Data for Covid-19 incidence and vaccination rate was collected during the time-period 19/11 to 26/11 (2021) for countries in Europe, regions in Germany and Check Republic and subregions in Sachsen, Sachsen-Anhalt, and Thüringen. The data was mainly retrieved before the partial lockdown in Germany, which started 24/11. As observed in Fig. 1, the incidence decreased with increasing vaccination rate. The data indicates that a vaccination rate around 92% (2 doses) is required to halt the Delta (B.1.617.2) variant of concern, which was dominant in Europe in November 2021. Several factors, including the choice of vaccine , the average time since the last vaccine dose and the average number of doses, strongly affect the required vaccination rate.
In December 2021, during the partial lockdown in Germany, the German incidence decreased rapidly, indicating that those restrictions were sufficient to halt the Delta-virus at a vaccination level around 70% (2 doses). The currently dominating Omicron (B.1.1.529) variant of concern is however more infectious and requires stronger measures to quench.
The original Covid-19 virus has R0 around 2-3 [5,6], Delta R0≈5  and Omicron R0≈7.5 [6,7]. The basic reproduction number increases with increasing population density and it becomes higher during the winter (on the northern hemisphere), probably because of lower humidity, less sunshine and more indoor-activities.
The efficacy of today’s vaccines is lower against Omicron than against Delta and the ancestral Covid-19 virus. For Pfizer (BNT162b2), the efficacy is 88.2% and 88.0% against Delta and Omicron, respectively, 2-9 weeks after dose 2 . 20-24 weeks after dose 2, the efficacy drop to 64.8% and 36. %. 2 weeks after dose 3 the efficacy increases to 92.6% and 75.5%, respectively. Similar trends were found for Moderna (mRNA-1273) .
In order to predict how large fraction of the population that must be vaccinated and how strong restriction that must be maintained to halt the current pandemic wave, equation 6 was plotted as heat-maps as function of basic reproduction number R0, vaccine efficacy, and restriction efficiency (Fig. 2).
For the initial Covid-19 version, with basic reproduction number R0 around 2-3 and vaccine efficiency around 80-90% (2 doses), the heat-maps indicate that medium-strong restrictions (50-70%) or a vaccination rate around 70% are sufficient to halt the pandemic. This has been confirmed empirically in many countries, both for the original Covid-19 virus and e.g. for the Alpha (B.1.1.7) and Beta (B.1.351) variants of concern.
For the Delta version, with R0≈5 and a vaccine efficiency around 65-90% (2 doses), very high vaccination rates (~90%), very strong restrictions (80%), or a combination of medium high restrictions (50-60%) and vaccination rates (70%) are required. This prediction is confirmed by the data in Fig. 1 and by e.g. the observation that the partial lockdown in Germany December 2021 succeeded to stop the Delta wave under these conditions.
For the Omicron version, with R0≈7.5 and vaccine efficiency around 0-88% (2 doses) or 54.6-75.5% (3 doses), extremely strong restrictions (90%) or a combination of strong restrictions (70-80%) and a high vaccination rate with three doses (70-85%) is required.
Note that the required vaccination rates in these graphs are the combined fraction of vaccinated or recovered persons. Even if the vaccination/restriction strategy fails, the virus wave will ultimately stop when a sufficient number of people has been infected and recovered, but to a larger cost in human lives and suffering. A high vaccination rate is strongly desired because it reduces both the virus transmission and the risk for serious health effects for the infected individual. Although the hospitalization probability for Omicron is 40-45% lower than for Delta, it is still high .