Analysis, control, and forecasting the dynamics of SIRD models with saturated treatment and nonlinear incidence

This study addresses the challenge of managing complex epidemic dynamics exhibited by a novel Susceptible-Infected-Recovered-Deceased (SIRD) model. The introduced model incorporates a biologically and behaviorally mediated nonlinear incidence rate alongside a saturated treatment function, realistically capturing both physical transmission risks and the human behavioral responses during an outbreak. Our comprehensive analysis establishes the mathematical rigor of the system through positivity, boundedness, and a detailed investigation of equilibrium points and bifurcations (transcritical, saddle-node, and Hopf), which reveal the potential for complex, oscillatory, and chaotic epidemic patterns. To mitigate the disease burden, we apply Pontryagin's Maximum Principle to derive optimal time-dependent prevention and treatment control strategies. Furthermore, we introduce a highly efficient machine learning framework based on a Logistic-Map Reservoir Computer (LMRC) for accurate, real-time forecasting of these complex dynamics. Numerical simulations validate the theoretical findings and reveal the model's rich dynamical behaviors, including stable limit cycles and chaotic regimes driven by parameter variations. The integrated analytical, control, and machine learning approach provides a robust toolkit for public health policymakers, offering data-driven insights crucial for designing adaptive, cost-effective interventions that directly support the achievement of Sustainable Development Goal 3 (Good Health and Well-being).