Optimal Control Model for The Spread of Respiratory Infectious Diseases Through Preventive and Treatment Measures

Abstract

Respiratory infectious diseases, including acute respiratory infections, pneumonia, MERS-CoV, and COVID-19, remain major global health concerns due to their high transmission rates and significant health and socioeconomic impacts. However, most existing models do not explicitly distinguish population groups based on behavioral factors such as mask usage nor integrate multiple interventions within a unified optimal control and cost-effectiveness framework. This study develops a modified seven-compartment epidemiological model that differentiates susceptible and infected individuals based on mask usage and incorporates four control variables: mask usage, vaccination, quarantine, and treatment. The optimal control problem is formulated using the Pontryagin Maximum Principle and solved numerically using the fourth-order Runge-Kutta method combined with the Forward-Backward Sweep Method. The results indicate that treatment is the most effective single intervention in rapidly suppressing infection, while the combination of all control measures provides the fastest overall reduction in transmission. In terms of cost-effectiveness, the four-control combination yields the lowest Average Cost-Effectiveness Ratio (ACER), indicating the highest efficiency, whereas the combination of mask usage, quarantine, and treatment produces a negative Incremental Cost-Effectiveness Ratio (ICER), implying greater health benefits at a lower additional cost compared to alternative strategies. These findings emphasize the importance of distinguishing between effectiveness and cost-efficiency in determining optimal intervention strategies.