Deep Neural Network Analysis and Optimal Control for a Fractional‐Order Model of HPV and Cervical Cancer

ABSTRACT Leveraging the Liouville–Caputo fractional derivative (LCFD), this work constructs a mathematical framework to examine the dynamics of Human Papillomavirus (HPV) transmission and its progression into cervical cancer. We propose a novel fractional‐order model (FOM) consisting of five coupled fractional differential equations (FDEs) that represent different population compartments and incorporate essential epidemiological characteristics of HPV infection. To ensure mathematical validity, we rigorously prove that the model's solutions exist, are unique, and remain positive and bounded. Our analysis involves deriving the basic reproduction number to establish a threshold for disease persistence, followed by a comprehensive examination of the stability of the disease‐free and endemic equilibrium states. Parameter sensitivity analysis identifies key factors influencing disease transmission dynamics. An extension of our framework involves the formulation of a fractional optimal control problem (FOCP), which includes three time‐dependent control measures: safe sexual practice promotion (), vaccination and immunity enhancement (), and comprehensive cancer screening and treatment implementation (). By employing Pontryagin's Maximum Principle (PMP), we rigorously derive the set of necessary conditions required for optimal control. To numerically solve and validate the model, a deep neural network (DNN) with Tanh and ReLU activations was implemented within a stochastic framework. The model's fidelity was verified through convergence testing, error distribution analysis, and regression metrics, using the Adams‐Bashforth‐Moulton (ABM) scheme and with data divided for training (70%), validation (15%), and testing (15%). The proposed methodology yields solutions that align precisely with benchmark data, achieving a best validation performance of and a minimum absolute error of . Numerical simulations of four control strategies demonstrate that while each intervention reduces infection and cancer progression to some extent, the combined strategy proves most effective. This study highlights the pivotal role of the fractional operator in disease dynamics and introduces a novel hybrid approach that integrates deep learning with fractional calculus, offering a computationally efficient and highly accurate tool for analyzing complex epidemiological systems.