As all those who can self-isolate to help reduce the spread of COVID-19 and all but essential services are closed in many locations, local, regional, national, and international commerce has declined precipitously, with calamitous effects for many workers, businesses – large and small – and even governments and their instrumentalities. There is fear, pain, and suffering in many places and in many forms, and this reality supersedes any claims to the contrary. (See Nature Cannot Be Fooled.)
This is a wicked problem, more complex in many ways than the Great Recession of 2008 because it convolves two complex issues – public health and economic vitality. The same actions that encourage commerce endanger human lives. Conversely, the actions that protect lives have profound, negative economic consequences. Our socioeconomic equilibrium has been disrupted, with an uncertain path to its restoration. It is, to use a common phrase, a global catastrophe -- an abrupt and unexpected event causing great and relatively sudden damage and suffering.
There’s a branch of mathematics that studies such abrupt and unexpected changes; it’s aptly named catastrophe theory. As Wikipedia explains, “catastrophe theory is a branch of bifurcation theory in the study of dynamical systems,” but a simpler description is that catastrophe theory characterizes small changes in the parameters of a nonlinear system that cause equilibria to appear or disappear and a system to jump from one macrostate to another. In that sense, it is a formalization of the colloquial meaning of a disruptive event.
My Geek Childhood
When I was a young teenager, the catastrophe theory work of René Thom had a brief run in the popular consciousness, as the country grappled with economic shocks engendered by the energy crisis of the 1970s. It was even featured in the print edition of Newsweek, due in part to its popularization by Christopher Zeeman. Catastrophe theory was also featured in that other must read periodical of my childhood, Scientific American, alas after I was already in college. (See Zeeman’s delightful April 1976 Scientific American article, entitled simply, Catastrophe Theory.)
That I both read Newsweek and Scientific American avidly and ravaged multiple libraries for books on the underlying mathematics of catastrophe theory perhaps says more about my young teenaged years in rural Arkansas than about catastrophe theory’s grip on the popular imagination. (See Libraries: Arms Too Short to Feed the Mind.) Let us now bow our heads in a brief acknowledgement of the self-evident reality that I was and am a geek or nerd. (I accept either appellation – but I take umbrage at dweeb or dork.) Needless to say, I had plenty of time for reading on Saturday nights. Having explicated the limitations of my teenaged social life, let us plunge deeper into catastrophe theory.
Catastrophe Theory 101 
This simple diagram on the right illustrates one of the most commonly used models, a cusp catastrophe. Here, two control variables (a, b) perturb a simple function y = u4/4 by the addition of a quadratic
yielding a potential function of the form
(Note that the constants are purely for mathematically simplicity. Any desired scaling factor can be applied to the control or internal variables.)
When the control variables (a, b) have fixed values, the system settles into an equilibrium state where the internal variable u minimizes the potential function. However, as the control variables change, a local minimum can disappear, and the internal variable can jump suddenly to a different equilibrium, the cusp in the diagram.
A little calculus – setting the first derivative to zero – shows that the equilibrium surface is
The line that marks the edges of the bend in the surface from the upper sheet to the lower sheet is called the fold curve. Finally, the bifurcation set (cusp) is the critical image of the projection (a, b, u) -> (a, b) from the equilibrium surface onto the control space.
Intuitively, small changes in the (a, b) control plane define situations where a stable solution (for values of u) will suddenly jump to an alternate stable solution. As the diagram also shows, there are hysteresis loops where crossing the cusp curve in one direction may cause a catastrophe, but immediately reversing direction does not restore the previous state.
The astute reader will have noted that the cusp catastrophe describes a deterministic system (i.e., the first derivative is time invariant), and most domains include random, time-dependent elements. The simple cusp catastrophe can be readily extended to stochastic environments (e.g., by adding a Wiener process, W(t)), resulting in a stochastic differential equation. This means the equilibrium plane and transition cusp vary over time, based on the underlying stochastic process.
