Artificial Intelligence–Assisted Fractional Nonstandard Finite Difference Scheme for a Noncommensurate Quarantine Reaction–Diffusion Epidemic Model

Younes Brahim Oumedjber, Adel Ouannas, Nadjette Debbouche and Giuseppe Grassi

International Journal of Differential Equations, 2026, vol. 2026, 1-23

Abstract: This paper develops a noncommensurate time-fractional SEIQR reaction–diffusion model for infectious disease dynamics with artificial intelligence (AI)–assisted inverse identification and forecasting. The susceptible, exposed, infected, quarantined, and recovered classes are governed by distinct Caputo fractional orders so that latency, infection progression, quarantine response, and recovery may carry different memory intensities, whereas spatial mobility is described by classical Laplacian diffusion. A positivity-preserving fractional nonstandard finite difference scheme is constructed by combining the L1 approximation of the Caputo derivative with an implicit Laplacian and nonlocal discretizations of nonlinear incidence terms. The scheme preserves nonnegative population densities and gives the forward reference solution with consistency order Ok2−ϑmax+h2. The qualitative structure of the model is discussed through positivity, finite-horizon admissibility, the disease-free equilibrium, the basic reproduction number, and noncommensurate fractional stability. The AI component is used as a data-assimilation and surrogate layer rather than as a replacement for the structure-preserving solver: it identifies epidemiological parameters, diffusion coefficients, and fractional orders and provides fast repeated forecasts once trained on NSFD-generated reference solutions. Numerical experiments demonstrate compartment-dependent memory effects, positivity preservation against a standard explicit scheme, recovery of model parameters and fractional orders, manufactured-solution verification, and neural-operator speedup for repeated forecasting.

Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnijde:1001115

DOI: 10.1155/ijde/1001115

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