Previous Article in Journal / Special Issue
Sustainable Water Treatment Through Fractional-Order Chemostat Modeling with Sliding Memory and Periodic Boundary Conditions: A Mathematical Framework for Clean Water and Sanitation
 
 
Search for Articles:
Title / Keyword
Author / Affiliation / Email
Journal
Article Type
 
 
Advanced Search
 
Section
Special Issue
Volume
Issue
Number
Page
 
Journals
Fractal Fract
Volume 10
Issue 1
10.3390/fractalfract10010003
fractalfract-logo
Submit to this Journal Review for this Journal Propose a Special Issue
► Article Menu

Article Menu

share Share announcement Help format_quote Cite
first_page
settings
Order Article Reprints
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
This is an early access version, the complete PDF, HTML, and XML versions will be available soon.
Article

The Fractional SI Reaction–Diffusion Model with Incommensurate Orders: Stability Analysis and Numerical Simulations

by
Ali Aloui
1,*,
Amel Hioual
2,
Omar Kahouli
1,
Adel Ouannas
3,
Lilia El Amraoui
4 and
Mohamed Ayari
5,*
1
Department of Electronics Engineering, Applied College, University of Ha’il, Ha’il 2440, Saudi Arabia
2
Laboratory of Dynamical Systems and Control, Department of Mathematics and Computer Science, University of Oum El Bouaghi, Oum El Bouaghi 04000, Algeria
3
Department of Mathematics and Computer Science, University of Oum El Bouaghi, Oum El Bouaghi 04000, Algeria
4
Department of Electrical Engineering, College of Engineering, Princess Nourah Bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
5
Department of Information Technology, Faculty of Computing and Information Technology, Northern Border University, Arar 91431, Saudi Arabia
*
Authors to whom correspondence should be addressed.
Fractal Fract. 2026, 10(1), 3; https://doi.org/10.3390/fractalfract10010003 (registering DOI)
Submission received: 6 October 2025 / Revised: 10 December 2025 / Accepted: 14 December 2025 / Published: 19 December 2025
(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations, 3rd Edition)
Versions Notes

Abstract

In this work, we present a fractional-order reaction–diffusion model for the spread of infectious diseases, incorporating incommensurate Caputo derivatives to capture memory effects and heterogeneous temporal behavior across compartments. Focusing on a generalized SI model with nonlinear incidence, we explore the local asymptotic stability of both disease-free and endemic equilibria. The model accommodates spatial diffusion, saturation effects, and varying fractional orders, yielding a more realistic depiction of epidemic propagation. Analytical techniques—ranging from linearization to spectral analysis—are employed to rigorously establish stability conditions. Numerical simulations support the theoretical findings, highlighting the impact of memory and spatial structure on long-term dynamics. This study offers a refined mathematical lens to understand the persistence or eradication of infectious diseases under memory-dependent and spatially heterogeneous environments.
Keywords: fractional-order differential equations; incommensurate systems; epidemic modeling; reaction–diffusion equations; Caputo derivative; stability analysis; fractional calculus in biology fractional-order differential equations; incommensurate systems; epidemic modeling; reaction–diffusion equations; Caputo derivative; stability analysis; fractional calculus in biology

Share and Cite

MDPI and ACS Style

Aloui, A.; Hioual, A.; Kahouli, O.; Ouannas, A.; Amraoui, L.E.; Ayari, M. The Fractional SI Reaction–Diffusion Model with Incommensurate Orders: Stability Analysis and Numerical Simulations. Fractal Fract. 2026, 10, 3. https://doi.org/10.3390/fractalfract10010003

AMA Style

Aloui A, Hioual A, Kahouli O, Ouannas A, Amraoui LE, Ayari M. The Fractional SI Reaction–Diffusion Model with Incommensurate Orders: Stability Analysis and Numerical Simulations. Fractal and Fractional. 2026; 10(1):3. https://doi.org/10.3390/fractalfract10010003

Chicago/Turabian Style

Aloui, Ali, Amel Hioual, Omar Kahouli, Adel Ouannas, Lilia El Amraoui, and Mohamed Ayari. 2026. "The Fractional SI Reaction–Diffusion Model with Incommensurate Orders: Stability Analysis and Numerical Simulations" Fractal and Fractional 10, no. 1: 3. https://doi.org/10.3390/fractalfract10010003

APA Style

Aloui, A., Hioual, A., Kahouli, O., Ouannas, A., Amraoui, L. E., & Ayari, M. (2026). The Fractional SI Reaction–Diffusion Model with Incommensurate Orders: Stability Analysis and Numerical Simulations. Fractal and Fractional, 10(1), 3. https://doi.org/10.3390/fractalfract10010003

Article Metrics

Back to TopTop