Hacks of System Engineering to Understand Exponential Growth and Vaccine Panic.
Okay! Today I get to play armchair epidemiologist! But first, let's all learn the Rule of 72!
The Rule of 72 is a way to understand exponential growth. Exponential growth is something we see all the time in information technologies. Moore's law [1] is an example. It says that every two years the number of transistors you can fit into a fixed area doubles.
This picture is updated from one of Kurzweill's books: [2] The Singularity is Near. It shows several things that will become very useful to be able to predict exponential growth rates. There's a linear X-axis that goes in terms of years (it could be any standard increment of time, days, hours, weeks, months, etc. Then there is a Y-axis that is given in powers of 10. Each tick represents 10 times the tick below it. Thus a straight line in this graph is actually an exponential growth curve. For every increment of time you get a multiplicative increase in what is measured.
It turns out this is also the way interest works and there's a well known hack for interest rates called the Rule of 72. You take the interest rate and divide it into 72 and that tells you how long it takes for your money to double. Lets do an example.
Assume you get 12% (Interest) on your $100 (Principle) deposited in a bank (it happened in the 80's, just sayin') that means after the first year your hundred dollars turns into $112 dollars. The formula is
P * (1 + I) = P',
where P' is the principle after one year (the same time the interest rate is quoted over.) To find out what happens after N years you just have to keep multiplying by 1 + I:
P'(N) = P * (1 + I) ^ N.
Where the " ^ N " (to the Nth power) just means you multiple (1 + I) together N times: (1 + I) * (1 + I) * (1 + I).... N times. 1.12 * 1.12 = 1.25, 1.25 * 1.12 = 1.40, 1.40 * 1.12 = 1.57, 1.57 * 1.12 = 1.76, 1.76 * 1.12 = 1,98 (essentially doubled!) How many times did we have to multiple 1.12 times itself to get to 2? 6. Six times. (You can count them.) And it continues into the future. If you want to double your money again, it's six more years. So now you have four times as much money. Or double it again, now we have 8 times as much money after six more years. How long did that take? We had three doublings (2, 4, 8) and each one took six years. So we have 8 times as much money in 3 * 6 = 18 years if we earn 12% interest.
Now where does the Rule of 72 come in? It's a shorthand way of figuring out he doubling time. To figure out how many periods it takes the return to double you can just divide the interest rate for a perioid into 72 and that tells you how many periods it takes to double. 72 divide by 12 is 6. So it takes six years (the period) to double your returns.
Lets' look at Moore's law = the thing that doubles takes 2 years. 72 divided by I = 2, so I must by 36%. Every year the engineers are forced to increase the number of transistors in a fixed area by 36%. Now Moore was being conservative as this means every year the width and length of a transistor must get smaller by only 20%. Why is this so? If a transistor's side is 80% as big as it was last year, then the area it takes up is 80% * 80% = 64% as much space, or 36% less area. So by making transistors 20% smaller we get twice as many in a fixed area every two years.
What else can use this exponential growth hacks on?
Well, the most disturbing thing in the last few years I used this rule on was the outbreak of the Covid-19 Virus last year.
What does exponential growth mean for the Covid-19 Virus?
Back in late January 2020, I saw the Chinese building field hospitals that could handle thousands of very sick people. They thought this virus was going to be very bad (and they were right.) But how bad? Even in China at this time there were only a few hundred deaths, but they obviously expected a whole lot more., on January 23 the Chinese reported 250+ cases confirmed in the laboratory. 2 ^ 8 = 256. This means that the number of infections had doubled 8 times. The earliest known cases were from a Doctor who reported the deaths of 4-8 patients in mid-December, and caught the disease and died himself. (Unbelievably sad that he was ignored by the authorities - one of the problems that a one-party state makes worse is hiding relevant information so they won't look bad...)
It turns out that cases were popping up in other places. Nobody had stopped travel from Wuhan China, and it was spreading across China (Beijing, Shanghai.) What would stop it from coming to the US. NOTHING. It was already here. You can just look at the flights out of Wuhan to the US, assume the infection rate was the same as in China and you would likely conclude that a few infected people from China were already here in the US. And living in a town that is more than half Chinese, it was likely someone had the virus and lived less than two miles from my house. So now the critical number becomes how fast are people infecting others? How many people do they typically infect over the course of the disease? The dreaded R0.
If you want to find out what the professionals were doing at this time, I suggest you read Michael Lewis's book "The Premonition: A Pandemic Story". He goes into great detail about how some of the smartest doctors in the land were looking at the tragedy that was about to unfold and what they were trying to do about it. Myself, I'm not a professional medical person, I'm just a lowly physicist who pretends to do system engineering. So I sat down in late January and said, "What's going to happen in my town?" I was already confident that there were infected people in my town. We already have an estimate of how fast it's spreading in China and they went into a serious lockdown about a week later, trying to stop the spread of the disease. And they did, but not until almost 10,000 people died.
