Rashid Saima, Shafique Rafia, Akram Saima, Elagan Sayed K
Department of Mathematics, Government College University, Faisalabad 38000, Pakistan.
Department of Mathematics, Government College for Women University Faisalabad, Faisalabad 38000, Pakistan.
Heliyon. 2024 Aug 13;10(16):e35955. doi: 10.1016/j.heliyon.2024.e35955. eCollection 2024 Aug 30.
Wireless sensor networks (WSNs) have attracted a lot of interest due to their enormous potential for both military and civilian uses. Worm attacks can quickly target WSNs because of the network's weak security. The worm can spread throughout the network by interacting with a single unsafe node. Moreover, the analysis of worm spread in WSNs can benefit from the use of mathematical epidemic models. We suggest a five-compartment model to characterize the mechanisms of worm proliferation with respect to time in WSN. Taking into account the transform convoluted with the Atangana-Baleanu-Caputo (ABC) fractional derivative operator, we employ it to analyze the characteristics and applications of the transformation using the Mittag-Leffler kernel. Moreover, we construct a new algorithm for the homotopy perturbation method (HPM) in conjunction with the transform technique to generate analytical solutions for the worm transmission model. Also, we address the qualitative aspects such as positivity, boundness, worm-free state, endemic state, basic reproduction number and worm-free equilibrium stability. Furthermore, we prove that the virus rate in sensor nodes is extinct if and the virus persists if . In addition, we develop analytical findings to evaluate the series of solutions. Furthermore, a detailed statistical analysis is conducted to verify the nonlinear dynamics of the system by verifying the test to determine whether uncertainty exists using approximation entropy and the data. An extensive analysis of the vaccination class with respect to the transmitting class as well as the susceptible class is being used to investigate the effects of stepping up precautions on WP in WSN. Moreover, the modeling of the WSN revealed that reducing the fractional-order from 1 requires that the recommended approach be implemented at the highest rate so that there is no long-lasting immunization; instead, nodes remain briefly defensive before becoming vulnerable to future worm attacks.
无线传感器网络（WSNs）因其在军事和民用方面的巨大潜力而备受关注。由于网络安全薄弱，蠕虫攻击可以迅速瞄准无线传感器网络。蠕虫可以通过与单个不安全节点交互在整个网络中传播。此外，对无线传感器网络中蠕虫传播的分析可以受益于数学流行病模型的使用。我们提出了一个五 compartment 模型来描述无线传感器网络中蠕虫随时间扩散的机制。考虑到与 Atangana - Baleanu - Caputo（ABC）分数阶导数算子卷积的变换，我们使用它来分析使用 Mittag - Leffler 核的变换的特征和应用。此外，我们结合变换技术构建了一种新的同伦摄动法（HPM）算法，以生成蠕虫传播模型的解析解。同时，我们还讨论了诸如正性、有界性、无蠕虫状态、地方病状态、基本再生数和无蠕虫平衡稳定性等定性方面。此外，我们证明如果 ，传感器节点中的病毒率将灭绝，如果 ，病毒将持续存在。另外，我们得出分析结果以评估解的级数。此外，通过验证测试来进行详细的统计分析，以使用近似熵和数据来验证系统的非线性动力学，从而确定是否存在不确定性。对疫苗接种类别相对于传播类别以及易感类别的广泛分析正在用于研究加强预防措施对无线传感器网络中蠕虫传播的影响。此外，无线传感器网络的建模表明，将分数阶从 1 降低要求以最高速率实施推荐方法，以便不存在持久免疫；相反，节点在变得易受未来蠕虫攻击之前仅短暂防御。