For those of you playing the game at home, a Weiner process is a real valued continuous time stochastic process that is the scaling limit of a random walk. This is yet another example of the interconnectedness of things, spanning topology, dynamic systems, random walks, probability theory, and diffusion processes.
Rebooting the Planet
Now that I have totally buried the lede of the current public health and economic crisis under childhood geekdom and the mathematics of catastrophe theory, here is the relevance and intuition. Think of our control variables (a, b) as social distancing and government economic policy, respectively, and our outcome as socioeconomic health and stability. Small changes in social distancing or government policy can trigger abrupt transitions in socioeconomic stability due to large declines in public health or economic function (e.g., a pandemic and recession). This is the important point: no simple reversal of those changes can immediately return society to its prior, healthy and prosperous state. All of which brings us to hysteresis and a restart of global society.
Shutting down a planet, a country, a city, or even a university is neither a small nor a simple thing. Critical infrastructure must be maintained and essential services must remain operational. More importantly, each of those has collateral dependences. Infrastructure and services are fractal and self-similar at multiple levels. They are also globally interconnected, with dependences unimagined even a generation ago. (See The World Is Small and Globally Connected, Globally Worried.)
How does one reboot a planet? First, an immediate return to normal is simply not possible; as the cusp catastrophe model shows, there is hysteresis in any individual and collective action set. Jobs and income have disappeared; businesses and industries have been damaged and destroyed; tax revenue has been lost; and families and communities have been devastated by illness and death.
Second, an overly rapid and aggressive restart – an “all is back to normal” ethos – risks a rebound pandemic, given inadequate herd immunity or a time to develop a future vaccine. Conversely, a timid, overly cautious resumption of business and social activity risks unnecessary economic loss and prolonged pain and suffering for hundreds of millions of individuals.
Resumption is not a binary process. Today, we are self-isolating and social distancing, alternating between sitting alone in sweatpants while anxiously watching talking heads on cable television and venturing out with trepidation for food and supplies, warily eyeing the person across the street, fearful they will not keep their distance. We cannot immediately shift to large group activities, attending mass sporting events and glad-handing colleagues, friends, and family with abandon. Rather, resumption must be phased, with careful monitoring of public health and economic activity and recognition that there may be multiple missteps and necessary backtracking. It is a thoughtful, careful process that must be driven by reason and data, not emotion and intuition. (See The Simple Things Matter, Most of All Now and Just the Facts, Ma’am: Reasoning is Not Dead, Jim.)
Nor are universities exempt, with their tens of thousands of students, faculty, and staff drawn from around the world, living, working, and eating in close proximity. As a practical matter, universities are small cities, with logistics and transportation systems, museums and performing arts centers, hospitals and healthcare systems, housing, dining, and recreation facilities, all interconnected with research, education, and community engagement.
These are complex dynamic systems with many co-moving stochastic components, self-similar at many, many scales. Individuals adapt continuously, but public policy is necessarily discrete. As my colleague, Michael Good noted, it is wise to view the post-pandemic reboot as consisting of red, orange, yellow, and green phases, each representing incremental resumption of social and economic activities, albeit with potential reversion to earlier phases based on observed outcomes.
Remember the key lesson of the stochastic cusp catastrophe. Disruptions cannot be immediately inverted; complex dynamic systems involve hysteresis. We will persevere and the planet will reboot, but it is more complex than a simple Ctrl+Alt+Del. The future will not look like the past for quite a while, but it can be both different and better if we work together in a collaborative spirit. (See Renaissance Teams: Reifying the School at Athens.)
Nice article. I was just ruminating on this with Barb over dinner the other night. As usual you have enlightened me.
I had no idea "wicked problem" was actually a thing and not some south Boston slang. I also didnt know about catastrophe theory. I hope our government is getting good advice that includes the wickedness of this catastrophe.
And, while you were searching the library for articles on catastrophe theory I was flipping through Model Airplane News, and Air Progress. :)
Thank you for the article.
Posted by: Dave Semeraro | April 13, 2020 at 07:48 AM