{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,17]],"date-time":"2026-03-17T19:34:54Z","timestamp":1773776094857,"version":"3.50.1"},"update-to":[{"DOI":"10.1371\/journal.pcbi.1009472","type":"new_version","label":"New version","source":"publisher","updated":{"date-parts":[[2021,11,11]],"date-time":"2021-11-11T00:00:00Z","timestamp":1636588800000}}],"reference-count":50,"publisher":"Public Library of Science (PLoS)","issue":"10","license":[{"start":{"date-parts":[[2021,10,25]],"date-time":"2021-10-25T00:00:00Z","timestamp":1635120000000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001659","name":"deutsche forschungsgemeinschaft","doi-asserted-by":"publisher","award":["GRK 2277"],"award-info":[{"award-number":["GRK 2277"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]},{"name":"google cloud platform","award":["Covid-19 Research Grant"],"award-info":[{"award-number":["Covid-19 Research Grant"]}]},{"DOI":"10.13039\/100000865","name":"Bill and Melinda Gates Foundation","doi-asserted-by":"publisher","award":["Project INV-006261"],"award-info":[{"award-number":["Project INV-006261"]}],"id":[{"id":"10.13039\/100000865","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100005156","name":"Alexander von Humboldt-Stiftung","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100005156","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012284","name":"Chica and Heinz Schaller-Stiftung","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100012284","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Informatics for Life funded by the Klaus Tschira Foundation and by the Cluster of Excellence STRUCTURES funded by DFG"},{"DOI":"10.13039\/100005156","name":"Alexander von Humboldt-Stiftung","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100005156","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["www.ploscompbiol.org"],"crossmark-restriction":false},"short-container-title":["PLoS Comput Biol"],"abstract":"<jats:p>Mathematical models in epidemiology are an indispensable tool to determine the dynamics and important characteristics of infectious diseases. Apart from their scientific merit, these models are often used to inform political decisions and interventional measures during an ongoing outbreak. However, reliably inferring the epidemical dynamics by connecting complex models to real data is still hard and requires either laborious manual parameter fitting or expensive optimization methods which have to be repeated from scratch for every application of a given model. In this work, we address this problem with a novel combination of epidemiological modeling with specialized neural networks. Our approach entails two computational phases: In an initial training phase, a mathematical model describing the epidemic is used as a coach for a neural network, which acquires global knowledge about the full range of possible disease dynamics. In the subsequent inference phase, the trained neural network processes the observed data of an actual outbreak and infers the parameters of the model in order to realistically reproduce the observed dynamics and reliably predict future progression. With its flexible framework, our simulation-based approach is applicable to a variety of epidemiological models. Moreover, since our method is fully Bayesian, it is designed to incorporate all available prior knowledge about plausible parameter values and returns complete joint posterior distributions over these parameters. Application of our method to the early Covid-19 outbreak phase in Germany demonstrates that we are able to obtain reliable probabilistic estimates for important disease characteristics, such as generation time, fraction of undetected infections, likelihood of transmission before symptom onset, and reporting delays using a very moderate amount of real-world observations.