Model parameters and their values in system (4).

Non-pharmaceutical interventions (NPIs) are effective in mitigating infections during the early stages of an infectious disease outbreak. However, these measures incur significant economic and livelihood costs. To address this, we developed an optimal control framework aimed at identifying strategies that minimize such costs while ensuring full control of a cross-regional outbreak of emerging infectious diseases. Our approach uses a spatial SEIR model with interventions for the epidemic process, and incorporates population flow in a gravity model dependent on gross domestic product (GDP) and geographical distance. We applied this framework to identify an optimal control strategy for the COVID-19 outbreak caused by the Delta variant in Xi’an City, Shaanxi, China, between December 2021 and January 2022. The model was parameterized by fitting it to daily case data from each district of Xi’an City. Our findings indicate that an increase in the basic reproduction number, the latent period or the infectious period leads to a prolonged outbreak and a larger final size. This indicates that diseases with greater transmissibility are more challenging and costly to control, and so it is important for governments to quickly identify cases and implement control strategies. Indeed, the optimal control strategy we identified suggests that more costly control measures should be implemented as soon as they are deemed necessary. Our results demonstrate that optimal control regimes exhibit spatial, economic, and population heterogeneity. More populated and economically developed regions require a robust regular surveillance mechanism to ensure timely detection and control of imported infections. Regions with higher GDP tend to experience larger-scale epidemics and, consequently, require higher control costs. Notably, our proposed optimal strategy significantly reduced costs compared to the actual expenditures for the Xi’an outbreak.