Stages of COVID-19 pandemic and paths to herd immunity by vaccination: dynamical model comparing Austria, Luxembourg and Sweden

Background Worldwide more than 72 million people have been infected and 1.6 million died with SARS-CoV-2 by 15th December 2020. Non-pharmaceutical interventions which decrease social interaction have been implemented to reduce the spread of SARS-CoV-2 and to mitigate stress on healthcare systems and prevent deaths. The pandemic has been tackled with disparate strategies by distinct countries resulting in different epidemic dynamics. However, with vaccines now becoming available, the current urgent open question is how the interplay between vaccination strategies and social interaction will shape the pandemic in the next months. Methods To address this question, we developed an extended Susceptible-Exposed-Infectious-Removed (SEIR) model including social interaction, undetected cases and the progression of patients trough hospitals, intensive care units (ICUs) and death. We calibrated our model to data of Luxem-bourg, Austria and Sweden, until 15th December 2020. We incorporated the effect of vaccination to investigate under which conditions herd immunity would be achievable in 2021. Results The model reveals that Sweden has the highest fraction of undetected cases, Luxembourg displays the highest fraction of infected population, and all three countries are far from herd immunity as of December 2020. The model quantifies the level of social interactions, and allows to assess the level which would keep R eff ( t ) below 1. In December 2020, this level is around 1/3 of what it was before the pandemic for all the three countries. The model allows to estimate the vaccination rate needed for herd immunity and shows that 2700 vaccinations/day are needed in Luxembourg to reach it by mid of April and 45,000 for Austria and Sweden. The model estimates that vaccinating the whole country’s population within 1 year could lead to herd immunity by July in Luxembourg and by August in Austria and Sweden. Conclusion The model allows to shed light on the dynamics of the epidemics in different waves and countries. Our results emphasize that vaccination will help considerably but not immediately and therefore social measures will remain important for several months before they can be fully alleviated. ### Competing Interest Statement The authors have declared no competing interest. ### Funding Statement FK is supported by the Luxembourg National Research Fund PRIDE17/12244779/PARK-QC.DP and SM is supported by the FNR PRIDE DTU CriTiCS, ref 10907093. A.H. work was partially supported by the Fondation Cancer Luxembourg. JG is partly supported by the 111 Project on Computational Intelligence and Intelligent Control, ref B18024. AA is supported by the Luxembourg National Research Fund (FNR) (Project code: 13684479). AS were supported by the Luxembourg National Research Fund (FNR) through the C14/BM/7975668/ CaSCAD project and by the National Biomedical Computation Resource (NBCR) through the NIH P41 GM103426 grant from the National Institutes of Health. ### Author Declarations I confirm all relevant ethical guidelines have been followed, and any necessary IRB and/or ethics committee approvals have been obtained. Yes The details of the IRB/oversight body that provided approval or exemption for the research described are given below: None as not applicable All necessary patient/participant consent has been obtained and the appropriate institutional forms have been archived. Yes I understand that all clinical trials and any other prospective interventional studies must be registered with an ICMJE-approved registry, such as ClinicalTrials.gov. I confirm that any such study reported in the manuscript has been registered and the trial registration ID is provided (note: if posting a prospective study registered retrospectively, please provide a statement in the trial ID field explaining why the study was not registered in advance). Yes I have followed all appropriate research reporting guidelines and uploaded the relevant EQUATOR Network research reporting checklist(s) and other pertinent material as supplementary files, if applicable. Yes All data employed are publicly available at the links indicated in the manuscript.