Constructive fractional models through Mittag-Leffler functions

In recent decades, there has been a growing interest in generalizing differential equations of arbitrary order by replacing integer derivatives with non-integer derivatives. This allows for the construction of new models that can replicate memory effects due to the non-locality of fractional operators. However, this approach can lead to models that suffer from physical misinterpretation, do not maintain mass balance, and alter the original units of model parameters. This paper introduces an approach based on the generalization of exponential by Mittag-Leffler decays, which is associated with waiting time distributions. The model remains physically interpretable and ensures that the dimensions are constructively corrected. We discuss the mean waiting time, the asymptotic behavior, and apply this theory to develop a novel fractional SIRS model. Additionally, we present numerical results.