Left unchecked, COVID-19 could have spread in a frightening flash; math can help us calculate just how fast

The novel coronavirus permeates all quarters of worldwide daily life in the first part of May 2020. We are hunkered down at home, trying to stay safe by minimizing the number of trips out into the world. Many of us wear masks when we do venture out. The objective in doing this, of course, is to try and keep COVID-19 from spreading further than it has already. Every night on the local or national news, the continuing spread of the coronavirus is the lead story. These stories often lead with a welter of numbers, charts and graphs. Two mathematical terms are also mixed into the stories: “exponential growth” and “flattening the curve.” At this point in the COVID-19 pandemic, the former is known to be bad and the latter to be good.

Exponential growth is the most rapid kind of growth that can theoretically continue forever, representing a growth in which a quantity is multiplied by a particular factor every time period. In particular, exponential growth comes with a constant doubling time — if a quantity experiences exponential growth, it takes the same amount of time for the quantity to go from, say, 100 to 200, as it does for the quantity to increase from 1 million to 2 million.

In mid-March, before social distancing was recommended, the number of coronavirus cases in the United States was increasing at a rate of 33.6% per day. … If this rate had remained constant, the doubling time for COVID-19 cases here would have been about 2.39 days. From one patient being diagnosed, without mitigation measures, the entire U.S. population — over 328 million people — could theoretically have been infected in just over 58 days.

An old mathematical problem asks: If a 64-square chessboard were to have rice placed upon each square such that one grain were placed on the first square, two on the second, four on the third, and so on (doubling the number of grains on each subsequent square), how many grains of rice would be on the chessboard at the finish? The answer is more than 18 *quintillion* (18 followed by 18 zeroes), well beyond the capacity of the world to produce in a year.

We see exponential growth in many aspects of everyday life. Financial institutions compound interest in our accounts by multiplying the balance by a set percentage every month — or increasing our balance on our loan that we have to pay back. Ponzi schemes show this type of growth, where small numbers of early investors make a lot of money, while large numbers of later investors tend to lose a lot. Bacteria grow exponentially until they either run out of room in their environment or exhaust their nutrition source. Internet memes and videos often spread this rapidly through social networks.

The term for the fast spread of internet content is “to go viral,” a reference to the exponential growth of viruses. Viruses like COVID-19 and severe acute respiratory syndrome, aka SARS, spread quickly, with an infected person passing the virus to a number of new people. The first case of coronavirus was detected in China in November 2019; as of the morning of May 3, there are nearly 3.45 million cases confirmed with 244,000 deaths resulting. Here in the United States, the first diagnosis of coronavirus was reported in Washington state in early January; in early May 2020, we have over 1.1 million confirmed cases and 67,000 deaths. By any measure, this spread has been exponential.

To put it another way, in mid-March, before various states or the Centers for Disease Control and Prevention started recommending staying at home or social distancing, the number of coronavirus cases in the United States was increasing at a rate of 33.6% per day, according to the Center for Economic and Policy Research. If this rate had remained constant, the doubling time for COVID-19 cases here would have been about 2.39 days. From one patient being diagnosed, without mitigation measures, the entire U.S. population — over 328 million people — could theoretically have been infected in just over 58 days.

The previous sentence points to both why mitigation measures are needed and perhaps why some resist such measures. Exponential growth is astoundingly fast and, in the absence of a vaccine, measures such as self-quarantining, social distancing, the wearing of face masks and the temporary shuttering of nonessential businesses are meant to slow the growth rate. Various scientists and government agencies are cautiously optimistic that this is occurring as the daily increase in cases looks to be moving from exponential growth toward an almost linear growth. (The caution exists because the percentage of the population tested has ebbed and flowed.)

**Why resist mitigation?**

Continued mitigation measures are likely necessary to turn the curve of the total cases into a logistic one, in which the increase in cases becomes more logarithmic and then flatlines. The overall total would level out at some quantity. This is behind the idea of “flattening the curve,” the notion that the daily increase in COVID-19 cases can, with mitigation, be stretched out over time so no particular day’s increase is too great and eventually will decrease to zero. Keeping the largest increase as small as possible and spreading out the increase over a longer time can mean, for instance, that the health care system can handle the number of incoming cases and not be overwhelmed.

So why do some people resist mitigation measures? Political discussions aside, it may well be that the very idea of exponential growth is too fantastic for some to believe. When we mention that 328 million people could be infected by the coronavirus a mere 58 days after the first patient is diagnosed, many of us immediately discount that as impossible or completely unrealistic. It’s absurd to think the entire U.S. population would be infected in two months’ time.

Why do some people resist mitigation measures? Perhaps the idea of exponential growth seems too fantastic to believe.

Most of us look at growth on a day-to-day basis, causing us to linearize the growth in our minds, to think things are not that bad. In reality, exponential growth tends to be a short-term phenomenon and so we have a tendency to downplay its effects. A 2009 paper in the Journal of Finance referred to this mindset as “exponential growth bias.” This has the effect in household finances of underestimating the effect of interest rates for loans, given facts about other loan terms, and also underestimating the future value of investments. Such biased households borrow more, save less and favor investment strategies with shorter-term payoffs. The move in some states to lift mitigation measures while COVID-19 cases still increase could be a result of this bias, a small reduction in the increase in cases being seen as a victory over the virus.

Technology futurist Ray Kurzweil coined the phrase “the second half of the chessboard,” a reference to the point where an exponentially growing quantity begins to affect an organization’s business strategy. Many would agree that we are in the second half of the chessboard with regard to the pandemic. Understanding exponential growth of viruses and the long-term effects can help us to see the continued usefulness of flattening the curve.

*Dr. Robert Poodiack is a mathematics professor at Norwich University.*

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