Optimal control of chaos in a novel SITR epidemic model with generalized incidence and adaptive treatment dynamics: A deep neural network analysis

This paper introduces a novel SITR (Susceptible-Infected-Treated-Recovered) epidemic model that incorporates a Holling Type III incidence rate and a saturated treatment function to capture superspreading dynamics and finite healthcare capacity. We establish the model’s well-posedness by proving the positivity and boundedness of solutions. A comprehensive bifurcation analysis reveals that the system exhibits rich dynamical behaviors, including Transcritical, Saddle–node, and Hopf bifurcations, which delineate thresholds between disease extinction, persistence, and oscillatory states. An optimal control framework is subsequently formulated to derive effective intervention strategies. The core methodological contribution is the development of a hybrid deep neural network (DNN) architecture, utilizing Tanh and ReLU activations, to serve as a high-fidelity surrogate for the model’s complex dynamics. This approach is validated within a stochastic numerical scheme, employing a 70%–15%–15% data split for robust training and testing. The DNN achieves exceptional predictive accuracy, with a mean squared error of 10−10 and a minimum absolute error of 10−8, demonstrating precise alignment with benchmark solutions. This work establishes a novel paradigm that integrates sophisticated dynamical systems theory with advanced deep learning, resulting in a computationally efficient and highly accurate framework for analyzing and controlling complex epidemic systems.