Piecewise combination of fractional and stochastic mathematical modeling to analyses the effect of HIV/AIDS on working-class population

Abstract

This study presents a piecewise mathematical model for HIV/AIDS transmission that integrates deterministic, fractional-order, and stochastic dynamics. Memory effects are modeled using the Atangana-Baleanu-Caputo (ABC) fractional operator, while stochastic differential equations capture inherent randomness, offering a realistic representation of HIV/AIDS spread within working-class populations. Numerical simulations employed the Runge–Kutta method (deterministic phase), the Toufik-Atangana scheme (fractional phase), and the Euler-Maruyama method (stochastic phase). The model was fitted using real-world HIV/AIDS data (2001–2023) by artificial neural network methods, yielding a strong fit (RMSE = 0.000160; MAE = 0.000130). A neural network was applied to forecast trends from 2024 to 2050. Results highlight the transmission rate ( $$\beta$$ ) as a key driver of infection dynamics and show that higher productivity rates significantly reduce disease burden. Projections indicate that all infected compartments may decline to zero before 2050. The model provides actionable insights for public health policy, particularly in reducing contact rates and addressing socioeconomic disparities, and offers a unified, flexible framework for long-term HIV/AIDS analysis.