A novel discrete fractional-order mathematical framework for modeling and forecasting the coupled progression of diabetes and cardiovascular complications, with emphasis on population dynamics in Saudi Arabia, offers valuable insights for public health decision-making and epidemic mitigation.
Abstract
This study presents a novel discrete fractional-order mathematical framework for modeling and forecasting the coupled progression of diabetes and cardiovascular complications, with emphasis on population dynamics in Saudi Arabia. Using the discrete Caputo fractional operator, the model captures memory effects and long-term disease dependence not represented by classical integer-order systems. The population is divided into five interacting compartments: susceptible, exposed, diabetic without major complications, diabetic with severe complications, and cardiovascular-affected individuals. A rigorous qualitative analysis establishes equilibrium points and examines their local stability under fractional discrete dynamics. Stability regions are identified in parameter space, clarifying the mechanisms governing disease persistence and progression. A feedback-based control strategy is proposed to restore stability of the disease-free equilibrium when destabilization occurs. To enhance predictive performance, a Reservoir Computing framework is integrated and compared with Long Short-Term Memory networks, demonstrating improved accuracy and lower computational cost. Numerical simulations validate theoretical findings and highlight key epidemiological influences. Future work will extend the model by incorporating optimal control strategies aimed at minimizing infections through targeted interventions such as vaccination, improved hygiene, and environmental sanitation. The integration of fractional-order dynamics with control theory offers valuable insights for public health decision-making and epidemic mitigation.
This study proposes a Fractional-Order Physics-Informed Neural Network (FPINN) framework for solving inverse parameter estimation problems in both fractional SIR and augmented SEIR epidemiological models and demonstrates that the proposed method accurately reconstructs epidemic trajectories and captures the influence of memory effects on disease evolution.
S. Naveen, V. Parthiban· Network Modeling Analysis in...· 0 citations
Empirical results based on monthly influenza data from Xinjiang show that the proposed framework can capture epidemic trends, seasonal peaks, and fitting uncertainty and provide a theoretically grounded and practically applicable approach for infectious disease modeling under memory effects and stochastic perturbations.
Ge Zhang, Zhi-Hao Wang, Zhiming Li et al.· Fractal and Fractional· 0 citations
Hybrid computational models can improve prediction when governing equations are incomplete, but their advantages are often evaluated on isolated systems and without simultaneous assessment of accuracy, stability, interpretability, and uncertainty. This study develops a modular hybrid mathematical model that combines a partially specified ordinary differential equation, a regularized neural residual, joint parameter calibration, physical constraints, and ensemble-based uncertainty quantification. The framework was evaluated through in silico experiments on five benchmark systems representing periodic, chaotic, stiff, ecological, and engineering dynamics: Van der Pol, Lorenz-63, Robertson kinetics, Lotka–Volterra, and a continuous stirred-tank reactor. Six hundred trajectories were generated using space-filling sampling, partitioned at the trajectory level, and tested under interpolation, extrapolation, measurement noise, data scarcity, and partial observability. The proposed model was compared with an incomplete mechanistic model, a neural ordinary differential equation, and sequential residual correction. In illustrative synthetic results, the hybrid model achieved a mean normalized root-mean-square error of 0.0678, reducing error by 37.3% relative to the strongest baseline. It also increased the mean time to divergence to 83.6% of the forecast horizon, reduced median mechanistic parameter error to 4.8%, limited physical-constraint violations to 0.6%, and attained 0.947 coverage for nominal 95% prediction intervals. Friedman and Holm-adjusted Wilcoxon tests indicated significant paired improvements with large effect sizes. Ablation analyses showed that residual regularization, physical constraints, and joint calibration each contributed materially. These findings illustrate how restricted data-driven correction can enhance heterogeneous dynamical-system prediction while preserving mechanistic meaning, although empirical execution and external validation are required before scientific claims are made
George Em Karniadakis, E. Torfs, L. Marchetti· Global Synthesis in Educatio...· 0 citations
It is demonstrated that fractional dynamics fundamentally reshape transient epidemic evolution, providing new insight into memory-driven processes in complex biological systems and nonlinear epidemiological modeling.
This paper presents a fractional-order mathematical model for describing the dynamics of cancer biomarkers based on the Caputo fractional derivative. The proposed model incorporates memory effects inherent in biological systems, which provides a more realistic framework than the corresponding classical integer-order model. The existence and uniqueness of solutions are established using fixed-point theory, while the local stability of the equilibrium point is investigated through suitable stability criteria. Numerical solutions are obtained using the Generalized Fractional Taylor Expansion (GFTE), and the effectiveness and accuracy of the proposed numerical scheme are demonstrated. Furthermore, Particle Swarm Optimization (PSO) is employed for determining the optimal model parameters and nanoparticle dosage to improve tumor size estimation. The numerical results reveal the influence of the fractional-order parameter on biomarker dynamics and show that the proposed fractional-order model offers a flexible and robust framework for cancer biomarker analysis, early tumor detection, and optimization-based parameter estimation.
Bushra M. Al-Smadi, S. Momani, I. Batiha et al.· An International Journal of...· 0 citations
This study develops a fractional-order mathematical model based on the Atangana--Baleanu--Caputo (ABC) operator to investigate the transmission dynamics of tuberculosis (TB). The framework incorporates memory and nonlocal effects to represent the spread and progression of TB within a population. The first derivative of a Lyapunov function is used to evaluate the infection locally and globally within the fractional-order model. The model satisfies the essential mathematical properties of positivity, boundedness, existence, and uniqueness of solutions, thereby establishing its biological and mathematical well-posedness. Fixed-point theory is used to analyze the model and bound its solution. The analysis establishes local and global stability conditions and identifies the parameters that most strongly affect disease transmission. An advanced numerical method is used to obtain approximate solutions of the fractional-order system and evaluate the effect of the fractional-order parameter. Numerical simulations show that decreasing the fractional-order parameter enhances memory effects and produces smoother convergence toward equilibrium states than the classical integer-order model. The results indicate that the fractional-order framework provides a useful representation of TB dynamics and may support the understanding and control of TB transmission.
M. Farman, K. S. Nisar· Journal of the Nigerian Soci...· 0 citations
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