Suppose that there is a lily pad on a pond that proliferates very quickly. Every day, the number of lily pads in the pond doubles. On the first day, the 1 lily pad doubles to 2, on the second day, the 2 become 4, the next day, 8, then 16–you get the idea. One month later, on the thirtieth day, the pond is completely filled with lily pads. So, the question is, on what day was the pond half filled with lily pads? As it turns out, the answer to this question is more important than one might think, as it leads us to a term that people have probably heard a lot more about ever since the COVID-19 pandemic started: exponential growth! When someone uses “exponential” in everyday conversation, they often mean a big change, like “That test was exponentially harder than the last one” or something that is growing or changing really fast, like “That situation grew out of hand exponentially.” However, when a mathematician uses the word “exponential” they mean something very precise.
In mathematics, something is exponential if the way it grows (or shrinks) is directly proportional to its size, that is to say, its growth depends on how big it is. With our lily pads at the beginning, we started with 1 lily pad, and we added 1 more to have 2. Then, since we had 2, we added 2 more, and ended up with 4. Then we added 4, ending up with 8. Do you see the pattern? At each step, we are adding how many lily pads we had on that day, that is, our growth in this case is actually equal to the number of lily pads! We could imagine situations where instead of adding an amount equal to the number of lily pads we had, we instead added twice as many lily pads as we have, or maybe, if we want to grow more slowly, half as many. Because exponentials relate the way something grows to its size, they can be used to predict things like population growth, where you expect the number of children born each year to be related to the total number of people.
Now, to understand why this is important when it comes to pandemics, we need to answer our question from the beginning. When I first saw this problem (Used by Professor Shane Frederick of Yale University), my first instinctive answer was that the pond would be halfway full half way through the time period, so on the fifteenth day. However, let’s think about it a little deeper. We are told that the amount of lily pads doubles every day, that means the day before the pond is completely full, it must have only been half full. So, if the pond is completely full on day 30, it must have been half full on day 29. This is kind of surprising, after all, that means that in just 1 day, the pond went from half full, to all the way full, when it took 29 days to go from a single lily pad to filling half the pond. The growth that occurred in a single day was greater than the growth from the previous 29 days combined! That is why it can be terrible when pandemics grow exponentially. Imagine instead that we had a disease that spread like our lily pads, with everyone being sick after 30 days. Surprisingly, after 23 days, less than 1% of the population will be sick. But, in the week following, that population will go from less than 1% sick, to 100% sick. Now thankfully, COVID does not spread this quickly, and things usually do not grow exponentially forever. But, this rapid change in growth is something that is often not realized when it comes to exponentials. While at the beginning the growth may seem really slow, it can suddenly explode faster than you would otherwise expect. Well, maybe not, now that you understand exponentials.
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