Mathematical biology models incorporating memory kernels via ordinary, delay, and fractional differential equations

Abstract

Biological systems frequently exhibit non-exponential dwell times and delay distributions. While chains of ordinary differential equations, delay systems, and fractional calculus formulations have been used to incorporate memory effects, a unified and biologically consistent framework connecting these paradigms remains lacking. This thesis presents a general memory kernel formulation that integrates ordinary, delayed, and fractional differential systems into a unified structure. By representing memory using integral kernels with statistically interpretable distributions, the study shows that these modeling approaches lie on a continuous spectrum determined by the shape of the time-delay kernel, while retaining physical interpretability. In particular, the generalized gamma Mittag-Leffler distribution is introduced as a versatile kernel that can capture both exponential and heavy-tailed fractional characteristics, while also describe statistical moments. The proposed structure is explored analytically and numerically in several applications, including population dynamics, epidemiology, pharmacokinetics, and electrophysiology. It is demonstrated that memory kernels induce complex behaviors in simple models. These effects can lead to phenomena such as multi-wave epidemic dynamics, and oscillations in cardiac action potentials, known as early afterdepolarizations. Rather than acting as an external periodic forcing, memory reshapes the phase-space geometry of trajectories, thus modulating the dynamics. By recognizing memory in biological models as more than a mathematical adjustment, but as a structural expansion of phase space through hidden variables and degrees of freedom, this thesis clarifies the relationship between Markovian and non-Markovian systems. Furthermore, it establishes consistent and physically grounded fractional formulations, contributing to the rigorous application of fractional calculus in biological modeling and thereby reducing common sources of misinterpretation. Taken together, these results provide a unified basis for the analysis of complex behaviors across diverse disciplines.