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

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...

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Vydáno v:	Differential equations Ročník 61; číslo 7; s. 1171 - 1195
Hlavní autoři:	Aqib Zafar, Shahid Hussain, Xinlong Feng, Sidorov, Denis
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Moscow Pleiades Publishing 01.07.2025
Témata:	Difference and Functional Equations Mathematics Mathematics and Statistics Numerical Methods Ordinary Differential Equations Partial Differential Equations computer virus model diffusion positive bounded numerical method
ISSN:	0012-2661, 1608-3083
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Abstract	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.
AbstractList	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.
Author	Sidorov, Denis Aqib Zafar Shahid Hussain Xinlong Feng
Author_xml	– sequence: 1 surname: Aqib Zafar fullname: Aqib Zafar email: aqibzafar76@gmail.com organization: College of Mathematics and System Sciences, Xinjiang University – sequence: 2 surname: Shahid Hussain fullname: Shahid Hussain email: shahid_math@xju.edu.cn organization: College of Mathematics and System Sciences, Xinjiang University – sequence: 3 surname: Xinlong Feng fullname: Xinlong Feng email: fxlmath@xju.edu.cn organization: College of Mathematics and System Sciences, Xinjiang University – sequence: 4 givenname: Denis surname: Sidorov fullname: Sidorov, Denis email: dsidorov@isem.irk.ru organization: Sino-Russian Joint Research Center for Advanced Energy and Power Systems, Energy Systems Institute of the Siberian Branch of the Russian Academy of Sciences, School of Electrical Engineering and Automation, Harbin Institute of Technology
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Cites_doi	10.1016/S0167-9473(03)00113-0 10.1016/j.chaos.2020.110127 10.1016/0025-5564(93)90018-6 10.1016/S0375-9601(02)00152-4 10.1016/j.bspc.2019.101584 10.1016/j.physa.2019.122372 10.13001/1081-3810.1374 10.1016/j.rinp.2021.104017 10.1145/262793.262811 10.1016/j.cose.2008.07.006 10.1016/j.nonrwa.2019.04.006 10.1016/0378-4371(77)90001-2 10.3844/jcssp.2005.31.34 10.1016/j.nonrwa.2011.07.048 10.1109/MSPEC.2013.6471059 10.1016/0167-4048(92)90192-T 10.1016/j.physa.2019.02.018 10.1109/TEC.2011.2162093 10.1016/0022-247X(84)90182-3 10.1016/j.cnsns.2012.05.030 10.1016/j.cmpb.2020.105350 10.1016/j.chaos.2011.10.003 10.1016/0022-0396(79)90088-3
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Title	Exploring the Stability Region and Designing the Operator Splitting Numerical Algorithm for the Virus Communication in Reaction Diffusion Environment
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