What is exponential growth anyway?

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With the COVID-19 pandemic dominating life right now, there are many discussions and debates over whether nearly complete shutdowns of economies are worth it. Undoubtedly, these shutdowns are having massive negative effects on the livelihoods of millions. Some of these shutdowns came relatively early, with local cases only numbering in the hundreds or thousands. Why would officials want to take such drastic methods when the numbers were so low? Surely, compared to the number of people killed in car accidents or influenza, this was an overreaction. But the real reason for these extreme and unprecedented actions is tied to the exponential growth of the disease. “Exponential growth” is often used in common speech as meaning fast-growing, or maybe even something with an increasing growth rate. The figure below shows these signs. The cases are rising fast, and the rate is clearly increasing too. This is a curve someone may call exponential. It’s not.

While not entirely wrong, describing a rate as exponential simply because it is growing fast and the rate of growth is increasing lacks the complete details of what exponential growth is and why it should raise concerns. This article will analyze in detail what exponential growth actually is, how it relates to real-world systems which ensure computer security around the world, and non-intuitive consequences of exponential growth.

Let’s go back to the first image. This curve is actually just quadratic. The rate is indeed increasing over time. The equation describing that curve is:

Let’s now compare that to an exponential growth function. Take a look at the next image.

Here it’s not clear that these two curves are structurally that different. The exponential is rising more slowly, but they have a similar shape. The example exponential function is described by:

Remember, these are two curves I made up. I could have made the exponential curve rise faster, but that would miss the point. The real concern is the rise over longer periods of time. Consider these same growth curves over 60 days instead of 20 days.

This is why an “exponential curve” is not just a phrase, it has a very strict definition, and is quite different from a “fast rising curve.” Now, it may be years or decades since you’ve needed to worry about exponents or logarithms. Maybe you last saw these in high-school mathematics, or perhaps you never even learned about them at all. As a Research Engineer, I am one of the crazy ones who actually enjoys math and uses it every day. Regardless, as a refresher, exponential functions hold a general form:

In this equation, b is called the base. In many scientific and economic fields, one of the most important exponential functions has base e, where e is approximately 2.7128. e is irrational, just like the constant π that is used most commonly to relate the diameter and circumference of a circle. This means that no matter how many digits are written out for e or π, it is impossible to write these numbers exactly. There are not even any eventual repeating sequences. e is important because exponential functions with base e often represent systems where the growth rate is proportional to its size. This occurs in compound interest, radioactive decay, and, unfortunately, in the spread of disease.

These functions have a curious and important feature. Namely, their rate-of-change is equal to their value at any point. In calculus, this rate-of-change is called the derivative. This is one way to understand why exponential growth curves rise so fast, seemingly so suddenly. The growth builds on itself. It rises some, which increases the rate of rise proportionally, further rising the curve.

While the focus of this article is clarifying exponential growth with respect to the spread of disease, exponential growth is not all bad. In fact, it is often quite useful. You are taking advantage of a feature of it right now, in fact. Modern classical encryption, used to secure and provide privacy to interactions online, often takes advantage of one-way-functions. These are functions that are easy to calculate in one direction, but extremely slow to calculate in the other. To understand this, think of multiplication and division. These are inverses of each other. Consider a case where we both know a number, let’s say 10. I multiply 10 by a secret number, and I don’t tell you what that number is. I only provide you with the solution 50. It is trivial for you to calculate that my number is 5, and it is not so secret after all. This is an example of a two way function — it is easy to calculate in both directions, multiplication and division. Scientists and mathematicians have discovered functions that they think (but have not yet proven!) are one-way functions. These are easy in one direction, and hard in the other. One example is the SHA-256 cryptographic hashing function. This function can take a phrase and calculate an output. Can you guess what I said here through SHA-256?

