Bifurcation and optimal control analysis for a fractional-order model of drug-resistant HBV infection

Hepatitis B Virus (HBV) remains a significant global health challenge, with control efforts complicated by the emergence of drug resistance and limitations in healthcare capacity. This study develops and analyzes a more realistic epidemiological model to investigate these complex dynamics and identify optimal intervention strategies. We formulate a novel six-compartment fractional-order model using the Caputo derivative, which incorporates memory effects, drug resistance, vertical transmission, and a crucial non-linear saturated treatment function to reflect healthcare system limitations. The methodology involves a rigorous mathematical analysis, including the calculation of the basic reproduction number (R₀), stability analysis of equilibria, sensitivity analysis, and an in-depth bifurcation analysis using center manifold theory. Furthermore, an optimal control problem is formulated using Pontryagin's Maximum Principle to evaluate the effectiveness of three time-dependent control mechanisms. A principal finding is the demonstration of a backward bifurcation, driven by the treatment saturation parameter, which implies that simply reducing the basic reproduction number below unity is insufficient for disease eradication. This creates a bistable region where the disease can persist. The optimal control analysis unequivocally demonstrates that an integrated strategy simultaneously implementing all three controls is synergistically superior and provides the most effective pathway to mitigate the HBV burden, significantly outperforming any single or paired intervention.