Stability analysis and numerical simulation of nonlocal extended epidemic models using positivity-preserving scheme.

In this paper we introduce a robust numerical framework for simulating the nonlocal extended epidemic models that incorporate the fractional diffusion to capture the complex spatial-temporal dynamics of disease spread. The presented numerical scheme uses the Fourier spectral approach for spatial discretization and a positivity-preserving exponential time differencing approach for temporal integration. Two fractional epidemic models-the SIR model with a modified saturated incidence rate and the SEIR model-are analyzed to investigate the impact of fractional diffusion on disease transmission. The transmission dynamics are explored using the developed L-stable, positivity-preserving numerical method. Numerical experiments demonstrate the influences of the fractional diffusion on the spatial distribution and temporal evolution of infectious diseases. These experiments also highlight the importance of fractional diffusion in epidemic models and provide a rigorous computational framework that can be used in future research in public health policy and epidemic control strategies. A benchmark test problem with a known exact solution is considered, which confirms that the proposed L-stable scheme achieves second-order accuracy for different values of α.