Memodelkan terapi ubat kelas NRTI dan PI pada dinamik jangkitan HIV dengan analisis data pesakit sebenar dan strategi kawalan yang dioptimumkan – Alam Semula Jadi

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HIV/AIDS remains a major global health concern, necessitating advanced therapeutic strategies to enhance immune response and suppress viral replication. This research formulates and examines a novel fractional-order mathematical model that integrates Nucleoside Reverse Transcriptase Inhibitors (NRTIs) and Protease Inhibitors (PIs) to evaluate their collective effectiveness in managing HIV infection. Unlike existing models, our framework incorporates memory effects via fractional calculus, explicitly accounts for macrophage-derived viral load as an external reservoir, and employs optimal control theory together with bifurcation analysis to investigate long-term treatment outcomes. In addition, the model is validated using real patient data, ensuring both biological relevance and practical applicability. We determine the biologically feasible steady states of the system and compute the basic reproductive ratio ( $${mathcal {R}}_0$$ ), which serves as a threshold parameter for infection persistence. Stability analysis is performed for each equilibrium point to derive conditions for disease eradication or persistence. Sensitivity analysis identifies key parameters influencing disease progression, and an optimal control strategy is derived using the Backward-Forward Runge-Kutta method to enhance CD $$4^+$$ T-cell counts while reducing infected CD $$4^+$$ cells and HIV viral load. Furthermore, the impact of drug efficacy parameters ( $$Pi _R$$ ) and ( $$Pi _P$$ ) is explored through extensive numerical simulations, demonstrating their role in mitigating viral replication and stabilizing the system. We also conduct a bifurcation analysis, including Hopf and transcritical bifurcations, to examine qualitative system dynamics. A novel Hopf bifurcation diagram illustrates the influence of fractional-order derivatives on oscillatory behavior, and a two-parameter bifurcation diagram reveals the combined effect of critical parameters on system stability. Our findings demonstrate that NRTIs and PIs effectively suppress oscillations and high infection rates, with combination therapy significantly reducing viral load and enhancing system stability. The fractional-order parameter ( $$alpha$$ ) plays a crucial role in suppressing oscillations and ensuring long-term stability. Finally, a comparative study with existing literature highlights the originality and significance of our results, reinforcing the applicability of fractional-order modeling with real data in HIV/AIDS therapy research.