Aplicação do cálculo de ordem arbitrária à epidemiologia

Abstract

The arbitrary order Calculus, known as Fractional Calculus, has applications in numerous areas, such as engineering, physics, and biology. In contrast to the integer-order Calculus, which is local, the operators we are studying in the non-integer Calculus are non-local. So, they include the possibility of considering the memory of the studied phenomenon, which means, the dependence on previous stages. However, its immense applicability coexists with the difficulty of a unified and coherent theory. In this work, we studied applications of arbitrary order Calculus to SIR-type models. We aimed to establish some bases, both for Fractional Calculus and for the SIR model as it was originally built and, also, we discussed the main difficulties in using an arbitrary order SIR model. Moreover, we reviewed the construction of an unusual model, in which arbitrary order derivatives arise in an epidemiological significant way from the dependence of time since infection on infectivity and removal, allied to the use of Mittag-Leffler functions. In both cases, we studied equilibrium points, positivity, and other questions of relevance, such as the meaning of the parameters. Some of these studies were carried out in an original way. The models presented were also analyzed numerically, by MATLAB implementation. Finally, we proposed extensions of the built model, with regard to the possibility of waves representation: oscillatory infectivity, quarantine compartment, and imitation dynamic (human behavior). The extensions were proposed in such a way that the meaning of the construction is maintained and were also analyzed numerically, compared, furthermore, to some real data from the COVID-19 pandemic.