A novel numerical solution of nonlinear stochastic model for the propagation of malicious codes in Wireless Sensor Networks using a high order spectral collocation technique
Abstract The open nature of Wireless Sensor Networks (WSNs) renders them an easy target to malicious code propagation, posing a significant and persistent threat to their security. Various mathematical models have been studied in recent literature for understanding the dynamics and control of the pr...
Saved in:
| Main Authors: | , , , , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Nature Portfolio
2025-01-01
|
| Series: | Scientific Reports |
| Subjects: | |
| Online Access: | https://doi.org/10.1038/s41598-024-82033-2 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1850281848888885248 |
|---|---|
| author | Junjie Zhu Misbah Ullah Saif Ullah Muhammad Bilal Riaz Abdul Baseer Saqib Atif M. Alamri Salman A. AlQahtani |
| author_facet | Junjie Zhu Misbah Ullah Saif Ullah Muhammad Bilal Riaz Abdul Baseer Saqib Atif M. Alamri Salman A. AlQahtani |
| author_sort | Junjie Zhu |
| collection | DOAJ |
| description | Abstract The open nature of Wireless Sensor Networks (WSNs) renders them an easy target to malicious code propagation, posing a significant and persistent threat to their security. Various mathematical models have been studied in recent literature for understanding the dynamics and control of the propagation of malicious codes in WSNs. However, due to the inherent randomness and uncertainty present in WSNs, stochastic modeling approach is essential for a comprehensive understanding of the propagation of malicious codes in WSNs. In this paper, we formulate a general stochastic compartmental model for analyzing the dynamics of malicious code distribution in WSNs and suggest its possible control. We incorporate the stochasticity in the classical deterministic model for the inherent unpredictability in code propagation, which results in a more appropriate representation of the dynamics. A basic theoretical analysis including the stability results of the model with randomness is carried out. Moreover, a higher-order spectral collocation technique is applied for the numerical solution of the proposed stochastic model. The accuracy and numerical stability of the model is presented. Finally, a comprehensive simulation is depicted to verify theoretical results and depict the impact of parameters on the model’s dynamic behavior. This study incorporates stochasticity in a deterministic model of malicious codes spread in WSNs with the implementation of spectral numerical scheme which helps to capture these networks’ inherent uncertainties and complex nature. |
| format | Article |
| id | doaj-art-e605fc838d704f4192501cd4acdba8a6 |
| institution | OA Journals |
| issn | 2045-2322 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | Nature Portfolio |
| record_format | Article |
| series | Scientific Reports |
| spelling | doaj-art-e605fc838d704f4192501cd4acdba8a62025-08-20T01:48:08ZengNature PortfolioScientific Reports2045-23222025-01-0115112110.1038/s41598-024-82033-2A novel numerical solution of nonlinear stochastic model for the propagation of malicious codes in Wireless Sensor Networks using a high order spectral collocation techniqueJunjie Zhu0Misbah Ullah1Saif Ullah2Muhammad Bilal Riaz3Abdul Baseer Saqib4Atif M. Alamri5Salman A. AlQahtani6School of Mathematics, Shandong UniversityDepartment of Mathematics, City University of Science and Information Technology PeshawarDepartment of Mathematics, University of PeshawarIT4Innovations, VSB-Technical University of OstravaFaculty of Education, Department of Mathematics, Nangrahar UniversitySoftware Engineering Department, College of Computer and Information Sciences, King Saud UniversityComputer Engineering Department, College of Computer and Information Sciences, King Saud UniversityAbstract The open nature of Wireless Sensor Networks (WSNs) renders them an easy target to malicious code propagation, posing a significant and persistent threat to their security. Various mathematical models have been studied in recent literature for understanding the dynamics and control of the propagation of malicious codes in WSNs. However, due to the inherent randomness and uncertainty present in WSNs, stochastic modeling approach is essential for a comprehensive understanding of the propagation of malicious codes in WSNs. In this paper, we formulate a general stochastic compartmental model for analyzing the dynamics of malicious code distribution in WSNs and suggest its possible control. We incorporate the stochasticity in the classical deterministic model for the inherent unpredictability in code propagation, which results in a more appropriate representation of the dynamics. A basic theoretical analysis including the stability results of the model with randomness is carried out. Moreover, a higher-order spectral collocation technique is applied for the numerical solution of the proposed stochastic model. The accuracy and numerical stability of the model is presented. Finally, a comprehensive simulation is depicted to verify theoretical results and depict the impact of parameters on the model’s dynamic behavior. This study incorporates stochasticity in a deterministic model of malicious codes spread in WSNs with the implementation of spectral numerical scheme which helps to capture these networks’ inherent uncertainties and complex nature.https://doi.org/10.1038/s41598-024-82033-2Spectral collocation techniqueStochastic compartmental modeling, Distribution density, Wireless sensor network, Malicious codes propagation, Stability Analysis |
