Evidence for Complex Fixed Points in Pandemic Data

Abstract

Mathematical models used in epidemiology to describe the diffusion of infectious diseases often fail to reproduce the recurrent appearance of exponential growth in the number of infections (waves). This feature requires a time-modulation of some parameters of the model. Moreover, epidemic data show the existence of a region of quasi-linear growth (strolling period) of infected cases extending in between waves. We demonstrate that this constitutes evidence for the existence of near time-scale invariance that is neatly encoded via complex fixed points in the epidemic Renormalization Group approach. As a result, we obtain the first consistent mathematical description of multiple wave dynamics and its inter-wave strolling regime. Our results are tested and calibrated against the COVID-19 pandemic data. Because of the simplicity of our approach that is organized around symmetry principles, our discovery amounts to a paradigm shift in the way epidemiological data are mathematically modelled. We show that the strolling period is crucial in controlling the emergence of the next wave, thus encouraging the maintenance of (non)pharmaceutical measures during the period following a wave.

Epidemiological theory of virus variants

“…where the gradient operator is defined in (2.22) and Φ is a quadratic or cubic function, respectively. In fact, taking these results together with the liberty of scheme-redefinitions and comparing with the general form of the eRG for single variants (see [34]) we are naturally led to choose for simplicity f i (I c,i ) = I c,i and model Φ as…”

“…This lead to the prediction, with few months of advance, of the advent of a second pandemic wave that started in the fall 2020 in Europe [33]. The framework has been extended to contain quasi-fixed points [34] to provide a first fully consistent mathematical description of multiwave pandemics. The extended framework, dubbed Complex eRG, related the interwave period to the timing of the insurgence of the next wave [35].…”

“…We also discover that virus mutations are at the heart of switching on relevant operators from the perspective of fixed points controlling the final stage of a single wave pandemic. These lend a natural and profound interpretation of the complex fixed points eRG (CeRG) [34,35], which mathematically models multi wave pandemics.…”

Effective Mathematical Modelling of Health Passes During a Pandemic

“…The eRG approach 20,21 describes the spread of a disease through flow equations (the so called β -functions) and characterises a wave as the flow between fixed points 5,20,26,27 . Concretely, for α = φ (I c ), with φ a continuous, differentiable and monotonic function, the β -function for a single wave (and a single variant of a disease) can be written as…”

“…a smooth monotonic function of the cumulative number of infected individuals I c ) as the flow between different fixed points. It has been demonstrated 25,26 that the eRG approach is indeed capable of describing accurately not only a full wave of COVID-19, but is also capable of modelling more complex multi-wave structures 27,28 even under changing conditions, like the appearance of new variants 5 , vaccination dynamics 29 , change in the social dynamics 30 , among the ones mentioned above.…”