Fractal Pattern in COVID-19 Case Growth

Fractal geometry is about spotting repeating patterns, analyzing them, quantifying them and manipulating them; it is a tool of both analysis and synthesis.

Benoit Mandelbrot [1]

Figure 1. Mandelbrot Set. Created by Wolfgang Beyer with the program Ultra Fractal 3. — Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=321973

I have been looking at the John Hopkins Coronavirus data since the inception of the pandemic. The key metrics being tracked and reported include cases, hospitalizations, hospital capacity, and deaths. The growth in cases can be modeled and estimated using a figure called R-naught that tells us how many persons an infected individual can infect. The growth pattern appears to be exponential. As of November 27, following a Thanksgiving holiday that saw record travel numbers since the pandemic started, the US recorded 205, 577 COVID-19 infections in a single day. After months of looking at the time series of U.S. COVID-19 daily cases, I eventually noticed what appeared to be a fractal pattern.

Figure 2. U.S. COVID-19 cases over time, Source: John Hopkins University https://coronavirus.jhu.edu/map.html

What is a fractal? Fractals are a visual mathematics to describe shapes that occur in nature but cannot be modelled using simple structures from Euclidean geometry or closed-form mathematical equations. Fractal patterns can be found in nature and include heartbeats, snowflakes, coastlines, clouds, and forests. Fractals are also a measure of roughness. The concept of a fractal dimension (FD) refers to the notion that objects in nature are not particularly smooth like shapes in Euclidean geometry. For instance, the FD of the Koch Snowflake is 1.26186 (See Figure 3). Fractal dimension can be calculated but that is beyond the scope of this discussion.

Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.

Benoit Mandelbrot [2]

A fractal is defined as a self-similar structure that is set over an extended, but finite, scale. Benoit Mandelbrot, the inventor of fractal geometry, described fractal patterns[1] as having 3 components: an initiator, a generator, and a rule of recursion (See Figure 3, Koch Snowflake). The initiator is a classical geometric object or scalable pattern. The generator is a simple geometric pattern (or the template from which the fractal will be made). Finally, the rule of recursion is the manner in which the patterns repeats over time and/or space.

Figure 3. Koch Snowflake. CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=1898291

I noticed a recurring pattern — a wave — like an ocean wave. I learned how to spot waves while surfing. I reminisce on taking surfing lessons at Maroubra Beach in Sydney, Australia on my 30th birthday. I didn’t immediately stand on the board — that happened on a trip to California where a 50-something former surfer from Newport suggested that I put my back to the shore, egg-paddle past the point where the waves “break”, and turn around, pop-up, and stand up after I spot the wave.

Figure 4. Ride the Wave, Source: SECOORA, https://secoora.org/education-outreach/waves/glossary/

A wave has key features that include the crest (top), trough (bottom), and wavelength. I took a deeper look at the John Hopkins COVID-19 data and started to notice peaks and troughs.

Figure 5. Wave Spotting, Source: John Hopkins University

The JHU COVID-19 Cases chart is a bar graph with the number of cases over time. Each bar represents one day. Figure 6 takes a closer look at the wave and each day gets a number. I developed a heuristic to define and form waves. The local minimum refers to the data point following the wave which is typically lower than the wave values and will be followed by a higher data point in the next wave.

Crest=highest value in the wave or final value before local minimum

Trough = data point following crest, followed by a higher value

Wave Length = days from crest to crest ranging from approx. 4–6

Figure 6. Taking a Closer Look, Source: John Hopkins University

Data Analysis Concerns

The troughs in the wave pattern can provide a momentary sense of fall hope. A steep one-day decline is certainly not a trend. Data analysts and media outlets have utilized a 7-day moving average model to adjust for the decreasing case values as the overall trend climbs. Furthermore, additional analysis can be done by focusing on analysis of peak-to-peak values as it’s easier to identify the exponential nature of COVID case growth. There are many methods to determine if data fits a particular statistical distribution, exhibits skew/kurtosis, or possess traits like autoregression or heteroskedasticity (primer for (G)ARCH modeling). However, the nature of fractals do not make closed-model fit tests a quick and easy approach for identification. Lastly, I can’t state with confidence that confirming that COVID growth follows a fractal pattern is any help with fighting the pandemic. Yet, great advances in science and mathematics do yield value with additional investment.

[1] Mandelbrot, Benoit and Richard L. Hudson. The Misbehavior of Markets: A Fractal View of Financial Turbulence. 2007.

[2] Mandelbrot, Benoit. The Fractal Geometry of Nature. 1986.

[3]Koch Snowflake https://en.wikipedia.org/wiki/Koch_snowflake

[4] Koch Snowflake Creative Commons License Information https://commons.wikimedia.org/wiki/File:KochFlake.svg