Looking at the doubling rate of 8 times in about 7 weeks scared the piss out of me. This disease was doubling every week. Every week. If we did nothing we can use the rule of 72 to figure out when everyone in the US would catch the disease.
Let's practice!
Assume that only one person from China got to my town with the infection. During that week they spread it to two people. Then the next week those two each spread it to two more people. How many doublings do we need to get to 350 Million? We can use some shorthand knowledge of powers of 2. 2 ^ 10 = 1024. If you double something ten times you get one thousand times more of it. Continue it ten more times and you get a million times more of it. (One thousand times one thousand is a million.) We need to double it to 350, which is about 9 more times. It doubles every week. In 29 weeks everyone in the United States would likely have gotten Covid-19 if we did nothing. So by the time my birthday rolled around in 6 months, 3.5 million people would be dead in the US. (The death rate at that time was estimated to be around 1-1.5%, but maybe as high as 3%...)
This did not make me very happy. In fact, it threw me into a panic. I had just returned from a trip to India through Singapore, and they weren't fooling around. Everyone was checked for a fever when they got off the plane. If you had one you were taken away and quarantined for two weeks. In the US, the government was laughably lackadaisical about the whole thing. "It'll just go away. It'll be a miracle. One day you'll wake up and it'll be gone." (And lots of your friends will be dead, I was thinking.)
But some instinct kicked in finally and California declared a lockdown in early March. A week or two later the New York area did the same. That two week delay caused havoc with people's lives in New York. While the doubling rate in California had been basically halved two weeks earlier, New York just missed the bus by that much and the virus raged through the population for multiple weeks until their lockdown stopped the spread and tens of thousands of people had died. For the entire month of February plus a week of January and a week of March went by and the federal government did nothing. 6 doubling times. 2, 4, 8, 16, 32, 64. Where there had been one case in my town, there were now 64. If we had waited that extra two weeks there would have been 256. If we waited as long as New York it would have been 1000 cases. If we had waited an additional 6 weeks, every one in my town would have been sick (2 ^ 6 is 64 or 64,000 cases, but there's only 50,000 people in my town.) It's just the mathematics of exponential growth.
The amazing thing is that for the first time in history a lockdown actually worked and prevented this disaster from happening. My entire town didn't get the virus (and thank god for that, my guts blew out the same day we declared a lockdown. If we hadn't, and my gut had held together a little longer, I'd have been out of luck as the hospitals would be full of Covid-19 patients and I'd be in the ground, dead from neglect.)
I didn't die. (the doctor said that if my disease had occurred thirty years earlier, I would have been dead. And the recovery time was about six weeks. When my grandmother had the same disease, they caught it earlier, so she lived, but her recovery time was a year. Thank the Lord for small incremental improvements in medical care over the last sixty years. It's all about information, and information changes can keep improving for a long time at a steady rate and the accumulated improvements really do save lives.)
So now what other equation can we learn how to use to help us in these perilous times? We can learn how to solve the birthday paradox.
Solving the Birthday Paradox, for Fun and Safety!
What is the birthday paradox? It's how you calculate the odds that anyone has the same birthday in a group of people. It's very similar to the Rule of 72. Here's the way to reason about this situation. If there are two people in a room, what are they odds that they have the same birthday? My birthday is on one particular day. What are the odds some random person has that same birthday? The probability, Pb, is one out of 365 or about a 0.3% chance. What are the chances that person doesn't have the same birthday as me? 364 / 365, or 99.7% chance our birthdays don't match or (1 - (1 / 365)) or in algebraic form:
P(no common birthday) = (1 - Pb)
for N people we can multiply the probabilities together, every time we introduce another person, we just have to multiply the odds together. This tells us the probability that no birthdays overlap, so the odds that anyone's birthdays are the same is just 1 - P(no common birthday). Actually, the birthday problem is worse than the Covid-19 problem because it has a limited number of dates for birthdays, so the odds of the third person is (1 - 2/365) that they don't have the same birthday as the first two, and the odds that the fourth person doesn't have the same birthday as the other three are (1 - 3/365) and so on.
For Covid-19 it's much easier. We can look at the infection rates in the area surrounding an event, we can see how many people will be at the event and we can calculate what are the odds that anyone has Covid-19 at the event. Let's try a sample!