<\/jats:p>","DOI":"10.1371\/journal.pcbi.1009472","type":"journal-article","created":{"date-parts":[[2021,10,25]],"date-time":"2021-10-25T17:31:25Z","timestamp":1635183085000},"page":"e1009472","update-policy":"https:\/\/doi.org\/10.1371\/journal.pcbi.corrections_policy","source":"Crossref","is-referenced-by-count":31,"title":["OutbreakFlow: Model-based Bayesian inference of disease outbreak dynamics with invertible neural networks and its application to the COVID-19 pandemics in Germany"],"prefix":"10.1371","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6702-9559","authenticated-orcid":true,"given":"Stefan T.","family":"Radev","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1198-6632","authenticated-orcid":true,"given":"Frederik","family":"Graw","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0555-2157","authenticated-orcid":true,"given":"Simiao","family":"Chen","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0156-9595","authenticated-orcid":true,"given":"Nico T.","family":"Mutters","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3259-2920","authenticated-orcid":true,"given":"Vanessa M.","family":"Eichel","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4182-4212","authenticated-orcid":true,"given":"Till","family":"B\u00e4rnighausen","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6036-1287","authenticated-orcid":true,"given":"Ullrich","family":"K\u00f6the","sequence":"additional","affiliation":[]}],"member":"340","published-online":{"date-parts":[[2021,10,25]]},"reference":[{"key":"pcbi.1009472.ref001","doi-asserted-by":"crossref","DOI":"10.2307\/j.ctvcm4gk0","volume-title":"Modeling infectious diseases in humans and animals","author":"MJ Keeling","year":"2011"},{"issue":"10225","key":"pcbi.1009472.ref002","doi-asserted-by":"crossref","first-page":"689","DOI":"10.1016\/S0140-6736(20)30260-9","article-title":"Nowcasting and forecasting the potential domestic and international spread of the 2019-nCoV outbreak originating in Wuhan, China: a modelling study","volume":"395","author":"JT Wu","year":"2020","journal-title":"The Lancet"},{"issue":"6489","key":"pcbi.1009472.ref003","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1126\/science.aba9757","article-title":"The effect of travel restrictions on the spread of the 2019 novel coronavirus (COVID-19) outbreak","volume":"368","author":"M Chinazzi","year":"2020","journal-title":"Science"},{"issue":"5","key":"pcbi.1009472.ref004","doi-asserted-by":"crossref","first-page":"553","DOI":"10.1016\/S1473-3099(20)30144-4","article-title":"Early dynamics of transmission and control of COVID-19: a mathematical modelling study","volume":"20","author":"AJ Kucharski","year":"2020","journal-title":"The lancet infectious diseases"},{"issue":"19-20","key":"pcbi.1009472.ref005","doi-asserted-by":"crossref","first-page":"w20271","DOI":"10.4414\/smw.2020.20271","article-title":"Reproductive number of the COVID-19 epidemic in Switzerland with a focus on the Cantons of Basel-Stadt and Basel-Landschaft","volume":"150","author":"J Scire","year":"2020","journal-title":"Swiss medical weekly"},{"issue":"7","key":"pcbi.1009472.ref006","doi-asserted-by":"crossref","first-page":"e1003189","DOI":"10.1371\/journal.pmed.1003189","article-title":"Estimation of SARS-CoV-2 mortality during the early stages of an epidemic: A modeling study in Hubei, China, and six regions in Europe","volume":"17","author":"A Hauser","year":"2020","journal-title":"PLoS medicine"},{"issue":"6491","key":"pcbi.1009472.ref007","doi-asserted-by":"crossref","first-page":"638","DOI":"10.1126\/science.abb6105","article-title":"An investigation of transmission control measures during the first 50 days of the COVID-19 epidemic in China","volume":"368","author":"H Tian","year":"2020","journal-title":"Science"},{"issue":"6500","key":"pcbi.1009472.ref008","doi-asserted-by":"crossref","DOI":"10.1126\/science.abb9789","article-title":"Inferring change points