Exploring the Stability Region and Designing the Operator Splitting Numerical Algorithm for the Virus Communication in Reaction Diffusion Environment - Aqib Zafar; Shahid Hussain; Xinlong Feng; Sidorov, Denis (Differential equations, 2025)
A computer virus poses significant risks to individual computer systems. To mitigate these risks, various mathematical models have been developed. Several techniques, including the installation of antivirus software and the implementation of preventive measures based on epidemiological studies, can reduce the impact of virus attacks. This study focuses on the reaction-diffusion computer virus model. A qualitative analysis of the model is conducted, and the positivity of the model is established. Additionally, the boundedness of the model’s solutions is demonstrated. The stability region is determined by analyzing the variational matrix across different parametric spaces for both temporal and diffusive components of the model. It is observed that the stability region expands under the influence of diffusion phenomena. For the numerical investigation of the model, three computational schemes are employed: the forward Euler scheme, the backward Euler operator splitting scheme, and the non-standard finite difference (NSFD) scheme. The NSFD scheme is analyzed in terms of positivity, consistency, and convergence, with positivity being proven using M-matrix theory. A test problem is utilized to obtain numerical solutions, and various parameter values are explored to identify virus-free and virus-endemic equilibrium states. The NSFD scheme is shown to preserve all essential characteristics of the continuous model.