A little bit every day adds up in the long run

Illustration by Nino Mekanarishvili

This article is an introduction to the concept of exponential growth. As several stories and adages tell us exponential growth has often been misunderstood or underestimated by people, leading to undesirable outcomes. In the wake of COVID19, it has become extremely important to understand this kind of growth for both self-preservation as well as policy-making.

The world has been changed by the COVID-19 pandemic. There are many complex predictive models created by expert virologists and epidemiologists. However, I have found it useful to apply a few simple Mathematical ideas to get a better understanding of the latest developments being reported in the news. The aim of this article is to enhance the reader’s intuition for exponential growth as it relates to the current pandemic.

My frame of reference is a classic puzzle in recreational Mathematics that goes something like this: “There is a pond with some lily-pads on it. On each day, the area taken up by the lily-pads doubles. If the whole lake gets covered in 48 days, how long will it take for half of the lake to be covered?” [1]. This puzzle often highlights how we struggle to understand exponential growth.

The thought process behind finding the answer is as follows: if half of the lake is covered on some day, then on the next day the area will double, so the whole lake will be covered. It takes 48 days to cover the pond, so half of it will be covered in 47 days. At first it seems counter-intuitive that the lily-pads should be able to cover so much of the pond in a single day, but constructing a mathematical model reveals that this is correct.

Consider a single lily-pad in this hypothetical pond: on each day, the lily-pad’s size increases by an amount equal to its size at the end of the previous day. Notice that the only factor affecting its size on some day is its size on the previous day. If it starts with a size of 1m2, then it can be shown that its size will be 2 n m2 after n days. This growth can be seen in Fig. 1.

Fig. 1. Area of the lily-pad on the first few days

The formula explicitly shows the exponential growth in the puzzle, but exponential growth is not limited to a hypothetical lily pond. In fact, any model where the amount by which a value changes is some constant multiple of its value on that day exhibits exponential growth. That constant multiple is the growth rate and knowing what it is makes it easy to find the function for the growth. If r is the growth rate of some population and the initial population size is P, then the size after n days will be

    P(1 + r)n

In the case of the lily-pads, we had P = 1m2 and r = 1. This simplified model has some shortfalls when applying it to more realistic systems. This can be seen by noting that the lily-pad model predicts that after 50 days, the lily-pads will cover an area of 1.13×1015 m2, which is about twice the surface area of the Earth [2]. This clearly cannot happen in the real world since whatever lake the lily-pads are in must have have a fixed, finite surface area that cannot be exceeded.

Fig. 2. Lily-pad growth if left unchecked

The model can be adjusted to take this into account. Instead of area increasing by a constant multiple of its value on a given day, we can let the multiple be a constant scaled by what portion of the lake remains that can be spread into. This adjustment changes the model growth from exponential to logistic [3]. Fig. 3. shows how logistic and exponential growth may look very similar initially.

Fig. 3. Comparison of an exponential and logistic function

At this point, an observation may be made. The same thought-process used in the lily-pad example can be applied to the spread of a virus. The virus SARS-CoV-2 that causes the disease COVID-19 [4] spreads from person to person through droplets produced when an infected person sneezes or coughs[5]. This is similar to previous coronaviruses such as SARS or MERS, but it is much more transmissible as it can survive on surfaces for many hours [6]. Unlike influenza, there is no vaccine as of yet, although researchers are hard at work trying to find one [7].

After making the simplifying assumptions that it affects all people equally and that all people interact with the same number of people every day, we can say that the number of new infections is a constant multiple of the number of people infected on that day. As seen above, this suggests that the number of infections will exhibit exponential growth. These simplifying assumptions actually turn out to be somewhat accurate when looking at the actual data on the number of infections.

Fig. 4. Log-linear graph of number of infections in South Africa
Data: Johns Hopkins University [8]
Analysis: Ralph McDougall

Fig. 4. shows the actual number of infections in South Africa up to 17 May 2020. The scale was chosen so that perfect exponential growth would be represented as a perfectly straight line. I have found this visualisation somewhat reassuring. Even though the news reports that the number of new infections is increasing every day, this is to be expected with exponential growth. Furthermore, since a logistic model is more accurate, it also means that the growth will eventually stop. Fig. 5 shows the same graph, but for Spain instead of South Africa.

