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We humans are pretty good at understanding linear change. It's all around us:
--eat more, gain some weight; eat less, lose some weight
--press on the gas pedal, speed up; press on the brake, slow down
--turn up the heat, feel warmer; turn down the heat, feel cooler
--someone can't hear you, talk louder; someone moves closer, talk more softly
--and thousands of other daily experiences in which a small change produces a small result and a larger change produces a larger result.
However, some events don't behave linearly. There are situations where a small change can quickly produce a big effect. We're living through at least two of them right now--climate change and the Covid-19 pandemic. We're failing miserably in understanding and dealing with both of them.
Let's do a little math--please bear with me:
Step number Linear growth (n x 2) Exponential growth (2 to the n)
1 2 2
2 4 4
3 6 8
4 8 16
5 10 32
6 12 64
7 14 128
8 16 256
9 18 512
10 20 1024
11 22 2048
Let's pretend that these numbers represent two different responses to the Covid-19 virus.
The steps might represent weeks. The first column might represent a situation where the virus is fairly well controlled (e.g. through some combination of social distancing, mask wearing, testing, tracing and isolating people likely to shed the virus). On average one infected person infects approximately one other person, with the result that the number of cases grows linearly. The third column represents exponential growth--where the virus truly is "going viral." All that takes is for one infected person to pass the virus on to more than one new victim.
Notice the enormous and rapidly expanding difference as the weeks tick by. At week three, exponential growth has infected just two more people than in the linear growth model, but at ten weeks the difference is more than 1,000, and one week later, more than 2,000.
So, our first takeaway is that when we're dealing with any situation involving exponential growth, at first it's hard to distinguish from a normal linear situation that we're used to, but sooner or later it takes off explosively.
Now let's examine that phrase "sooner or later."
Week Doubling stopped sooner Doubling stopped later
1 2 2
2 4 4
3 8 8
4 16 16
5 18 32
6 20 64
7 22 128
8 24 256
9 26 258
10 28 260
11 30 262
Total Cases: 198 1290
What a difference a delay makes. This might represent two states, one of which sees the exponential handwriting on the wall after just four weeks, institutes effective controls and ends up with 198 cases, while the other delays control measures one month more and ends up with with 6.5 times as many cases. (Or these numbers could equally well represent deaths, in which case that one month delay would have caused over a thousand needless deaths.
Obviously the numbers above are meant to be an illustration; they don't match exactly with the real world. However, you can see this in the real world if you compare the impact of a one-week delay in responding to the pandemic in New York vs. California. Here's a zerospinzone.blogspot post from March 30:
In this comparison of coronavirus cases in New York vs California, you can see what difference even a few days lag in imposing strict stay-at-home orders makes. California--population 39.5 million, first state to impose strict controls: current # of cases 6358, deaths 132. New York--population 19.5 million, delayed a week before imposing controls: current # of cases 60,679, deaths 1063. The number of cases per population in NY is 19 times greater than in California, and the number of deaths per population 17 times greater. What a tragic difference a week's delay makes.
A lot has happened in New York and California in the subsequent three months, with start-and-stop openings and closings in both states. Through all that, however, California has managed to keep a tighter lid on the explosive potential of the coronavirus. As of today, June 28, New York has documented 416,769 cases and 31,483 deaths while Callifornia has had 215,575 cases and just 5,934 deaths. Adjusted for population, late-starting New York has suffered nearly four times as many cases and more than ten times as many deaths as California. Clamping down just one week earlier in California almost certainly saved tens of thousands of lives.
Our second takeaway, then, is that when dealing with the potential for exponential growth, getting control of the rate of growth as early as possible--before the number of cases explodes-- makes a huge difference.
It's pretty clear that we, as a species, aren't very good at dealing with things that have the potential for exponential growth, and so can blast out of control at any time. We're seeing the results of this globally as we try to put a cap on the coronavirus. Similarly, experts have been warning us for decades that, because of a number of looming tipping points, climate change too has the potential to spin wildly out of control.
However, we don't have a good excuse for continued ignorance; it's not as though the potential for dangerous exponential growth has been a secret. Consider this old proverb, a version of which goes like this:
For want of a nail a horseshoe was lost,
for want of a horseshoe a horse went lame,
for want of a horse a rider never got through,
for want of a rider a message never arrived,
for want of a message an army was never sent,
for want of an army a battle was lost,
for want of a battle a war was lost,
for want of a war a kingdom fell,
and all for want of a nail.
Clearly we've had folk knowledge about exponentially large consequences from a small event for a lot time.
Even more ancient is a story from India about a chess-loving king who promised to reward a visitor with whatever he wanted. The visitor "modestly" asked for one grain of rice on the first square of a chessboard, 2 grains on the second square, 4 on the third, 8 on the fourth, etc. It didn't take the King many squares before he realized he'd taken on an impossible task. Sixty-four doublings would have required some 18,000,000,000,000,000,000 grains of rice, enough to cover all of India a meter deep.
That ancient king learned a striking and costly lesson about exponential growth. It seems as though we--and our current leaders--need to learn it all over again, and quickly. That is, if we don't want our kingdom to fall for want of an adequate response to this microscopic but immensely dangerous virus.