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I doubt it, and encryption relies on reversing this being very difficult, in fact exponentially difficult. If the amount of computer calculations you must do to reverse the function are large enough, it is either not financially beneficial due to the cost of power, or there is not a powerful enough computer available to perform the calculation before the sun burns out. That is what is meant by “secure” classical encryption — it is not impossible to break, it just would take far too long. This method of using SHA-256 or similar “cryptographic hash” functions is often used by websites to store login information. Rather than store your password as ‘1234568’, it puts the password through the SHA-256 function and stores that. Then, every time you login, it hashes your input again and compares the outputs. If a hacker were to access their database, they would only find the hash of your password, and would be unable to try it on another website. In reality, this process is more complicated, but the overall concept holds.

But it’s not just encryption where exponential growth comes in. One of the most famous problems in the world is the Traveling Salesman problem. Pretend you need to travel the country visiting a large number of cities. By looking at a map, you can determine how far the drive is between each city. To save gas, you want to chose a path between the cities that minimizes the amount of driving you need to do. In daily life, delivery companies like UPS, FedEx, and Amazon want to solve this problem all the time. At first, it seems that this problem is easy to solve. Why don’t we just calculate all the routes we could take and pick the fastest one? It turns out that the number of possible routes increases extremely quickly with the number of locations to visit. This seemingly simple problem does not have a known solution that takes less than an exponential amount of time. Even crazier, if a solution was found, it would be one of the most important discoveries ever. Sure, it might make deliveries faster, but more importantly this problem is NP-hard. In solving this, you may have to show that P=NP. Essentially, the P=NP problem asks whether every problem whose solution is easily confirmed can also be quickly solved. It would show that the SHA-256 hash function mentioned above actually isn’t a one-way function, or maybe even that one-way functions cannot and do not exist. But let’s bring the discussion back to disease.

Disease transfer modeling is incredibly hard. People with mild symptoms may not be tested, there may not be enough tests available, and countries may have reasons to not be truthful with their statistics. The rules and laws enforcing social distancing and restricting travel change every day, and models must account for that. But one quick way to check if the growth rate is exponential is performing a logarithmic plot of known cases. Instead of a linear scale, a logarithmic plot separates the scale by factors of 10. It may go 10, 100, 1000, etc. On a logarithmic plot, an exponential curve looks linear. The plot below is the same data as previously shown, but now on a logarithmic scale.

With this in mind, consider the world’s confirmed cases of COVID-19. Consider these results as of April 2, 2020. At least since late March, the rise certainly looks exponential.

Consider the cases in the USA since the end of February. The cases on a logarithmic scale are nearly linear, indicating exponential growth. Towards today, the rate appears to be slowing somewhat, but remember this is on a logarithmic chart, and the growth rate is still incredibly high, the highest it has been yet. Worldwide, the number of cases has doubled in the past week. Not a huge concern when the world goes from 200 to 400 cases, but a major concern with 500,000 new cases in a week.

Exponential growth in a pandemic like this also has odd numerical effects. Studies have estimated the death rate in the range of 0.66–3%. However, of the cases which are “closed,” with a patient either recovering or passing away, the rate of death is approximately 20% as of today. This absolutely does not contradict those studies. A disease takes time to spread, and the number of people infected is rapidly increasing. As of today, approximately 75% of everyone who has ever had COVID-19 still has it, even though it has been around for a couple months and the typical person recovers much faster than that. The exponential growth rate causes these odd numerical phenomenons to occur.

We couldn’t complete this article without mentioning the “Flatten the Curve” push. By avoiding contact with other people, this effort hopes to slow the exponential growth rate, so that people will recover in time to open up hospital resources for future waves of patients. Perhaps flatten isn’t the right word for it, let’s reduce the exponent.

This article focused on what exponential growth rates are, how they relate to the COVID-19 pandemic, and how exponential growth rates are encountered in everyday life all the time. Governments acted early in some cases because of their concern about exponential growth. If not acted upon early, it is very difficult to recover. Stay safe, stay inside, and save lives.