Threshold Dynamics, Endemic Stability, and Degree-Targeted Optimal Control of HIV/AIDS on Heterogeneous Contact Networks

Contact heterogeneity can substantially alter HIV/AIDS transmission thresholds and intervention efficiency, yet many control models retain homogeneous mixing. We develop a conservative degree-structured susceptible–infectious–removed model on correlated contact networks with demographic replacement. Positivity, forward invariance, and global well-posedness are established, and the basic reproduction number is expressed through the spectral radius of the degree-mixing matrix. The disease-free equilibrium is globally asymptotically stable for R0≤1; above threshold, interior trajectories cannot converge to the disease-free state. For uncorrelated networks, a unique locally stable endemic equilibrium emerges through a forward transcritical bifurcation. Preventive and treatment controls are formulated within Pontryagin’s maximum principle, yielding explicit uniform and degree-targeted necessary conditions and a short-horizon uniqueness result for the coupled optimality system. Simulations on a truncated scale-free network illustrate the analytical threshold, quantify the effects of assortative mixing and control-cost weights, and show that degree-targeted control substantially reduces cumulative infection burden under the baseline weights. Calibration to 192 monthly HIV/AIDS reports from Beijing during 2005–2020 favors a trend-plus-seasonality transmission model, with monthly and annual coefficients of determination of approximately 0.835 and 0.970. The framework links topology, persistence, cost-weighted intervention, numerical robustness, and surveillance-scale validation.