A Positivity-Preserving Numerical Scheme for Reaction–Diffusion SIRA Models of Cyber-Epidemics

The growing complexity of computer viruses has necessitated the development of more sophisticated mathematical models to understand and control cyber-epidemics. Building on classical compartmental approaches, we extend the Susceptible-Infected-Removed-Antidotal (SIRA) model into a reactiondiffusion framework to capture both local infection dynamics and spatial transmission across networks. The resulting system presents significant analytical challenges due to stiffness, positivity constraints, and nonlinear couplings. To address these issues, we propose a semi-implicit numerical scheme that treats diffusion terms implicitly for stability and reaction terms explicitly for computational efficiency, coupled with Nonstandard Finite Difference (NSFD) techniques to ensure solution positivity. Rigorous stability analysis based on M-matrix theory establishes conditions for boundedness and convergence of the discrete system. Compared to existing methodologies, our approach enables accurate, efficient simulation of large-scale malware propagation, offering critical insights for cybersecurity strategies such as antivirus deployment optimization and outbreak containment. Numerical experiments demonstrate the effectiveness of the proposed scheme in capturing complex spatiotemporal dynamics of cyber threats.