Optimal control of COVID-19 transmission model with breakthrough infection and immune failure

Abstract To explore the evolutionary characteristics of COVID-19 disease and prevention and control measures after the relaxation of the epidemic, we establish a COVID-19 transmission model with breakthrough infection and immune failure. Firstly, dynamic analysis of the model is conducted, the basic reproduction number is obtained using the next-generation matrix method, and the global stability of disease-free equilibrium is proven. Secondly, we use actual case data from the fifth wave of COVID-19 in Hong Kong before and after the policy relaxation to estimate parameters. We also conduct a local sensitivity analysis of basic reproduction number and various parameters using elasticity indices. The results suggest that minimizing the infection rate in the early stages after the policy relaxation and promptly isolating individuals upon discovery of symptoms can quickly control the spread of the epidemic. Finally, optimal control research is conducted using Bang-Bang control, and the existence of optimal solution is proven. Through the simulation study of optimal control, different control strategies are compared. The results indicate that to prevent a resurgence of the disease as quickly as possible after the policy is relaxed, it is necessary to rapidly increase vaccination coverage at all doses before the policy is implemented, ensure prompt isolation and treatment of infected individuals upon developing symptoms, and simultaneously reduce the transmission rate by wearing masks and reducing outdoor activities. This will allow the spread of the disease to be quickly controlled and economic life to return to normal.