Going Viral: A Brief Primer on Exponential Growth

NOTE: Please join us in a Good Friday WORLDWIDE DAY OF PRAYER AND FASTING for relief from the COVID-19 pandemic.

In the early weeks of the quarantine I spent a lot of time looking at the pandemic stats, refreshing the websites several times a day to keep up with the rising numbers of coronavirus infections around the world. My kids were also interested, being the little geeks that they are. We would always look at the latest numbers before going to bed. Maybe that’s not the best thing to rest your thoughts on before retiring for the night.

One evening instead of loading the coronavirus tracker websites I showed them a schematic picture of the life cycle of a virus. I explained the different parts of the picture, how the virus enters the cell and inserts its genome, and how it hijacks all of the cell’s machinery to turn it into a virus factory. The final picture in the series showed the cell rupturing and half a dozen new viruses emerging to infect other cells.

“Each of these baby viruses can infect another cell, so every time we go through this cycle the number of viruses is multiplied,” I said.

I opened up a spreadsheet and made a doubling function. (You can easily do this yourself by typing the following formulae in the cells: A1=2, A2=A1*2, A3=A2*2, etc. Or download a copy of my spreadsheet here.)

“Let’s say we had a virus that would double in number with every cycle. How many viruses would we have after 50 cycles?” Doubling numbers grow quickly; here are the first 10 values in the series: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024. After just 50 doublings you arrive at the astonishingly large number of 1,125,899,906,842,624. The kids were moderately impressed.

Blue line: doubling function. Red line: discreet derivative of the doubling function. Yellow line: doubling function multiplied by a random number between 0.25 and 1.5.

Then I plotted the numbers on a graph, and we saw the upward curve. “It doesn’t matter what scale you’re looking at — the line will always have a similar shape.” We looked at different parts of the curve at different zoom levels, and we saw what happened when we added one number at a time, and when we plotted the rate of change (the discreet derivative).

“This is called an exponential function,” I said. I grabbed a piece of paper and wrote down some equations as I lectured. “Notice that every number can be written out as the number of 2’s you are multiplying together. So the n^th number in the series is 2ⁿ.” Again, they were moderately impressed.

Then I opened up the COVID-19 data, and one of my kids had the “a-ha!” moment. His eyes got big as he pointed to the screen and said, “Exponential growth!”

“Yeah!” I said. “This is what it means to ‘go viral.’ And the same process happens at every scale: the number of virus particles over time within one infected person, or the number of infected people in the population of a city, a state, a country, or the whole world! The overall shape of this curve is the same at every zoom level and at every time point.”

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Of course, viruses aren’t the only things which go viral, or else that term wouldn’t have entered our vernacular. Every content creator these days wants their stuff to expand exponentially on social media. The mathematics of this phenomenon are essentially the same as what we see in biology: an incredibly funny cat video passes from person to person, reaching more and more people every time it is shared. The same kind of thing can happen with hairstyles, clothing, popular music, children’s toys, etc. Fads have probably been happening in every culture at every scale of society for as long as humans have existed.

Social fads are a lot like viral epidemics. Some people are immune to cat videos and will not watch them no matter how funny they are. And other people will laugh and enjoy the crazy cat’s antics without sharing the video with all of their own contacts. And some cat videos are not as funny as others. And even the funniest ones get tiresome if you have seen too many of them recently. Not every cat video will go viral (thankfully), and not every virus becomes a pandemic (thankfully).

And even if something does go viral it will eventually have to stop. In practical terms, there is not an infinite supply of anything. SARS-CoV2 will eventually run out of human beings to infect. Facebook does not have an infinite number of active users to keep sharing that cat video. Silly Bandz eventually saturated the population of elementary school students. Even in a computer simulation of exponential growth you eventually run out of memory space.

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New religions can also experience exponential growth. We don’t have the actual numbers to plot on a graph, but it is clear from the New Testament that Christianity started as a small sect of Judaism within the land of Israel, but within a generation had spread to every major city in Asia Minor and southern Europe.

The membership of The Church of Jesus Christ of Latter-day Saints grew exponentially for the first 150 years or so after its organization in 1830, and has been taking a more linear course since then. This growth can also be seen in the number of church units and buildings, particularly temples. When I was an undergraduate student I spent many hours making an animation to show the building of temples across the world. This was painstaking work, done a single frame and a single pixel at a time, and by today’s standards the results are fairly primitive-looking. But I remember being filled with wonder and gratitude to the Lord when I watched this animation over and over again back then. (Tip: It’s a little better at half speed, I think.) It is a great joy to be a servant of the Lord and to participate in his marvelous work and a wonder.

In 1834, when the Church was still small and struggling, the Prophet Joseph Smith stood up in a small church meeting and prophesied: “This Church will fill North and South America—it will fill the world.” You can see for yourself the fulfillment of that prophecy in the little video I made 17 years ago.

Last week in the April 2020 General Conference, Elder David A. Bednar said,

“As members of the Lord’s restored Church, we stand all amazed at the ever-accelerating pace of His work in the latter days.”

We were all watching this together in the living room, and after hearing this line my son — the same one who had the “a-ha!” moment during my impromptu lecture — smiled at me and said, “Exponential growth.”

I smiled back and thought, “That’s my boy!”

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Alan B. Sanderson, MD is a member of The Church of Jesus Christ of Latter-day Saints and is a practicing neurologist.