Containment efforts appear to step COVID-19 spread down from exponential norm

Summary: Community actions taken to slow the spread of COVID-19 may help to contain the virus and stop the exponential growth of cases. Researchers say the viral spread in China occurred in a fractal pattern. When social distancing was implemented, viral spread slowed. With more communities adopting social distancing measures, the spread became smaller in each new area exposed.

Source: University of Michigan

The actions many communities are taking to slow the spread of COVID-19 may be shrinking deaths in a pattern that brings together nature, art and math, according to a new study from the University of Michigan.

The findings, described in a preprint paper that has not yet been peer-reviewed, indicate that containment works and could help us snuff out the virus faster.

While the spread of a virus is often represented with exponential growth and then decay as a population becomes immune, the study suggests that in China, the containment efforts toned down the high-speed exponential increase of COVID-19 deaths into a spread better represented by a power law.

“It seems that they are succeeding in containing the virus in China, but in other places, the virus is still growing exponentially,” said Robert Ziff, a professor of chemical engineering at U-M, senior author of the preprint paper.

The power law equation suggests that the new coronavirus, known as SARS-CoV-2, spread in China through a fractal pattern—or a pattern repeated at smaller and smaller scales, like the compound leaves of ferns or the branches of a tree.

In effect, the virus slows its spread within areas where social distancing measures have been undertaken. These include quarantines; the closures of schools, restaurants and shops; the cancellation of events; and restrictions on leaving home. The virus can still leap to communities that don’t have these measures in place, but with each location quickly implementing containment measures, the spread in each new area is smaller.

This shows a graph
The deaths in China changed from exponential (curved) growth to the straight line of a power law before tapering off. Meanwhile, deaths outside China continue to follow an exponential curve. This shows that while the virus is being successfully managed in China, it is essentially uncontrolled in the rest of the world. Robert and Anna Ziff applied logarithmic scales to both the days axis and the death axis to make the difference between exponential and power law growth easier to see. The image is credited to Steve Alvey/University of Michigan Engineering.

“Further work is needed to understand the different trends in the death counts, but these results underline the importance of public health guidance—and the following of those orders,” said Anna Ziff, a doctoral student in economics at Duke University and first author on the paper.

The father-daughter team posted their analysis to Medrxiv February 17 and then updated it March 1. They use death data because it appears to be much more reliable than infection data, especially as many of those carrying the SARS-CoV-2 virus show no symptoms of COVID-19.

Deaths in China have tailed off considerably in recent weeks, with recovered patients now outnumbering active cases by nearly two to one. However, the team is concerned that COVID-19 deaths in the rest of the world currently follow an exponential curve.

“In most of the rest of the world, the number of daily cases has not peaked, so unfortunately there may be many more deaths,” said Robert Ziff.

While the team acknowledges that their study depends on the accuracy of the data provided by China, they believe the power law equation is a reason for optimism.

“It should be possible to stop the virus with prompt identification and containment,” said Robert Ziff.

About this coronavirus research article

University of Michigan
Media Contacts:
Nicole Casal Moore – University of Michigan
Image Source:
The image is credited to Steve Alvey/University of Michigan Engineering.

Original Research: Closed access
“Fractal kinetics of COVID-19 pandemic”. Anna L. Ziff, Robert M. Ziff.
medRxiv doi:10.1101/2020.02.16.20023820.


Fractal kinetics of COVID-19 pandemic

We give an update to the original paper posted on 2/17/20 — now (as of 3/1/20) the China deaths are rapidly decreasing, and we find an exponential decline to the power law similar to the that predicted by the network model of \citet{vazquez_polynomial_2006}. At the same time, we see non-China deaths increasing rapidly, and similar to the early behavior of the China statistics. Thus, we see three stages of the spread of the disease in terms of number of deaths: exponential growth, power-law behavior, and then exponential decline in the daily rate. (Original abstract) The novel coronavirus (COVID-19) continues to grow rapidly in China and is spreading in other parts of the world. The classic epidemiological approach in studying this growth is to quantify a reproduction number and infection time, and this is the approach followed by many studies on the epidemiology of this disease. However, this assumption leads to exponential growth, and while the growth rate is high, it is not following exponential behavior. One approach that is being used is to simply keep adjusting the reproduction number to match the dynamics. Other approaches use rate equations such as the SEIR and logistical models. Here we show that the current growth closely follows power-law kinetics, indicative of an underlying fractal or small-world network of connections between susceptible and infected individuals. Positive deviations from this growth law might indicate either a failure of the current containment efforts while negative deviations might indicate the beginnings of the end of the pandemic. We cannot predict the ultimate extent of the pandemic but can get an estimate of the growth of the disease.

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