I wanted to go to a concert in Napa last month. The venue was outside (smaller chance of Covid-a9 spreading over the air as the molecules get blown away quickly when you're outside.) The venue was small, about 500 people. The number of cases per day that were being found in Napa was about 10 per 100,000 people. A case lasts bout ten days, during which time we will assume the person was infectious the whole time. This means there were 100 people with Covid-19 in Napa at that time. So the probablility of someone having Covid-19, Pc, was about 100/100,000 or 0.1%. (Yes, that's only the tested people, so probably more had it than that... but I was planning on wearing my mask which reduces the odds of catching the virus by about a factor of 10, so I felt my estimates were safe.) So the odds of any one of the concert goers not having covid-19 was (1 - Pc). The odds of two people not having Covid-19 was (1 - Pc) * (1 - Pc). The odds of nobody in N people having Covid-19, P(N), is:
P(N) = (1 - Pc) ^ N.
And the odds of somebody (or more than one person) having Covid-19 is 1 - P(N).
Look familiar? Look at the first equation in this blog. Remember the caret symbol is exponentiation. You multiply the stuff in parenthesis with itself N times. These equations are just two sides of the same coin. So how does this work out for the concert? I can use my handy-dandy scientific calculator to give the answer:
P(500) = (1 - 0.001) ^ 500 = (0.999) ^ 500
1 - P(500) in Napa at that time was 40%. The odds that someone at the concert had Covid-19 was 40% if they let random people in the concert. However, they did not let random people in to the concert, you had to show your proof of vaccination. How does this affect the odds that someone at the concert has Covid-19. For that we have to calculate what are the odds a vaccinated person has Covid-19, in other words, how prevalent breakthrough cases are. There's a lot of conflicting data on this, but the odds of a vaccinated person having Covid-19 started out as 20 to one when compared to a non-vaccinated person. This is what the vaccine trials showed. When they say the vaccine is 95% effective it means that for every 100 unvaccinated people that would have got the virus only 5 people actually got it. Twenty to 1. At this time the delta variation had started to spread across the US and the vaccines are much less effective against that variant. Looking at the numbers it was from 6 times to 2.5 times less likely to catch the delta variant (depending on whose numbers you believe, how much of the delta variant is actually out there when you do the test and lots of other issues. The number that got me, though, was that the hospitalization rates and death rates were still down by a factor of 10, even with the Delta variant. You might catch it, but you weren't going to die from it.) So what are the odds of someone having Covid-19 at the concert if everyone is vaccinated? The odds of a single vaccinated person having it are 10x less (with masks and not so much Delta around...) than a vaccinated person. So we do the same calculation, only this time the odds of having Covid-19 are one out of a hundred thousand. This means the P(500) for vaccinated people, the chance that anyone at the concert has Covid-19 is not 40% but it's actually closer to 1 - (0.9999) ^ 500 = 1 - 95% = 5%. The odds that one or more vaccinated people at the concert had Covid-19 was 5%. So 95% of the time nobody who went to the concert had Covid-19. Even if a few people at the concert had Covid-19, I was wearing a mask which reduced the odds even more. Even if the one person who had Covid-19 was sitting close to me, I was unlikely to get the disease since we were outdoors and the wind was quickly blowing any virus they were emitting away. I felt safe.
And I was safe. This is why California is looking to reduce it's mask mandates. So few people have Covid-19 that the odds you are going to catch it if most people are vaccinated is miniscule. Would I got to a concert that had ten times as many people? I have. Would I got to a theme park that had another ten times as many people? I'm planning on doing it in a few weeks when it looks like the infection rate will be down by another factor of ten. Have we finally whipped this virus? If everyone got the vaccine, then yes we could whip this virus. If Covidiots continue to resist the vaccine they will continue to die at a higher rate than the vaccinated are dying. It's a chance to see how evolution actually works. Those who insist on virtue signaling (My immune system is strong enough to naturally survive the virus! I don't need any help... Yeah, you do, and the vaccine basically helps your body make it's own mono-clonal antibodies that help you fight the infection.) Get Vaxed.
Get Vaxed! Take the vaccination. Don't try to 'man up' and show how 'fit' you are and end up infected and dying like a whole host of right-wing radio talk show hosts did last month. And if you think it's better to not get vaccinated because your god given immune system should protect you... Please remember, God also gave you a brain, and we've used it to figure out how to create and test for safety in record time a vaccine that works. To not take advantage of this scientific breakthrough is why evolution works at all. If you aren't adaptable, nature will just take you out and your descendants will be non-existent. That's how evolution works. Be smarter than the dinosaurs, don't let a preventable accident cause your death (in their case their space program was woefully inadequate to prevent dying from a meteor impact, in our case our modern medicine used science to invent a vaccine in record time.)
Get vaccinated. Save lives.
Not to mention, the sooner we limit the number of cases we limit the number of variations that can evolve and be more infectious or more deadly. We need to reduce the dreaded R0 below 1. The further below the better.
Get vaccinated. Stop mutations. Save lives.
Thanks for reading!
-DrMike
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