Mathematical modeling of HIV/AIDS dynamics with optimal control analysis

HIV/AIDS continues to pose a serious global health threat, with working-class populations in low-resource settings particularly vulnerable due to unequal access to healthcare and greater exposure to risk factors. This vulnerability, intensified by socio-economic inequalities and recent disruptions such as the COVID-19 pandemic, threatens both public health and economic productivity. To address this challenge, we developed a novel nonlinear integral-order differential equation model that explicitly incorporates working-class productivity dynamics alongside HIV/AIDS transmission, an approach not commonly addressed in earlier studies. Model parameters were estimated using twenty-three years of epidemiological data from Ethiopia through least-squares fitting, ensuring strong policy relevance. The disease-free equilibrium was analyzed for local stability via the basic reproduction number ($R_0$), while forward and backward bifurcation analyses were conducted to reveal the possibility of multiple endemic equilibria. Numerical solutions were obtained using MATLAB, and the optimal control problem was solved using the forward–backward sweep method with a fourth-order Runge–Kutta algorithm. Unlike many previous models focusing on single interventions, our study evaluated simultaneous optimal control strategies, including education, testing, treatment, and behavioral interventions. Results showed that combined interventions substantially reduced infection levels and enhanced productivity, while cost-effectiveness analysis demonstrated that the integrated approach yielded the highest benefit to cost ratio. These findings emphasize the novelty and importance of linking socio-economic productivity with epidemiological modeling, providing new insights for policymakers seeking efficient and targeted HIV/AIDS intervention programs. By bridging health dynamics and workforce outcomes, this study advances current modeling approaches and highlights integrated strategies as the most effective means of reducing HIV/AIDS burden in vulnerable populations.