Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) is currently causing an unprecedented pandemic. Although vaccines and antivirals are limiting the spread, SARS-CoV-2 is still under selective pressure in human and animal populations, as demonstrated by the emergence of variants of concern. To better understand the driving forces leading to new subtypes of SARS-CoV-2, we infected an ex vivo cell model of the human upper respiratory tract with Alpha and Omicron BA.1 variants for one month. Although viral RNA was detected during the entire course of the infection, infectious virus production decreased over time. Sequencing analysis did not show any adaptation in the spike protein, suggesting a key role for the adaptive immune response or adaptation to other anatomical sites for the evolution of SARS-CoV-2.