染，

so the same underlying and exponential inducing laws apply.

因此同样适用引起指数级增长的底层规律。

Fortunately, saturating the whole population

幸运的是，让全部人口达到饱和

is not the only thing that can cause 語法

并不是唯一能导致

the two factors we care about to go down.

我们关心的两个因素下降的原因。

The amount of exposure can also go down

当人们停止聚会和旅行时，

when people stop gathering and traveling,

接触量也可以随之减少，

and the infection rate can go down

只要人们多洗手，

when people just wash their hands more.

感染率也会下降。

The other thing that's counterintuitive about exponential growth,

关于指数增长的另一个违反直觉之处在于，

this time in a more optimistic sense,

这一次是在比较乐观的意义上，

is just how sensitive it is to this constant. 語法

它是多么地对这个常数敏感。

For example, if it's 15%,

例如，如果它是 15%，

like it is as I'm recording this,

就像我录制视频时这样，

and we're at 21,000 cases now,

而我们现在的病例数是 21000，

that would mean that 61 days from now

那就意味着从现在起 61 天后

you hit over 100 million.

会突破 1 亿。

But, if through a bit less exposure and infection,

但是，如果通过稍微减少一点接触和感染，

that rate drops down to 5%,

这个比例降到了 5%，

it doesn't mean the projection also drops down

这并不意味着预测数字也随之下降到

by a factor of three. 用詞

三分之一。

It actually drops down to around 400,000.

它实际上会降至 40 万左右。

So, if people are sufficiently worried, 表達脈絡

所以，如果人们有足够的忧患意识，

then there's a lot less to worry about.

那么担心的事就会少很多。

But if no one is worried,

但如果没人担心，

that's when you should worry.

那才是你该担心的时候。