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<p>We present a fractional order model for the spread of malware in wireless sensor networks that builds memory directly into the dynamics through the proportional Hadamard–Caputo operator. The network population is organized into six groups, namely susceptible, exposed, infectious, quarantined, recovered, and vaccinated devices. We recast the system as an integral equation using a logarithmic change of time and we prove two fixed point results, where existence follows from the nonlinear Leray–Schauder alternative and uniqueness is obtained by Banach’s contraction principle. We then establish stability in the sense of Ulam–Hyers and in its extended form, showing that small modeling or data errors lead to proportionally small changes in the solutions. For computation, we build a predictor and corrector scheme in the modified Adams Bashforth Moulton framework adapted to the proportional Hadamard Caputo kernel with exponential memory in logarithmic time. Simulations show that stronger memory or a lower fractional order slows decay and extends spread while values near the classical case bring rapid stabilization.</p>

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Artificial neural network analysis of a fractional cyber-epidemic model in wireless sensors under the proportional Hadamard–Caputo operator

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  • 发表日期：

    2026年3月28日

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  • Mohamed A. Barakat,
  • Abd-Allah Hyder,
  • Tarek Aboelenen,
  • Naveed Iqbal,
  • Ahmed Shafee,
  • Noorullah Noori

摘要

We present a fractional order model for the spread of malware in wireless sensor networks that builds memory directly into the dynamics through the proportional Hadamard–Caputo operator. The network population is organized into six groups, namely susceptible, exposed, infectious, quarantined, recovered, and vaccinated devices. We recast the system as an integral equation using a logarithmic change of time and we prove two fixed point results, where existence follows from the nonlinear Leray–Schauder alternative and uniqueness is obtained by Banach’s contraction principle. We then establish stability in the sense of Ulam–Hyers and in its extended form, showing that small modeling or data errors lead to proportionally small changes in the solutions. For computation, we build a predictor and corrector scheme in the modified Adams Bashforth Moulton framework adapted to the proportional Hadamard Caputo kernel with exponential memory in logarithmic time. Simulations show that stronger memory or a lower fractional order slows decay and extends spread while values near the classical case bring rapid stabilization.