Incommensurate fractional epidemic dynamics with compartment-specific memory, treatment response, and physics-informed forecasting

Zied Elleuch; Younes Brahim Oumedjber; Omar Kahouli; Adel Ouannas; Sulaiman Almohaimeed; Lilia El Amraoui; Mohamed Ayari; Zied Elleuch; Younes Brahim Oumedjber; Omar Kahouli; Adel Ouannas; Sulaiman Almohaimeed; Lilia El Amraoui; Mohamed Ayari

doi:10.3934/math.2026903

AIMS Mathematics

2026, Volume 11, Issue 7: 22316-22353. doi: 10.3934/math.2026903

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Incommensurate fractional epidemic dynamics with compartment-specific memory, treatment response, and physics-informed forecasting

1.
Computer Science Department, Applied College, University of Ha'il, 2440, Ha'il, Saudi Arabia
2.
Laboratory of Materials Science for Applications and the Environment (LaSMAE), Department of Sciences, Teacher Education College of Sétif–Messaoud Zeghar, P.O. Box 556, El Eulma, Sétif 19600, Algeria
3.
Department of Electronics Engineering, Applied College, University of Ha'il, Ha'il 2440, Saudi Arabia
4.
Department of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah, Saudi Arabia
5.
Department of Electrical Engineering, College of Engineering, Qassim University, 52571, Saudi Arabia
6.
Department of Electrical Engineering, College of Engineering, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
7.
Department of Information Technology, Faculty of Computing and Information Technology, Northern Border University, Arar 91431, Saudi Arabia

Received: 26 May 2026 Revised: 22 June 2026 Accepted: 14 July 2026 Published: 27 July 2026
MSC : 92D30, 34A08, 34D20, 65L05

This paper develops an incommensurate fractional-order susceptible-exposed-symptomatic infectious-asymptomatic infectious-treated-recovered (SEIATR) model with compartment-specific Caputo memory. We establish local well-posedness, positivity, boundedness, biological feasibility, disease-free and endemic equilibria, and the basic reproduction number. Local stability is analyzed through the rational-order incommensurate Matignon criterion. A componentwise multi-order L1 scheme is then used for simulation, refinement, and sensitivity analysis. Finally, a physics-informed neural network is employed as a synthetic benchmark for hidden-state reconstruction, parameter and order identification, residual checking, and short-horizon forecasting. The results show that compartment-specific memory mainly affects peak timing, persistence, and treatment load, whereas epidemiological rates govern amplification and threshold behavior.
- incommensurate fractional order,
- Caputo derivative,
- SEIATR model,
- epidemic dynamics,
- L1 scheme,
- physics-informed neural network,
- forecasting
Citation: Zied Elleuch, Younes Brahim Oumedjber, Omar Kahouli, Adel Ouannas, Sulaiman Almohaimeed, Lilia El Amraoui, Mohamed Ayari. Incommensurate fractional epidemic dynamics with compartment-specific memory, treatment response, and physics-informed forecasting[J]. AIMS Mathematics, 2026, 11(7): 22316-22353. doi: 10.3934/math.2026903

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Abstract

This paper develops an incommensurate fractional-order susceptible-exposed-symptomatic infectious-asymptomatic infectious-treated-recovered (SEIATR) model with compartment-specific Caputo memory. We establish local well-posedness, positivity, boundedness, biological feasibility, disease-free and endemic equilibria, and the basic reproduction number. Local stability is analyzed through the rational-order incommensurate Matignon criterion. A componentwise multi-order L1 scheme is then used for simulation, refinement, and sensitivity analysis. Finally, a physics-informed neural network is employed as a synthetic benchmark for hidden-state reconstruction, parameter and order identification, residual checking, and short-horizon forecasting. The results show that compartment-specific memory mainly affects peak timing, persistence, and treatment load, whereas epidemiological rates govern amplification and threshold behavior.

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