To predict how large fraction of the population that must be vaccinated and how strong restrictions that must be maintained to cease the pandemic, Eq. 6 was plotted as function of basic reproduction number R0, vaccine efficacy, and restriction efficiency (Fig. 1).
The original Covid-19 virus has R0 around 2-3 [4,5], Delta R0≈5  and Omicron R0≈7.5 [5,6]. 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 efficiency 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 . The efficacy drops to 64.8 % and 36%, 20-24 weeks after dose 2. A third dose increases the efficacy (after 2 weeks) to 92.6 % and 75.5 %, respectively. Similar trends were found for AstraZeneca (ChAdOx1-S with BNT162b2 booster-dose)  and Moderna (mRNA-1273) .
For the original Covid-19 variant, with basic reproduction number R0 around 2-3 and a vaccine efficiency around 80-90 % (2 doses), Fig. 1 indicates that medium-strong restrictions (50-70%) or a vaccination rate around 70 % is 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 e.g., by 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 a 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 rate in Fig. 1 is the combined fraction of vaccinated or recovered persons. Even if the vaccination/restriction strategy fails, the transmission will stop when enough people have 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 individuals. Although the hospitalization probability for Omicron is 40-45% lower than for Delta, it is still high .
To confirm the modelling results, data for Covid-19 incidence and vaccination rate was collected 19-26 November 2021 for countries in Europe and regions in Germany and the Czech Republic. Data was mainly retrieved before the partial lockdown in Germany, which started 24th November. The 7-incidence (per 100 000) clearly decreased with increasing vaccination rate (Fig. 2). A triangular pattern is evident, indicated by two enveloping black dashed lines. The logic behind these lines is that each country, city, or region has a specific reporting/testing procedure, resulting in different degrees of underreporting. Therefore, a huge scatter is evident for low or moderate vaccination rates. When approaching a specific rate of vaccination, the scatter must be smaller and reach (almost) zero. Thus, the two dashed lines must have a common intersection with the x-axis, and this point is the required vaccination rate to halt the pandemic.
The data indicates that a vaccination rate around 92 % (2 doses) would have been sufficient to suppress the pandemic in Europe late November 2021, without serious restrictions. At that time, the Delta (B.1.617.2) variant of concern was dominant. Since the required vaccination rate is affected by season, currently dominant Covid variant, typical vaccine-type , average time since the last dose, maximum number of doses etcetera, the value of 92 % is obviously not generally applicable. In December 2021, during the partial lockdown in Germany, the German incidence decreased, 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. All these results agree with the modelling predictions.