Numerical approximation of fractional order HIV/AIDS model with treatment using Levenberg-Marquardt backpropagation approach.

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HIV specifically targets the immune system's T cells, mainly the CD4+ T cells, leading to a chronic and severe illness characterized by a long incubation period. In this study, we propose a novel nonlinear fractional-order model for analyzing and controlling HIV proliferation dynamics. The model categorizes the HIV-infected population into four distinct compartments: susceptible individuals (S), acutely infected (I 1 ), asymptomatically infected (I 2 ), and individuals with AIDS (A). Mathematical properties like positivity and boundedness provide for model validation. The analytical strategy blends practical wisdom with accurate mathematical analysis. We calculated two equilibrium states - disease-free equilibrium (DFE) and endemic equilibrium (EE)- and established their local stability regions. The analysis demonstrates that DFE is stable when the basic reproduction number R 0 <1, whereas EE becomes stable when R 0 >1. The Normalized Sensitivity Index (NSI) technique evaluates the model's sensitivity to key biological parameters. The fractional derivative is defined in the Caputo sense, and numerical solutions are obtained via the Adams-Bashforth-Moulton (ABM) method. To enhance computational efficiency and accuracy, an artificial neural network (ANN) with 10 hidden layers is trained using Levenberg-Marquardt backpropagation (LMBNN). Three distinct cases of the fractional derivative order (θ=0.55,0.7,0.9) are investigated. The LMBNN framework minimizes the Mean Squared Error (MSE) across training (80%), validation (10%), and testing (10%) phases using the ABM-generated solutions as the reference dataset. The numerical results demonstrate that the LMBNN approach achieves high precision, with MSE values as low as 10 -3, supported by regression analysis (R 2 ≈1). Simulation results suggest that the proposed model, coupled with the LMBNN technique, effectively predicts the control of HIV progression to AIDS, thereby offering significant potential for optimizing treatment strategies. Reducing contact with infected individuals and accelerating disease management interventions can significantly lower the burden of infection. Ultimately, our study offers a bridge to improve the forecasting and controlling of HIV/AIDS disease challenges, effectively connecting surrogate mathematics formulation with actual disease treatment measures.