Bang-bang control optimization in infectious disease model with incorporating breakthrough and reinfection

Breakthrough infections and reinfection are key factors leading to recurrent epidemic waves. However, sustained control strategies can lead to unnecessary resource wastage when tacking these issues. There is an urgent need to establish dynamic intervention systems capable of rapid response and efficient resource utilization. To address the question of how breakthrough infections and reinfections affect the dynamics of the pandemic, this study develops an infectious disease model that incorporates both breakthrough infections and reinfections, and while introduces bang-bang optimal control as an efficient public health intervention strategy to provide a new perspective and solutions. In theoretical analysis, we derive basic reproduction number via next-generation matrix method, prove the global stability of the disease-free equilibrium, and establish sufficient conditions for the existence of multiple endemic equilibria and the occurrence of backward bifurcation. Numerical simulations further confirm the critical role of breakthrough infections and reinfection in disease persistence and recurrent outbreaks. In control strategy research, we prove the existence of bang-bang optimal solutions based on optimal control theory and demonstrate their distinct advantages in rapidly outbreaks while minimizing operational costs. Simulation results show that a combined strategy implemented under the bang-bang control-reducing transmission rates, expanding vaccine coverage, and enhancing vaccine protection-most effectively contains disease spread. This results provide both theoretical foundation and practical guidance for developing efficient control strategies against recurrent infectious disease outbreaks.