in the spread of COVID-19 reveals the effectiveness of interventions","volume":"369","author":"J Dehning","year":"2020","journal-title":"Science"},{"issue":"6490","key":"pcbi.1009472.ref009","doi-asserted-by":"crossref","first-page":"489","DOI":"10.1126\/science.abb3221","article-title":"Substantial undocumented infection facilitates the rapid dissemination of novel coronavirus (SARS-CoV-2)","volume":"368","author":"R Li","year":"2020","journal-title":"Science"},{"issue":"4","key":"pcbi.1009472.ref010","doi-asserted-by":"crossref","first-page":"521","DOI":"10.1111\/j.1524-4733.2008.00484.x","article-title":"Calibration of disease simulation model using an engineering approach","volume":"12","author":"CY Kong","year":"2009","journal-title":"Value in health"},{"key":"pcbi.1009472.ref011","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4899-7612-3","volume-title":"An introduction to mathematical epidemiology","author":"M Martcheva","year":"2015"},{"key":"pcbi.1009472.ref012","doi-asserted-by":"crossref","DOI":"10.1201\/b16018","volume-title":"Bayesian data analysis","author":"A Gelman","year":"2013"},{"issue":"9","key":"pcbi.1009472.ref013","doi-asserted-by":"crossref","first-page":"1025","DOI":"10.1016\/S1473-3099(20)30361-3","article-title":"Individual quarantine versus active monitoring of contacts for the mitigation of COVID-19: a modelling study","volume":"20","author":"CM Peak","year":"2020","journal-title":"The Lancet Infectious Diseases"},{"key":"pcbi.1009472.ref014","doi-asserted-by":"crossref","first-page":"100368","DOI":"10.1016\/j.epidem.2019.100368","article-title":"Approximate Bayesian Computation for infectious disease modelling","volume":"29","author":"A Minter","year":"2019","journal-title":"Epidemics"},{"issue":"6","key":"pcbi.1009472.ref015","doi-asserted-by":"crossref","first-page":"613","DOI":"10.1007\/s40273-017-0494-4","article-title":"Bayesian methods for calibrating health policy models: a tutorial","volume":"35","author":"NA Menzies","year":"2017","journal-title":"Pharmacoeconomics"},{"issue":"48","key":"pcbi.1009472.ref016","doi-asserted-by":"crossref","first-page":"30055","DOI":"10.1073\/pnas.1912789117","article-title":"The frontier of simulation-based inference","volume":"117","author":"K Cranmer","year":"2020","journal-title":"Proceedings of the National Academy of Sciences"},{"key":"pcbi.1009472.ref017","article-title":"BayesFlow: Learning complex stochastic models with invertible neural networks","author":"ST Radev","year":"2020","journal-title":"IEEE Transactions on Neural Networks and Learning Systems"},{"key":"pcbi.1009472.ref018","article-title":"Estimate of the development of the epidemic reproduction number Rt from Coronavirus SARS-CoV-2 case data and implications for political measures based on prognostics","author":"S Khailaie","year":"2020","journal-title":"medRxiv"},{"key":"pcbi.1009472.ref019","doi-asserted-by":"crossref","unstructured":"Szegedy C, Liu W, Jia Y, Sermanet P, Reed S, Anguelov D, et al. Going deeper with convolutions. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition; 2015. p. 1\u20139.","DOI":"10.1109\/CVPR.2015.7298594"},{"issue":"10","key":"pcbi.1009472.ref020","doi-asserted-by":"crossref","first-page":"2451","DOI":"10.1162\/089976600300015015","article-title":"Learning to forget: Continual prediction with LSTM","volume":"12","author":"FA Gers","year":"2000","journal-title":"Neural computation"},{"issue":"7","key":"pcbi.1009472.ref021","doi-asserted-by":"crossref","first-page":"3797","DOI":"10.1109\/TIT.2014.2320500","article-title":"R\u00e9nyi divergence and Kullback-Leibler divergence","volume":"60","author":"T Van Erven","year":"2014","journal-title":"IEEE Transactions on Information Theory"},{"issue":"2","key":"pcbi.1009472.ref022","doi-asserted-by":"crossref","first-page":"462","DOI":"10.3390\/jcm9020462","article-title":"Estimation of the transmission risk of the 2019-nCoV and its implication for public health interventions","volume":"9","author":"B Tang","year":"2020","journal-title":"Journal of Clinical Medicine"},{"key":"pcbi.1009472.ref023","unstructured":"Papamakarios G, Sterratt D, Murray I. Sequential neural likelihood: Fast likelihood-free inference with autoregressive flows. In: The 22nd International Conference on Artificial Intelligence and Statistics. PMLR; 2019. p. 837\u2013848."