Achraf Zinihi

Achraf Zinihi, Matthias Ehrhardt, Moulay Rchid Sidi Ammi

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.

Achraf Zinihi

We develop a physics-informed neural network (PINN) framework for parameter estimation in fractional-order SEIRD epidemic models. By embedding the Caputo fractional derivative into the network residuals via the L1 discretization scheme, our method simultaneously reconstructs epidemic trajectories and infers both epidemiological parameters and the fractional memory order $α$. The fractional formulation extends classical integer-order models by capturing long-range memory effects in disease progression, incubation, and recovery. Our framework learns the fractional memory order $α$ as a trainable parameter while simultaneously estimating the epidemiological rates $(β, σ, γ, μ)$. A composite loss combining data misfit, physics residuals, and initial conditions, with constraints on positivity and population conservation, ensures both accuracy and biological consistency. Tests on synthetic Mpox data confirm reliable recovery of $α$ and parameters under noise, while applications to COVID-19 show that optimal $α\in (0, 1]$ captures memory effects and improves predictive performance over the classical SEIRD model. This work establishes PINNs as a robust tool for learning memory effects in epidemic dynamics, with implications for forecasting, control strategies, and the analysis of non-Markovian epidemic processes.