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.

Computer Engineering Department College of Computer and Information Sciences King Saud University Riyadh Saudi Arabia

Department of Computer Science and Mathematics Lebanese American University Byblos Lebanon

Department of Mathematics City University of Science and Information Technology Peshawar Peshawar KPK 25000 Pakistan

Department of Mathematics University of Peshawar Peshawar KPK 25000 Pakistan

Faculty of Education Department of Mathematics Nangrahar University Jalalabad Nangrahar Afghanistan

IT4Innovations VSB Technical University of Ostrava Ostrava Czech Republic

School of Mathematics and Data Sciences Changji University Changji 831100 China

School of Mathematics Shandong University Jinan 250100 China

Software Engineering Department College of Computer and Information Sciences King Saud University Riyadh Saudi Arabia

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