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Abstract <p> A computer virus poses significant risks to individual computer systems. To mitigate theserisks, various mathematical models have been developed. Several techniques, including theinstallation of antivirus software and the implementation of preventive measures based onepidemiological studies, can reduce the impact of virus attacks. This study focuses on thereaction-diffusion computer virus model. A qualitative analysis of the model is conducted, and thepositivity of the model is established. Additionally, the boundedness of the model’s solutions isdemonstrated. The stability region is determined by analyzing the variational matrix acrossdifferent parametric spaces for both temporal and diffusive components of the model. It isobserved that the stability region expands under the influence of diffusion phenomena. For thenumerical investigation of the model, three computational schemes are employed: the forwardEuler scheme, the backward Euler operator splitting scheme, and the non-standard finitedifference (NSFD) scheme. The NSFD scheme is analyzed in terms of positivity, consistency, andconvergence, with positivity being proven using M-matrix theory. A test problem is utilized toobtain numerical solutions, and various parameter values are explored to identify virus-free andvirus-endemic equilibrium states. The NSFD scheme is shown to preserve all essentialcharacteristics of the continuous model.</p>

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Exploring the Stability Region and Designing the Operator Splitting Numerical Algorithm for the Virus Communication in Reaction Diffusion Environment

  • 期刊论文

  • 发表日期：

    2025年7月1日

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  • Aqib Zafar,
  • Shahid Hussain,
  • Xinlong Feng,
  • Denis Sidorov

摘要

Abstract

A computer virus poses significant risks to individual computer systems. To mitigate theserisks, various mathematical models have been developed. Several techniques, including theinstallation of antivirus software and the implementation of preventive measures based onepidemiological studies, can reduce the impact of virus attacks. This study focuses on thereaction-diffusion computer virus model. A qualitative analysis of the model is conducted, and thepositivity of the model is established. Additionally, the boundedness of the model’s solutions isdemonstrated. The stability region is determined by analyzing the variational matrix acrossdifferent parametric spaces for both temporal and diffusive components of the model. It isobserved that the stability region expands under the influence of diffusion phenomena. For thenumerical investigation of the model, three computational schemes are employed: the forwardEuler scheme, the backward Euler operator splitting scheme, and the non-standard finitedifference (NSFD) scheme. The NSFD scheme is analyzed in terms of positivity, consistency, andconvergence, with positivity being proven using M-matrix theory. A test problem is utilized toobtain numerical solutions, and various parameter values are explored to identify virus-free andvirus-endemic equilibrium states. The NSFD scheme is shown to preserve all essentialcharacteristics of the continuous model.