What is it about?
Evolution equations involving convolution-type integral operators have been extensively studied in the literature. However, there is a gap in understanding the connection between specific convolution kernels and new models, such as delayed and fractional differential equations. In this study, we show that classical, delayed, and fractional models can be unified under a framework using a gamma Mittag-Leffler memory kernel, starting from the logistic model structure. We explore different types of this general kernel, analyze the asymptotic behavior of the model, and present numerical simulations. A detailed classification of the memory kernels is provided through parameter analysis. The fractional models we develop have unique characteristics, maintaining dimensional balance and explicitly linking fractional orders to past data points. Furthermore, we demonstrate how our models can accurately capture the dynamics of COVID-19 infections in Australia, Brazil, and Peru. This research contributes to mathematical modeling by offering a unified framework that allows for the integration of historical data using integro-differential equations, fractional or delayed differential equations, and classical ordinary differential equations.
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Why is it important?
This study focuses on integrating memory effects (the influence of past events on current states) into mathematical models for population dynamics. It presents a framework using the gamma Mittag-Leffler memory kernel. The key features of this framework include developing dimensionally correct models, offering biological insights into fractional orders, and the ability to incorporate past data using ordinary, delayed, and fractional differential equations.
Perspectives
The ad hoc fractional models expand on the original ODE model and can replicate important phenomena. However, these conventional practices may introduce limitations and challenges. While there are well-constructed previous approaches in the literature, they are often isolated, coexisting with the ongoing use of controversial results. The motivation for this research is to discuss and propose approaches for constructing biologically applicable fractional models that are consistent in terms of dimension and physical meaning.
Noemi Zeraick Monteiro
Federal University of Juiz de Fora
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This page is a summary of: Bridging the gap between models based on ordinary, delayed, and fractional differentials equations through integral kernels, Proceedings of the National Academy of Sciences, May 2024, Proceedings of the National Academy of Sciences,
DOI: 10.1073/pnas.2322424121.
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