-----
REA
--eat more, gain some weight; eat less, lose some weight
--press on the gas pedal, speed up; press on the brake, slow down
--turn up the heat, feel warmer; turn down the heat, feel cooler
--someone can't hear you, talk louder; someone moves closer, talk more softly
--and thousands of other daily experiences in which a small change produces a small result and a larger change produces a larger result.
However, some events don't behave linearly. There are situations where a small change can quickly produce a big effect. We're living through at least two of them right now--climate change and the Covid-19 pandemic. We're failing miserably in understanding and dealing with both of them.
A sneeze--a few tiny droplets can cause massive disruption
Photo credit: CDC
Let's do a little math--please bear with me:
Step number Linear growth (n x 2) Exponential growth (2 to the n)
1 2 2
2 4 4
3 6 8
4 8 16
5 10 32
6 12 64
7 14 128
8 16 256
9 18 512
10 20 1024
11 22 2048
Let's pretend that these numbers represent two different responses to the Covid-19 virus.
The steps might represent weeks. The first column might represent a situation where the virus is fairly well controlled (e.g. through some combination of social distancing, mask wearing, testing, tracing and isolating people likely to shed the virus). On average one infected person infects approximately one other person, with the result that the number of cases grows linearly. The third column represents exponential growth--where the virus truly is "going viral." All that takes is for one infected person to pass the virus on to more than one new victim.
Notice the enormous and rapidly expanding difference as the weeks tick by. At week three, exponential growth has infected just two more people than in the linear growth model, but at ten weeks the difference is more than 1,000, and one week later, more than 2,000.
So, our first takeaway is that when we're dealing with any situation involving exponential growth, at first it's hard to distinguish from a normal linear situation that we're used to, but sooner or later it takes off explosively.
Now let's examine that phrase "sooner or later."
Week Doubling stopped sooner Doubling stopped later
1 2 2
2 4 4
3 8 8
4 16 16
5 18 32
6 20 64
7 22 128
8 24 256
9 26 258
10 28 260
11 30 262
Total Cases: 198 1290
What a difference a delay makes. This might represent two states, one of which sees the exponential handwriting on the wall after just four weeks, institutes effective controls and ends up with 198 cases, while the other delays control measures one month more and ends up with with 6.5 times as many cases. (Or these numbers could equally well represent deaths, in which case that one month delay would have caused over a thousand needless deaths.
Obviously the numbers above are meant to be an illustration; they don't match exactly with the real world. However, you can see this in the real world if you compare the impact of a one-week delay in responding to the pandemic in New York vs. California. Here's a zerospinzone.blogspot post from March 30:
In this comparison of coronavirus cases in New York vs California, you can see what difference even a few days lag in imposing strict stay-at-home orders makes. California--population 39.5 million, first state to impose strict controls: current # of cases 6358, deaths 132. New York--population 19.5 million, delayed a week before imposing controls: current # of cases 60,679, deaths 1063. The number of cases per population in NY is 19 times greater than in California, and the number of deaths per population 17 times greater. What a tragic difference a week's delay makes.
A lot has happened in New York and California in the subsequent three months, with start-and-stop openings and closings in both states. Through all that, however, California has managed to keep a tighter lid on the explosive potential of the coronavirus. As of today, June 28, New York has documented 416,769 cases and 31,483 deaths while Callifornia has had 215,575 cases and just 5,934 deaths. Adjusted for population, late-starting New York has suffered nearly four times as many cases and more than ten times as many deaths as California. Clamping down just one week earlier in California almost certainly saved tens of thousands of lives.
Our second takeaway, then, is that when dealing with the potential for exponential growth, getting control of the rate of growth as early as possible--before the number of cases explodes-- makes a huge difference.
It's pretty clear that we, as a species, aren't very good at dealing with things that have the potential for exponential growth, and so can blast out of control at any time. We're seeing the results of this globally as we try to put a cap on the coronavirus. Similarly, experts have been warning us for decades that, because of a number of looming tipping points, climate change too has the potential to spin wildly out of control.
However, we don't have a good excuse for continued ignorance; it's not as though the potential for dangerous exponential growth has been a secret. Consider this old proverb, a version of which goes like this:
For want of a nail a horseshoe was lost,
for want of a horseshoe a horse went lame,
for want of a horse a rider never got through,
for want of a rider a message never arrived,
for want of a message an army was never sent,
for want of an army a battle was lost,
for want of a battle a war was lost,
for want of a war a kingdom fell,
and all for want of a nail.
Clearly we've had folk knowledge about exponentially large consequences from a small event for a lot time.
Even more ancient is a story from India about a chess-loving king who promised to reward a visitor with whatever he wanted. The visitor "modestly" asked for one grain of rice on the first square of a chessboard, 2 grains on the second square, 4 on the third, 8 on the fourth, etc. It didn't take the King many squares before he realized he'd taken on an impossible task. Sixty-four doublings would have required some 18,000,000,000,000,000,000 grains of rice, enough to cover all of India a meter deep.
That ancient king learned a striking and costly lesson about exponential growth. It seems as though we--and our current leaders--need to learn it all over again, and quickly. That is, if we don't want our kingdom to fall for want of an adequate response to this microscopic but immensely dangerous virus.
-----
REA
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