Optimal Strategies for a Controlled SIR Model with Dynamic Reproduction Number and Economic Feedback

Résumé

Mathematical models play a critical role in analyzing infectious disease dynamics, yet existing frameworks often overlook the interplay between epidemiological and socio-economic factors. This study develops the SIRK$\rho$ model, a novel mathematical framework that integrates time-varying transmission dynamics with economic feedback mechanisms. The model incorporates optimal control theory to determine vaccination v and public health intervention c strategies that simultaneously minimize disease prevalence and economic losses while maintaining the effective reproduction number below unity. Through analytical derivation using Pontryagin's Maximum Principle and numerical validation with Hepatitis B (HBV) parameters, we demonstrate the model's effectiveness in outbreak control. Simulation results show that optimized intervention strategies can reduce HBV infections while supporting economic recovery. The SIRK$\rho$ framework provides a comprehensive approach for public health decision-making that balances epidemiological control with economic considerations.