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

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.