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A nonstandard finite difference scheme for an SEIQR epidemiological PDE model

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  • Zinihi, Achraf
  • Ehrhardt, Matthias
  • Sidi Ammi, Moulay Rchid

Abstract

This paper introduces a nonstandard finite difference (NSFD) approach to a reaction-diffusion SEIQR epidemiological model, which captures the spatiotemporal dynamics of infectious disease transmission. Formulated as a system of semilinear parabolic partial differential equations (PDEs), the model extends classical compartmental models by incorporating spatial diffusion to account for population movement and spatial heterogeneity. The proposed NSFD discretization is designed to preserve the continuous model’s essential qualitative features, such as positivity, boundedness, and stability, which are often compromised by standard finite difference methods. We rigorously analyze the model’s well-posedness, construct a structure-preserving NSFD scheme for the PDE system, and study its convergence and local truncation error. Numerical simulations validate the theoretical findings and demonstrate the scheme’s effectiveness in preserving biologically consistent dynamics.

Suggested Citation

  • Zinihi, Achraf & Ehrhardt, Matthias & Sidi Ammi, Moulay Rchid, 2026. "A nonstandard finite difference scheme for an SEIQR epidemiological PDE model," Applied Mathematics and Computation, Elsevier, vol. 520(C).
    • Handle: RePEc:eee:apmaco:v:520:y:2026:i:c:s0096300326000056
    DOI: 10.1016/j.amc.2026.129953
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    References listed on IDEAS

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    1. Zhang, Chao & Gao, Jianguo & Sun, Hongquan & Wang, Jinliang, 2019. "Dynamics of a reaction–diffusion SVIR model in a spatial heterogeneous environment," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 533(C).
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    3. Zhang, Chao & Gao, Jianguo & Sun, Hongquan & Wang, Jinliang, 2019. "Dynamics of a reaction–diffusion SVIR model in a spatial heterogeneous environment," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 533(C), pages 1-1.
    4. Mickens, Ronald E., 2003. "A nonstandard finite difference scheme for the diffusionless Burgers equation with logistic reaction," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 62(1), pages 117-124.
    5. Gao, Shujing & Teng, Zhidong & Xie, Dehui, 2009. "Analysis of a delayed SIR epidemic model with pulse vaccination," Chaos, Solitons & Fractals, Elsevier, vol. 40(2), pages 1004-1011.
    6. Pasha, Syed Ahmed & Nawaz, Yasir & Arif, Muhammad Shoaib, 2023. "On the nonstandard finite difference method for reaction–diffusion models," Chaos, Solitons & Fractals, Elsevier, vol. 166(C).
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