How Math Predicts Pandemics

Updated: Jun 26

By Urvashi Balasubramaniam


Sitting behind a desk plotting graphs may seem like the last thing to do in a health crisis, but it may surprise you how often math is on the front lines in our fight against deadly diseases.


Our world runs on data, from the stock market and targeted ads to making scientific discoveries that push the human race forward. So what can data predict about our increasingly unpredictable futures?


Flatten the Curve


In 2007, the Centre for Disease Control and Prevention released a paper containing a now-infamous chart titled 'Flatten the Curve'. It put across three basic ideas to fight a pandemic: delay the outbreak, reduce the number of cases at its peak, and ease the burden on healthcare resources.


After being rediscovered amid the COVID-19 outbreak, 'Flatten the Curve' has become a popular fighting slogan against the pandemic. Without being rooted in real-world data, it has a single clear message for us: reduce daily infections.


Both curves represent the same number of COVID-19 cases. The only striking difference is that one curve is much higher than the other, which means a lot of people get the disease really quickly. The second curve is flatter and spread over a longer time, which means people get the disease much slower.


So why does it matter how fast the virus spreads?


Imagine you're holding an empty cup and I'm pouring water into it. Your task is to catch every drop in your cup. Would you prefer if I poured it drop by drop or all at once? If I poured the water all at once, you would struggle to catch them all. It would overflow! Your cup can only hold so much water at a time, and I'm being pretty inconsiderate by dumping all the water on you at once.


The cup represents our healthcare resources: every doctor, nurse and hospital we have all over the world to fight the outbreak. If fewer cases trickle in every day, they can handle it. But if everyone falls sick at once, there wouldn't be enough doctors to save them, and so many more people would die.


In the graph above, there's a dotted line running horizontally that indicates how many cases hospitals can handle at a time. Flattening the curve helps us stay safely under that line.


Exponential Growth


Let's take a little detour with a riddle. Imagine a pond with a single lily pad on it. Every day, the number of lily pads doubles. In 60 days, the entire pond is covered in lily pads.

How many days does it take to cover half the pond?


You might think Day 30, but the answer is actually Day 59! The lily pads cover half of the pond's surface on Day 59, and double to cover it completely on Day 60.


What's especially interesting is the day when the lily pads cover a mere 1% of the pond surface: Day 54.


The pond is practically empty until it's suddenly full! This is called exponential growth because the number of lily pads tomorrow depends on the number of lily pads today. Don't worry if you didn't get the answer right away. Our brains aren't great at intuitively understanding exponential growth.


In exponential growth, the numbers keep doubling and tripling forever, but we don't see that in reality. Eventually, The lily pads run out of space to grow. The virus just can't find any more people to infect. We could do this by creating a vaccine to protect people from it or isolating ourselves so it can't get to us. That's when the curve flattens.


You can plot this on a graph and see this squiggly shape, called a sigmoid curve. When it naturally slows down, it's called logistic growth.

A sigmoid curve flattens at the top before falling


Exponential growth shows us why we should take extreme measures when we only have a few infections, and why governments are shutting down entire cities when they only have a couple of cases.


Michael O. Leavitt, the former governor of Utah, once said, "Everything we do before a pandemic will seem alarmist. Everything we do after a pandemic will seem inadequate."


Predicting the End


In such uncertain times, it's nice to know there's a light at the end of the tunnel. Surprisingly enough, we can find hope in numbers once again.


When you plot the number of cases against time, all you have is an upward curve. You don't have a map to check if you've reached the peak (or the most cases in a day), or if the worst is still ahead. That's where growth factor and logarithmic charts come in to save the day.


The logarithmic scale is the natural scale for exponential growth. In a linear scale, the tick marks would be 1, 2 and 3 or 10, 20 and 30. But on a logarithmic scale, the numbers grow in multiples of 10, so the tick marks would be 10, 100, 1000 and so on.


The logarithmic chart scales up small numbers and scales down large numbers so the growth appears more equal. This way, we can easily compare the growth rate between countries with very different numbers of cases.


The line in a logarithmic chart is much straighter and a lot more pessimistic, so when it starts curving downwards, you can definitively say that it's getting better.


Another great strategy to know when we're near the end is calculating the growth factor. All you have to do is divide the number of new cases today by the number of new cases yesterday.


If your growth factor is higher than one, the cases are still growing. Seeing a growth factor of roughly one means you've hit the inflexion point (when the curve is falling).


Hope


To better understand the pandemic numbers, I interviewed Amit Godiyal, a data analyst professional in Ahmedabad, India. One of the most comforting facts he told me was the fundamental statistics principle that every curve upward will also curve downward. The cases can't grow forever. The curve will fall eventually, and thanks to quarantines and thousands of medical professionals working on vaccines and cures, it's likely to stay flat.


Data doesn't always tell the whole story. When there are under-reported numbers, asymptomatic cases and gaps in testing, it might seem like basing major decisions on data is a terrible idea.


"Whatever data you’re getting here is the only data you can analyse. You don’t have any other option. You can’t do a primary or secondary search on this data, because whatever you’re getting, the government gets from health facilities or from the civil administration," says Godiyal.


But what we can do right now is focus on the story the data does tell us, whether it's to prepare for the worst or hope for the best.


Though it may sound strange, math is one of our best weapons in the fight against COVID-19. It's also our biggest source of hope when the curve flattens, and it will.


Until then, stay hopeful and stay home.

And of course, stay curious.



References

  1. Sanderson, G. (2020). Exponential growth and epidemics. Retrieved May 2020, from https://www.youtube.com/watch?v=Kas0tIxDvrg

  2. Reich, H. (2020). How To Tell If We're Beating COVID-19. Retrieved May 2020, from https://www.youtube.com/watch?v=54XLXg4fYsc

  3. Chang, K. (2020). A Different Way to Chart the Spread of Coronavirus. (2020). Retrieved 9 May 2020, from https://www.nytimes.com/2020/03/20/health/coronavirus-data-logarithm-chart.html

  4. Hanson, PhD, J. (2020). What This Chart Actually Means for COVID-19. Retrieved 9 May 2020, from https://www.youtube.com/watch?v=fgBla7RepXU&

  5. Wilson, M. (2020). The story behind ‘flatten the curve,’ the defining chart of the coronavirus. Retrieved May 2020, from https://www.fastcompany.com/90476143/the-story-behind-flatten-the-curve-the-defining-chart-of-the-coronavirus

  6. Markel, D. (2020). What history revealed about cities that socially distanced during a pandemic. Retrieved 9 May 2020, from https://www.pbs.org/newshour/health/what-history-can-teach-us-about-flattening-the-curve

Image Credits

  1. Sigmoid Curve: https://www.geeksforgeeks.org/implement-sigmoid-function-using-numpy

  2. Wiles, S. (2020). Flatten the Curve. Retrieved May 2020, from https://twitter.com/siouxsiew/status/1236721200291655680?lang=en

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