Fractional derivatives as weighted average of historical values: an application to COVID-19 in Brazil

Abstract

The memory effect is an interesting tool that can be seen in fractional differential equations. To show this clearly, in this paper we prove that the Caputo derivative of a function 𝑓, as well as the Riemann-Liouville integral and derivative, are proportional to a weighted average of the historical values of 𝑓 or 𝑓 ′. For this, we use the statistical expectation of functions, whose random variable follows a beta distribution. Moreover, through the respective probability density functions, for each operator we specify the weight of the historical values of the function to determine its current value, according to the values of the fractional order of the derivative. Furthermore, to prove the effectiveness of the memory effect to describe real phenomena, we compared a classic model with its fractional version to model COVID-19 in Brazil.