Understanding the spread of coronavirus disease 2019 (COVID-19) requires understanding nonlinear growth. Whereas linear growth is intuitive, nonlinear growth is not. People’s predictions for nonlinear patterns tend to be closer to linear projections, assuming that future growth will be similar to that of the past. For a pandemic, this can lead to dangerous underestimations in the time to reach a critical value.
Most physical phenomena, such as the accumulation of snow or spread of a spill, have linear growth rates or rates that only increase slowly. Exponential growth, in which the rate of growth itself rapidly increases, may be less frequently observed, in part because such growth often quickly ceases after reaching some boundary condition. Pandemics such as COVID-19 grow exponentially, which can challenge understanding. Simulations can help build intuitions about exponential growth; The Washington Post recently produced an excellent simulation to illustrate the effects of quarantines and social distancing on the spread of infections. But nonlinear growth patterns can be so counterintuitive that even when presented within simulations, people will extrapolate a smooth function from a handful of samples over time, dramatically underestimating the increase in the growth rate.
Another challenge is that graphs of nonlinear growth are subject to perceptual distortions. The figure to the left illustrates a hypothetical situation where there is exponential growth in the number of infected individuals. The orange curve shows growth in one country; the dark gray curve shows growth in a second country, which started later but proceeded at the same rate. The gray curve is simply a shifted version of the orange curve, as shown by the yellow reference lines, but it appears that the gray curve is rising more slowly. This illusion is likely caused by our visual system’s habit of seeking the shortest distance between two lines instead of the correct vertical distance between them.
We also have a hard time understanding visual depictions of uncertainty. When we see point estimates that reflect averages or best guesses, we tend to view them as the only possibility. When we see error bars and error intervals, we often misinterpret depictions of larger error as indicating larger quantity. For example, in plots of hurricane trajectories, meteorologists often use error intervals to indicate that the uncertainty of a forecasted storm path increases as you get farther ahead in time, but people can interpret this as indicating that the storm will increase in size. Simulations can help here too, depicting a sample of potential outcomes rather than an interval or an area of plausible outcome.
What can be done to explain the exponential growth of COVID-19 cases more clearly? One approach is to use a logarithmic scale for the numbers of infected individuals or deaths. The advantage of this method is that exponential growth is visualized as a line, making it easier to extrapolate accurately. The disadvantage is that people lacking experience with logarithmic scales don’t always see that linear functions reflect powerfully accelerating growth.
Another strategy to convey more about a nonlinear process is to plot it in multiple ways. Below, we plot the reciprocal of the rate of occurrence of infection cases in addition to the rate itself, using recent data from Johns Hopkins Coronavirus Resource Center, inspired by Wade Fagen-Ulmschneider’s interactive visualizations. The conceptually tricky aspect of exponential growth is that increases in the number of cases appear modest during early stages (left), even when the number of infections is destined to increase rapidly. But if we plot the reciprocal, which is the number of uninfected people per infected person (right), the rate of change is high early in the growth curve.
Unfamiliar visualizations such as the one on the right often require explanation. Here is an explanation for this format: On 11 March, in Turkey, there was a single case in a population of about 83 million putatively uninfected people. Ten days later, that had shrunk to about 124,000—a small city. Ten days after that, the number of cases had shrunk to about 6000—the equivalent of a small town.
Visualizations that avoid perceptual distortions and play to cognitive strengths can improve public understanding of the evolving pandemic.
Jeffrey M. Zacks is professor and associate chair of Psychological and Brain Sciences at Washington University in Saint Louis, MO, USA. firstname.lastname@example.org
Steven L. Franconeri is a professor of Psychology and director of the Northwestern Cognitive Science Program at Northwestern University, Evanston, IL, USA. email@example.com