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

Oustaloup, Alain; Levron, Franccois; Victor, St'ephane; Dugard, Luc

This paper proposes a very simple deterministic mathematical model, which, by using a power-law, is a \emph{non-integer power model} (or \emph{fractional power model (FPM)}). Such a model, in non-integer power of time, namely $t^m$ up to constants, enables representing at each day, with a good precision, the totality of the contaminated individuals. 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, $t^m$, 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 paper, the spreading phenomena by contamination. The proposed model is indeed validated with the official data of Ministry of Health on the COVID-19 spreading. Used in prediction, it well enables justifying the choice of a lockdown, without which the spreading would have highly worsened. The comparison of this model in $t^m$ 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 \emph{internal complexity} that the internal structure and its stochastic behavior present, and the \emph{global simplicity} that the model in $t^m$ offers in a deterministic manner: it is true that the non-integer power of a power-law is well a marker of complexity.