Scaling Dynamics of Growth Phenomena: from Epidemics to the Resilience of Urban Systems

Defining optimal COVID-19 mitigation strategies remains at the top of public health agendas around the world. It requires a better understanding and refined modeling of the intrinsic dynamics of the epidemic. The common root of most models of epidemics is a cascade paradigm that dates to their emergence with Bernoulli and d’Alembert, which predated Richardson’s famous quatrain on the cascade of atmospheric dynamics. However, unlike other cascade processes, the characteristic times of a cascade of contacts that spread infection and the corresponding rates are believed to be independent on the cascade level. This assumption prevents having cascades of scaling contamination.

In this presentation, we theoretically argue and empirically demonstrate that the intrinsic dynamics of the COVID-19 epidemic during the phases of growth and decline, is a cascade with a rather universal scaling, the statistics of which differ significantly from those of an exponential process. This result first confirms the possibility of having a higher prevalence of intrinsic dynamics, resulting in slower but potentially longer phases of growth and decline. It also shows that a fairly simple transformation connects the two phases. It thus explains the frequent deviations of epidemic models rather aligned with exponential growth and it makes it possible to distinguish an epidemic decline from a change of scaling in the observed growth rates. The resulting variability across spatiotemporal scales is a major feature that requires alternative approaches with practical consequences for data analysis and modelling. We illustrate some of these consequences using the now famous database from the Johns Hopkins University Center for Systems Science and Engineering.

Due to the significant increase over time of available data, we are no longer limited to deterministic calculus. The non-negligible fluctuations with respect to a power-law can be easily explained within the framework of stochastic multiplicative cascades. These processes are exponentials of a stochastic generators Γ(t), whose stochastic differentiation remains quite close to the deterministic one, basically adding a supplementary term σdt to the differential of the generator. When the generator Γ(t) is Gaussian, σ is the “quadratic variation”. Extensions to Lévy stable generators, which are strongly non-Gaussian, have also been considered. To study the stochastic nature of the cascade generator, as well as how it respects the above-mentioned symmetry between the phases of growth and decline, we use the universal multifractals. They provide the appropriate framework for joint scaling analysis of vector-valued time series and for introducing location and other dependencies. This corresponds to enlarging the domain, on which the process and its generator are defined, as well as their co-domain, on which they are valued. These clarifications should make it possible to improve epidemic models and their statistical analysis.

More fundamentally, this study points out to a new class of stochastic multiplicative cascade models of epidemics in space and time, therefore not limited to compartments. By their generality, these results pave the way for a renewed approach to epidemics, and more generally growth phenomena, towards more resilient development and management of our urban systems.