Can Catastrophe Theory explain expansion and contagious of Covid-19?

Caetano, M. A. L.

Since SARS-Cov-2 started spreading in China and turned into a pandemic disease called Covid-19, many articles about prediction with mathematical model have appeared in the literature. In addition to models in specialized journals, a significant amount of software was made available, presenting with dashboards spreading of the pandemic for each new. These models are solved by computer simulation of traditional exponential models as a representation of the growth of cases and deaths. Some more accurate models are based on existing variations of SIR model (Susceptible, Infected and Recovered). A third class of study is developed in spatial or probabilistic models as a way of forecasting the effect of Covid-19 around the world. Data on the number of positive cases in all countries, show that SARS-Cov-2 shows great resistance even after strategies of lockdown or social distancing. The purpose of this article is to show how the bifurcation theory, known as Catastrophe Theory, can help to understand why Covid-19 expansion rates change so much and even with low values for a longtime trigger contagion quickly and abruptly. The Catastrophe Theory was conceived by the mathematician Rene Thom in the 60s with wide applications in works in the 70s. The outbreak of spruce budworm in Canada revealed a very interesting opportunity to test Catastrophe Theory whose explanation for the phenomenon was widely debated in the academic world. Inspired by the same mathematical approach to this phenomenon in Canada in the 1970s, we applied the Catastrophe Theory in the current Covid-19 pandemic. We observed that sudden outbreaks occur when the carrying capacity and the rate of expansion of the virus reach a region of bifurcation on the cusp surface. With actual Covid-19 data obtained from WHO, we fitted the dynamic model using the particle swarm technique and compared the results in the bifurcation plan with the Covid-19 outbreaks in different regions of the world. It is possible in many cases to observe the trajectory of the parameters between limit points in the bistable region and the consequent explosion of cases observed for each country assessed.