From the study published by the CDC yesterday
The understanding and epidemiology of COVID-19 has shifted substantially over time with the emergence and circulation of new SARS-CoV-2 variants, introduction of vaccines, and changing immunity as a result. Similar to the early period of this study, two previous U.S. studies found more protection from vaccination than from previous infection during periods before Delta predominance (3,7). As was observed in the present study after July, recent international studies have also demonstrated increased protection in persons with previous infection, with or without vaccination, relative to vaccination alone†††, §§§ (4). This might be due to differential stimulation of the immune response by either exposure type.¶¶¶ Whereas French and Israeli population-based studies noted waning protection from previous infection, this was not apparent in the results from this or other large U.K. and U.S. studies**** (4,8). Further studies are needed to establish duration of protection from previous infection by variant type, severity, and symptomatology, including for the Omicron variant.