| spellingShingle | Junjie Zhu Misbah Ullah Saif Ullah Muhammad Bilal Riaz Abdul Baseer Saqib Atif M. Alamri Salman A. AlQahtani A novel numerical solution of nonlinear stochastic model for the propagation of malicious codes in Wireless Sensor Networks using a high order spectral collocation technique Scientific Reports Spectral collocation technique Stochastic compartmental modeling, Distribution density, Wireless sensor network, Malicious codes propagation, Stability Analysis |
| title | A novel numerical solution of nonlinear stochastic model for the propagation of malicious codes in Wireless Sensor Networks using a high order spectral collocation technique |
| title_full | A novel numerical solution of nonlinear stochastic model for the propagation of malicious codes in Wireless Sensor Networks using a high order spectral collocation technique |
| title_fullStr | A novel numerical solution of nonlinear stochastic model for the propagation of malicious codes in Wireless Sensor Networks using a high order spectral collocation technique |
| title_full_unstemmed | A novel numerical solution of nonlinear stochastic model for the propagation of malicious codes in Wireless Sensor Networks using a high order spectral collocation technique |
| title_short | A novel numerical solution of nonlinear stochastic model for the propagation of malicious codes in Wireless Sensor Networks using a high order spectral collocation technique |
| title_sort | novel numerical solution of nonlinear stochastic model for the propagation of malicious codes in wireless sensor networks using a high order spectral collocation technique |
| topic | Spectral collocation technique Stochastic compartmental modeling, Distribution density, Wireless sensor network, Malicious codes propagation, Stability Analysis |
| url | https://doi.org/10.1038/s41598-024-82033-2 |
| work_keys_str_mv | AT junjiezhu anovelnumericalsolutionofnonlinearstochasticmodelforthepropagationofmaliciouscodesinwirelesssensornetworksusingahighorderspectralcollocationtechnique AT misbahullah anovelnumericalsolutionofnonlinearstochasticmodelforthepropagationofmaliciouscodesinwirelesssensornetworksusingahighorderspectralcollocationtechnique AT saifullah anovelnumericalsolutionofnonlinearstochasticmodelforthepropagationofmaliciouscodesinwirelesssensornetworksusingahighorderspectralcollocationtechnique AT muhammadbilalriaz anovelnumericalsolutionofnonlinearstochasticmodelforthepropagationofmaliciouscodesinwirelesssensornetworksusingahighorderspectralcollocationtechnique AT abdulbaseersaqib anovelnumericalsolutionofnonlinearstochasticmodelforthepropagationofmaliciouscodesinwirelesssensornetworksusingahighorderspectralcollocationtechnique AT atifmalamri anovelnumericalsolutionofnonlinearstochasticmodelforthepropagationofmaliciouscodesinwirelesssensornetworksusingahighorderspectralcollocationtechnique AT salmanaalqahtani anovelnumericalsolutionofnonlinearstochasticmodelforthepropagationofmaliciouscodesinwirelesssensornetworksusingahighorderspectralcollocationtechnique AT junjiezhu novelnumericalsolutionofnonlinearstochasticmodelforthepropagationofmaliciouscodesinwirelesssensornetworksusingahighorderspectralcollocationtechnique AT misbahullah novelnumericalsolutionofnonlinearstochasticmodelforthepropagationofmaliciouscodesinwirelesssensornetworksusingahighorderspectralcollocationtechnique AT saifullah novelnumericalsolutionofnonlinearstochasticmodelforthepropagationofmaliciouscodesinwirelesssensornetworksusingahighorderspectralcollocationtechnique AT muhammadbilalriaz novelnumericalsolutionofnonlinearstochasticmodelforthepropagationofmaliciouscodesinwirelesssensornetworksusingahighorderspectralcollocationtechnique AT abdulbaseersaqib novelnumericalsolutionofnonlinearstochasticmodelforthepropagationofmaliciouscodesinwirelesssensornetworksusingahighorderspectralcollocationtechnique AT atifmalamri novelnumericalsolutionofnonlinearstochasticmodelforthepropagationofmaliciouscodesinwirelesssensornetworksusingahighorderspectralcollocationtechnique AT salmanaalqahtani novelnumericalsolutionofnonlinearstochasticmodelforthepropagationofmaliciouscodesinwirelesssensornetworksusingahighorderspectralcollocationtechnique |