Fig. 5. Log-linear graph of the number of infections in Spain
Data: Johns Hopkins University
Analysis: Ralph McDougall

Fig. 4. also shows that something interesting must have happened just before day 10. The strict national lock-down started at that time and the slope of the line decreased significantly. In theory, this is exactly what one would expect with the simple exponential model since fewer human interactions means fewer people get infected every day, so the growth rate is much lower. This is crucial for our fight against the virus since we can reduce the overall growth rate by following social-distancing rules and consistently washing our hands. This will take a huge burden off of our healthcare systems since hospitals will not be overloaded by patients all at once.

On the other hand, humans are not merely points of data in a model. Since humans are social beings, it is difficult to keep an entire population isolated for extended periods of time [9]. I have seen stories on social media of people struggling with their mental health during the lock-down. Businesses are struggling and it becomes much more difficult to get essential goods to healthcare workers and those that are unable to stay away from work for extended periods. These issues need to be managed by governments in a human manner, even if it does not make sense economically or politically in the short term.

COVID-19 has affected everyone, directly or indirectly. In fact, it has brought the world to a halt. As individuals, we cannot stop the virus on our own. However, by taking action to slow the spread of the virus, we help to reduce the rate at which it spreads. If everyone does a little bit, we buy health experts time to find a cure. Maybe go and wash your hands again after reading this, just to be sure.

Ralph McDougall

Undergraduate Engineering Student,
STELLENBOSCH UNIVERSITY

Illustration by Liani Malherbe

[1]  “recreational mathematics – Question about logic riddle (patch of lily pads that doubles every day),” Mathematics Stack Exchange, 2016. https://math.stackexchange.com/questions/1892365/question-about-logic-riddle-patch-of-lily-pads-that-doubles-every-day (accessed Mar. 31, 2020).

[2]  “Earth.” http://abyss.uoregon.edu/~js/ast121/lectures/lec09.html (accessed May 16, 2020).

[3]  “Exponential growth & logistic growth (article),” Khan Academy. https://www.khanacademy.org/science/biology/ecology/population-growth-and-regulation/a/exponential-logistic-growth (accessed Mar. 31, 2020).

[4]  “Q & A on COVID-19,” European Centre for Disease Prevention and Control. https://www.ecdc.europa.eu/en/covid-19/questions-answers (accessed Apr. 14, 2020).

[5]  CDC, “Coronavirus Disease 2019 (COVID-19) – Transmission,” Centers for Disease Control and Prevention, Mar. 17, 2020. https://www.cdc.gov/coronavirus/2019-ncov/prevent-getting-sick/how-covid-spreads.html (accessed Mar. 31, 2020).

[6]  “A virology expert answers key questions on COVID-19,” World Economic Forum. https://www.weforum.org/agenda/2020/03/covid-19-explained-virology-expert/ (accessed Apr. 14, 2020).

[7]  J. Gallagher, “What is the most promising coronavirus drug?,” BBC News, Apr. 30, 2020.

[8]  CSSEGISandData, CSSEGISandData/COVID-19. 2020.

[9]  N. Morgan, “We Humans Are Social Beings – And Why That Matters For Speakers and Leaders,” Forbes. https://www.forbes.com/sites/nickmorgan/2015/09/01/we-humans-are-social-beings-and-why-that-matters-for-speakers-and-leaders/ (accessed Apr. 14, 2020).

Another story of the exponential growth

Once upon a time, there was a king who loved puzzles and games. One day a man came to him with an invention – the game of chess. King, happy with the man, offered him any reward that he might want. The inventor said that he he wants rice. When the king’s advisors asked him for the quantity the man said – “place a single grain of the first square of the chessboard. Two grains on the second square, four grains on the third, and so on doubling the number of grains each time.” 

The king was perplexed. The inventor could have asked for anything! Thinking this to be a small price to pay, king agreed and granted the reward. However, soon the king and all his advisors realized that the amount of grain required to fulfil king’s promise was too huge and would make the kingdom bankrupt!

Neeraja Sahasrabudhe

Department of Mathematics,
INDIAN INSTITUTE OF SCIENCE EDUCATION AND RESEARCH MOHALI

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