Caputo Derivative as Weighted Average of Historical Values: some consequences illustrated via COVID-19 data

Abstract

. This paper uses the formula previously proposed by the authors themselves, in which the Caputo fractional derivative is written proportionally to the weighted average of historical values of the classical derivative (Equation (2)). Three consequences of this formula are treated in this work. The first explicitly shows the dimension of the Caputo derivative, the second indicates which historical values of the classical derivative have greater/lower weight for the Caputo operator at the current instant, and finally, the third shows that the Caputo derivative is zero at instants after the critical point occurred (allowing interpretations for the order of the derivative, for example in the dynamics of some disease). To illustrate these three results, we used examples previously obtained by the authors themselves, modeling the curve of active COVID-19 cases with the SIR model. This approach captures the memory effect well in epidemiological models.