Fractional order modeling of ecological and epidemiological systems: ambiguities and challenges

Abstract

Fractional calculus is as old as the classical calculus. The classical calculus has been far enriched compared to the fractional calculus. However, in the last few decades, researchers have established new theories with non-trivial mathematical analysis in Fractional Calculus, and particularly the mathematicians continue their speedy progress in the field. Fractional differential equations have received immense popularity in modeling real-world events last few years, especially in modeling ecological and epidemiological systems. In this Survey Paper, we shall show that several population dynamics models in the existing literature are dimensionally inconsistent, and hence, those cannot be considered as realistic models. We shall demonstrate that the dimensions of the Caputo derivative and the conformal derivative are defined, and hence, these can be applied to build dimensionally homogeneous models. This survey paper, for the first time, uncovers that the Caputo-Fabrizio, Atangana-Baleanu-Caputo, and Caputo-Hadamard derivatives do not have any well-defined dimensions. Therefore, these derivatives should be rejected in describing ecological interaction and epidemiological dynamics. Often a time, several articles implemented real data in dimensionally inhomogeneous fractional order systems. Hence, it is questionable whether such fractional-order models could capture well coherent information about real events. We shall also highlight some other related ambiguities in the existing contributions and then present a few cautions (especially parameter readjustment and fixing parameter values) that should be taken care of in developing fractional order models. Finally, some challenges and open problems are proposed that might enrich the fractional order modelling framework significantly. © The Author(s), under exclusive licence to The Forum D’Analystes 2024.