},{"key":"pcbi.1009472.ref024","unstructured":"Greenberg D, Nonnenmacher M, Macke J. Automatic posterior transformation for likelihood-free inference. In: International Conference on Machine Learning. PMLR; 2019. p. 2404\u20132414."},{"issue":"7","key":"pcbi.1009472.ref025","doi-asserted-by":"crossref","first-page":"410","DOI":"10.1016\/j.tree.2010.04.001","article-title":"Approximate Bayesian computation (ABC) in practice","volume":"25","author":"K Csill\u00e9ry","year":"2010","journal-title":"Trends in Ecology & Evolution"},{"issue":"1","key":"pcbi.1009472.ref026","doi-asserted-by":"crossref","first-page":"e1002803","DOI":"10.1371\/journal.pcbi.1002803","article-title":"Approximate bayesian computation","volume":"9","author":"M Sunn\u00e5ker","year":"2013","journal-title":"PLoS computational biology"},{"issue":"10","key":"pcbi.1009472.ref027","doi-asserted-by":"crossref","first-page":"1720","DOI":"10.1093\/bioinformatics\/bty867","article-title":"ABC random forests for Bayesian parameter inference","volume":"35","author":"L Raynal","year":"2018","journal-title":"Bioinformatics"},{"issue":"1","key":"pcbi.1009472.ref028","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1111\/bmsp.12159","article-title":"Towards end-to-end likelihood-free inference with convolutional neural networks","volume":"73","author":"ST Radev","year":"2020","journal-title":"British Journal of Mathematical and Statistical Psychology"},{"key":"pcbi.1009472.ref029","unstructured":"Talts S, Betancourt M, Simpson D, Vehtari A, Gelman A. Validating Bayesian inference algorithms with simulation-based calibration. arXiv:180406788. 2018;."},{"issue":"10","key":"pcbi.1009472.ref030","doi-asserted-by":"crossref","first-page":"e1004452","DOI":"10.1371\/journal.ppat.1004452","article-title":"Detecting differential transmissibilities that affect the size of self-limited outbreaks","volume":"10","author":"S Blumberg","year":"2014","journal-title":"PLoS Pathog"},{"issue":"20","key":"pcbi.1009472.ref031","doi-asserted-by":"crossref","first-page":"3591","DOI":"10.1093\/bioinformatics\/bty361","article-title":"pyABC: distributed, likelihood-free inference","volume":"34","author":"E Klinger","year":"2018","journal-title":"Bioinformatics"},{"key":"pcbi.1009472.ref032","article-title":"SARS-CoV-2 seroprevalence in Germany-a population based sequential study in five regions","author":"M Harries","year":"2021","journal-title":"medRxiv"},{"issue":"7","key":"pcbi.1009472.ref033","doi-asserted-by":"crossref","first-page":"3572","DOI":"10.3390\/ijerph18073572","article-title":"Prevalence and risk factors of infection in the representative COVID-19 cohort Munich","volume":"18","author":"M Pritsch","year":"2021","journal-title":"International Journal of Environmental Research and Public Health"},{"issue":"1","key":"pcbi.1009472.ref034","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1038\/s41467-020-19509-y","article-title":"Infection fatality rate of SARS-CoV2 in a super-spreading event in Germany","volume":"11","author":"H Streeck","year":"2020","journal-title":"Nature communications"},{"issue":"47","key":"pcbi.1009472.ref035","doi-asserted-by":"crossref","first-page":"2001752","DOI":"10.2807\/1560-7917.ES.2020.25.47.2001752","article-title":"Serology-and PCR-based cumulative incidence of SARS-CoV-2 infection in adults in a successfully contained early hotspot (CoMoLo study), Germany, May to June 2020","volume":"25","author":"C