A mathematics professor explained COVID-19 growth to The Great Courses. The disease has disrupted daily life around the world; new cases and deaths haven’t stopped adding up in most countries. Exponential change is the key to understanding it.
The coronavirus is spreading along a path of “exponential growth.” This means that if you look at the spread of the disease over a standard increment of time multiple times—for example, every day for a week or a month—it doesn’t simply show more cases being reported over that increment of time. The rate at which the cases grow increases as well.
Dr. David Kung is Professor of Mathematics at St. Mary’s College of Maryland. He has also taught two lecture series for The Great Courses—How Music and Mathematics Relate and Mind-Bending Math: Riddles and Paradoxes. Now, Dr. Kung has teamed up with The Great Courses again, only this time, he hosts a new video that helps explain the disease spread of the novel coronavirus, SARS-CoV-2. You can watch his new video by clicking here. Here are some of the biggest takeaways.
Understanding Linear Change and Exponential Change
“The simplest change is linear—just adding the same number every day,” Dr. Kung said. “If some hypothetical illness started with just two cases, and grew by 100 every day, the math to predict the number of infected i during time t is just 2 + 100 t. Graphically, it’s a straight slope.”
The second thing to look at, beyond the number of cases, is how that number changes. Dr. Kung said that in linear change, the rate of change is constant, with 100 new infections per day in this example.
“But the coronavirus outbreak is definitely not linear: early numbers were 15, 16, then 24, then 53,” he said. “It wasn’t growing at the same rate each day—instead, the rate of change kept getting faster and faster. Any outbreak like that means the number of people infected already influenced the number who get infected next. This isn’t linear; it’s exponential growth.”
Dr. Kung offered a purely hypothetical example of the number of infected doubling every day. If we start the first day with just two infections, like the linear growth model, the second day the number of total infections doubles to four. The following day it doubles again to eight. Early on, 100 new infections a day seems far worse. After a while, though, exponential growth rears its ugly head.
“Shortly after a week passes, the numbers in our exponential model pass those in our linear model,” Dr. Kung said. “Every day the linear model adds just 100 infections, but the exponential one’s rate keeps increasing—adding about 1,000, then 2,000, then 4,000. By day 20, the linear model straggles along with about 2,000 cases while the exponential model soars to over one million.”
Why Reproduction Numbers Matter
One common term when discussing infectious disease is its reproduction number, often referred to verbally as “R zero” or “R naught” and written out as R0. The Centers for Disease Control and Prevention (CDC) defines it as “the contagiousness or transmissibility of infectious agents.” Dr. Kung says that in simpler terms, the R0 tells us, on average, “how many additional people each infected person will infect through the duration of their infection.”
“If the infected people have an average [R0] number that is more than one, they are collectively throwing more gasoline on the fire of exponential growth,” he said. “Remember: If you infect two people, and then those two people [each] infect two more, and so on, then we’re back to exponential growth—we’ve seen how bad that gets. And exponential growth like that happens anytime the average reproduction number is even slightly more than one, like 1.5 or even 1.1.”
The other side of that coin means that if people work to reduce the R0 number below one even a little bit, like 0.8, the infection will die off. This is because as the infected either recover or, in the worst-case scenario, pass away, either way they will fail to transmit the disease further along.
Dr. Kung said that the best way to lower the reproduction number is by controlling our contact rate, which, in terms of the coronavirus, means the rate at which the virus potentially comes in contact with us.
“If everyone washes their hands, with soap, for at least 20 seconds, the contact rate goes down a little bit, and the infected curve flattens,” he said. “If everyone avoids touching their face, then the contact rate goes down a little bit more, and the curve flattens even more.
“And most importantly—and hardest—if we isolate ourselves, we can drop the contact rate really low. That’s what social distancing is about; that’s what shelter-in-place orders are about.”
Dr. David Kung contributed to this article. Dr. Kung is Professor of Mathematics at St. Mary’s College of Maryland. He earned his B.A. in Mathematics and Physics and his Ph.D. in Mathematics from the University of Wisconsin, Madison.