Non-integer (or fractional) power model to represent the complexity of a viral spreading: Application to the COVID-19

Abstract

This article proposes a very simple deterministic mathematical model, which, by using a power-law, is a non-integer power model (or fractional power model (FPM)). Such a model, in non-integer power of time, namely tm up to constants, enables representing the totality of the contaminated individuals at each day, with a good precision, thus expressing the interest of this tool for time series. Despite being enriched with knowledge through an internal structure based on a geometric sequence “with variable ratio”, the model (in its non-integer representation) has only three parameters, among which the non-integer power, m, that determines on its own, according to its value, an aggravation or an improvement of the viral spreading. Its simplicity comes from the power-law, tm, which simply expresses the singular dynamics of the operator of non-integer differentiation or integration, of high parametric compactness, that governs diffusion phenomena and, as shown in this article, the spreading phenomena by contamination. The representativity of the proposed model is indeed validated with the official data of French Ministry of Health on the COVID-19 spreading, notably the time series of the contaminations and the hospitalizations. Used in prediction, the model well enables justifying the choice of a lockdown, without which the spreading would have highly worsened. Its predictivity is validated by verified predictions in lockdown and vaccination phases, and even for the vaccination itself; its simplicity enables a very simple implementation of the prediction technique. The comparison of this model in tm with two known models having the same number of parameters, well shows that its representativity of the real data is better or more general. Finally, in a more fundamental context and particularly in terms of complexity and simplicity, a self-filtering action enables showing the compatibility between the internal complexity that the internal structure and its stochastic behavior present, and the global simplicity that the model in tm offers in a deterministic manner: it is true that the non-integer power of a power-law, that an internal dynamics dispersion can justify, is well a marker of complexity, and this, beyond viral spreading phenomena.