Exponential Growth 4/11/20

Math by Richard Bleil

As I write this (yesterday), the governor of one isolated state finally declared a state of emergency, but only for one county. Her argument is that “one size does not fit all” for the Coronavirus national shutdown, and for the most part she is right. Unfortunately, in following the “fine” example of our president, while she knew the virus was approaching her state, she also failed to prepare. It’s one thing to say that a state of emergency isn’t warranted yet, and it’s another not to take advantage of the slow approach to put things in place to protect the state before it hits. Now it’s probably too late. Theirs will be one of the last states to be hit hard and will likely be shut down as the rest of the nation begins to remove to normalcy.

The clear problem is that these politicians (I think I’ll refrain from calling them “leaders”, political or otherwise, as they have failed to show any leadership throughout this crisis) don’t understand exponential growth. As they are looking at the initial numbers and proudly exclaiming how low the count is right now (back when it was low), they failed to see what their earliest critics and those ringing the early alarms were looking at; the rate of increase. If you only have one case, these numbers will look small at first as they continue to double: 2, 4, 6, 8, 10, 12, 24, 48. The problem is that these numbers suddenly explode: 96, 192, 384, 768, 1,536, 3,072, 6,144, 12,288. If these numbers double daily, then in just two weeks we’ve gone from one case to over 12,000. After that it begins growing even faster.

It’s worse than this, I’m afraid. Today, an article out of Johns Hopkins university said that their research indicates that only about 6% of the actual cases have been identified. As such, 94% of those infected with the virus will never know, and neither will we. So, in two weeks, 12,288 detected cases could be around 205,000 in reality. As of the writing of this article, there are around 100 cases in that one county in the state to which I am referring, so, again in that one county, there are likely actually about 1,700 cases. There are fewer than 190,000 in that county, so this is roughly 1% of the total population.

Yup, one in a hundred. Less than three days ago, it was only one in five hundred. And yet the governor decided against declaring a shutdown, or even making preparations to slow the spread.

There is an old story from China that is worth mentioning here to help illustrate the point. The story, no doubt, is a fable but it’s interesting. The emperor owed a general a great favor. The general suggested that all he wanted was rice. He asked the emperor to put a single grain of rice on the first square of a chessboard (8×8, so a total of 64 squares). On the adjacent square, double that amount, and double again on the next and so on until there is rice on each square. The emperor acquiesced and ordered that it be done, but by the time the final square was reached, it would have contained 2 to the power 64 grains of rice, or 18,450,000,000,000,000,000 (over eighteen thousand trillion grains). One grain of rice weighs about 0.029 grams. If you work this out, it is about 590,000,000,000, or 590 billion tons of rice on the final square (English tons, it would be over 530 billion metric tonnes). Recognizing that this is more rice than the entire nation of China could produce, but being honor bound, the emperor turned the nation over to the general.

In calculus, acceleration is known as the second derivative. Looking at the number at one point is static, but the number is moving. Looking at two points defines a line and is known as velocity. If the velocity were linear, then 2 becomes 4 becomes 6 becomes 8. This doesn’t look nearly as bad because the assumption is that the velocity is constant, that is, it increases by just to each time. Unfortunately, the velocity of cases increases because the second derivative, the acceleration, is exponential. Thus, 2 becomes 4 becomes 8 becomes 16.

The true leaders in this crisis are emerging, but sadly, it’s not the national leader it should be (the president), and in states like this, it’s not even the state leader (the governor). The leaders are the people who recognize just how critical this crisis is. There is a company in this county that has been striking because of unsafe conditions in response to the Coronavirus. Leaders such as those who organized that strike are the true leaders in the state. The owners of restaurants who are following national recommendations are the leaders in that state. But it’s not the governor, and it’s certainly not the president who elected to play golf when he was hearing the warnings from multiple sources.

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