The COVID-19 pandemic posed a global challenge, prompting different countries to adopt distinct control strategies. This study focuses on Montreal, Quebec, and Hong Kong, China, both with similar topologies and populations but divergent approaches to tackling the pandemic. Montreal enforced strict internal control measures with lenient border policies, while Hong Kong opted for rigorous external control and relaxed internal measures. This study builds two comprehensive mobility-based epidemic simulation models for both cities. These models are then utilized to test three aspects concerning the external and internal control policies used to mitigate the outbreak of the Omicron variant: sensitivity of internal and external control measures and lag time in implementing on-demand internal control measures. Numerical experiments on the Montreal model suggest the existence of a critical value for internal restrictions and implementation lag, leading to sudden geometric increases in infections. Regardless of internal strictness, some form of external control was found necessary. Further experiments can shed light on the complex interaction between control measures in restricting the Omicron outbreak.