Mathematical models, sophisticated graphs and curves can take time to comprehend and often intimidate people. This is exactly why today we are going to tackle some curves and mathematical principles that are extremely important to understand when following the COVID-19 story.
In this post we are still discussing the coronavirus, but this time we’re going to focus on math. Mathematical models, sophisticated graphs and curves can take time to comprehend and often intimidate people. This is exactly why today we are going to tackle some curves and mathematical principles that are extremely important to understand when following the COVID-19 story. If we learn how to interpret a few basic graphs, it will be easier to take the appropriate actions as a community to fight the virus and save lives. And right now, it’s time to step up because every day counts. So let’s dive into math and curves.
“If we learn how to interpret a few basic graphs, it will be easier to take the appropriate actions as a community to fight the virus and save lives.”
Exponential Curves
In some countries coronavirus is spreading at an exponential rate. What exactly does this mean? Basically, anything changing at an exponential pace is doing so extremely fast. To help you picture an exponential growth rate, we are going to talk about algae and chess. Probably, the one time you can see these two words in one sentence.
Let’s imagine that a fast growing species of algae was introduced to a lake. Its rate of growth is so rapid that every day the area of the lake that it covers doubles. When it covers the entire lake, the fish in the lake will die. At the rate it’s growing, it will take 30 days to cover the whole lake. Can you guess by what day the lake will be 50% covered? The answer is somewhat counter intuitive but very logical because, you know, it’s math. Since the area covered by the algae doubles every day and it will be completely covered on day 30, it means that the lake will be half covered by day 29. But the story continues, on the 29th day a clean up crew appears and removes a majority of the algae - only 1% remains.
This part is important, because the crew left a small portion of the algae in the lake, it will continue to double in size. Since the algae grows at an exponential rate, this time it will only take 7 days for the lake to be completely covered. From 1% to 100% in 7 days, that’s how fast exponential growth can occur.
Another example of exponential growth involves the famous wheat and chessboard problem. Once upon a time, the person who invented chess taught a king how to play the game. After learning, the king enjoyed it so much he demanded that the inventor sell him the chessboard. Eventually, the chess inventor and the king made a deal. The inventor requested to be paid in wheat (or rice, according to other versions) in such a way that one grain would be placed on the first square of the board, two on the second, four on the third, and so on (the number of grains would be doubled on each subsequent square). By the way, there are 64 squares on a chessboard.
The king thought that this offer was a great deal and happily agreed to the price. However, his math-savvy master of coin (accountant) knew that this rate of doubling grains per square was something their kingdom couldn’t afford. And he was right, the total price for the board ended up being 18,446,744,073,709,551,615 grains. Exponential growth is not always intuitive, and surprises many of us (even kings) by its results.
These two stories aim to illustrate how fast exponential growth really is. When you see curves like the one below on the news - these are exponential curves! The idea behind showing these curves is to display how quickly COVID-19 is spreading around the world. However, without a full comprehension of exponential curves/growth rates, the impact of these plots is lost.
Exponential Growth of COVID-19 Infections Globally - 31/03/2020
The bad news is that COVID-19 is STILL spreading at an exponential rate and we don’t know when it’s going to start slowing down. Experts employ caution when asked this question but they tend to agree that it’ll be “a while”. The good news is that no exponential curve can go on endlessly, the curve (or the number of new infections) has to flatten out eventually. This happens either because the virus runs out of new people to infect OR because appropriate measures slow the spread of the virus and we eventually find a treatment or vaccine.
Why flatten the curve?
To understand why it’s so important to flatten the curve that we see everyday on the news, let’s look at it from a different angle, football and food.
Let’s imagine that we are at a football game and there are 50 thousand people there cheering for their teams. Some of these 50 thousand people will go to the bathroom during the game. Usually we don’t experience any major problems with this because it happens over the course of the game. And the bathroom capacity at the stadium can handle the flow of people. But what if all these people decide to go to the bathroom at the same time, or let’s say, within 10-15 minutes? It’s easy to imagine that it would be an uncomfortable mess because there would be huge lines of people all trying to access the same limited number of bathrooms over a short period of time (the red dotted curve on the graph below).
For another example, let’s assume that the cafeteria at your work is open for lunch for three hours every day, from 12 PM to 3 PM. There are some people who came to work early or skipped breakfast and they are ready for lunch at noon. Then there are others who had a mid morning snack and they are not hungry until 2:30 PM. Which means that, generally speaking, if you decide to go have lunch at any given time within these three hours, you will be able to find a table and enjoy your salad. But what if everyone decided to go to the cafeteria at the same time? Again, huge lines, panic, and potential injuries (our imaginary co-workers get aggressive when hungry). If you were making a graph of the number of people going to the cafeteria over time, this would result in a ‘flattened curve’, which in the figure above is the green solid curve.
The important thing to understand here is that the total number of people going to the bathroom at a football game or to the cafeteria for lunch is generally the same. The key difference is the time over which this will happen. If it happens over a longer period of time (green flatten curve), no major problems would arise. If it occurs at the same time or within a short period of time the system may collapse (red dotted curve).
THIS is exactly what the situation with the coronavirus is. However, instead of lunch tables and toilet stalls we are dealing with intensive care beds, ventilators, and healthcare workers. It is possible that the total number of infected people will be the same with or without the quarantine order; BUT the only way to ensure that everyone gets the care that they need is to slow the pace at which people are getting infected and subsequently admitted to the hospital. Simply put, by staying at home you will help healthcare workers save lives.
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