Santos-H\u00f6vener","year":"2020","journal-title":"Eurosurveillance"},{"issue":"6","key":"pcbi.1009472.ref036","doi-asserted-by":"crossref","first-page":"1118","DOI":"10.3390\/v13061118","article-title":"Estimates and determinants of SARS-CoV-2 seroprevalence and infection fatality ratio using latent class analysis: the population-based Tirschenreuth study in the hardest-hit German county in spring 2020","volume":"13","author":"R Wagner","year":"2021","journal-title":"Viruses"},{"issue":"18","key":"pcbi.1009472.ref037","doi-asserted-by":"crossref","first-page":"1708","DOI":"10.1056\/NEJMoa2002032","article-title":"Clinical characteristics of coronavirus disease 2019 in China","volume":"382","author":"Wj Guan","year":"2020","journal-title":"New England journal of medicine"},{"key":"pcbi.1009472.ref038","unstructured":"World Health Organization. Coronavirus disease 2019 (COVID-19): situation report, 73. 2020;."},{"issue":"1","key":"pcbi.1009472.ref039","doi-asserted-by":"crossref","first-page":"12","DOI":"10.1016\/j.jmii.2020.05.001","article-title":"A systematic review of asymptomatic infections with COVID-19","volume":"54","author":"Z Gao","year":"2021","journal-title":"Journal of Microbiology, Immunology and Infection"},{"issue":"8","key":"pcbi.1009472.ref040","doi-asserted-by":"crossref","first-page":"1193","DOI":"10.1038\/s41591-020-0949-6","article-title":"Seroprevalence of immunoglobulin M and G antibodies against SARS-CoV-2 in China","volume":"26","author":"X Xu","year":"2020","journal-title":"Nature Medicine"},{"issue":"10250","key":"pcbi.1009472.ref041","doi-asserted-by":"crossref","first-page":"535","DOI":"10.1016\/S0140-6736(20)31483-5","article-title":"Prevalence of SARS-CoV-2 in Spain (ENE-COVID): a nationwide, population-based seroepidemiological study","volume":"396","author":"M Poll\u00e1n","year":"2020","journal-title":"The Lancet"},{"issue":"10247","key":"pcbi.1009472.ref042","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1016\/S0140-6736(20)31304-0","article-title":"Seroprevalence of anti-SARS-CoV-2 IgG antibodies in Geneva, Switzerland (SEROCoV-POP): a population-based study","volume":"396","author":"S Stringhini","year":"2020","journal-title":"The Lancet"},{"issue":"1","key":"pcbi.1009472.ref043","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1038\/s43856-021-00007-1","article-title":"High SARS-CoV-2 seroprevalence in children and adults in the Austrian ski resort of Ischgl","volume":"1","author":"L Knabl","year":"2021","journal-title":"Communications Medicine"},{"issue":"7844","key":"pcbi.1009472.ref044","doi-asserted-by":"crossref","first-page":"134","DOI":"10.1038\/s41586-020-03095-6","article-title":"Underdetection of cases of COVID-19 in France threatens epidemic control","volume":"590","author":"G Pullano","year":"2021","journal-title":"Nature"},{"issue":"1112","key":"pcbi.1009472.ref045","article-title":"Potential impact of seasonal forcing on a SARS-CoV-2 pandemic","volume":"150","author":"RA Neher","year":"2020","journal-title":"Swiss medical weekly"},{"issue":"6501","key":"pcbi.1009472.ref046","doi-asserted-by":"crossref","first-page":"315","DOI":"10.1126\/science.abc2535","article-title":"Susceptible supply limits the role of climate in the early SARS-CoV-2 pandemic","volume":"369","author":"RE Baker","year":"2020","journal-title":"Science"},{"issue":"7101","key":"pcbi.1009472.ref047","doi-asserted-by":"crossref","first-page":"448","DOI":"10.1038\/nature04795","article-title":"Strategies for mitigating an influenza pandemic","volume":"442","author":"NM Ferguson","year":"2006","journal-title":"Nature"},{"issue":"7","key":"pcbi.1009472.ref048","doi-asserted-by":"crossref","first-page":"e375","DOI":"10.1016\/S2468-2667(20)30133-X","article-title":"Effects of non-pharmaceutical interventions on COVID-19 cases, deaths, and demand for hospital services in the UK: a modelling study","volume":"5","author":"NG Davies","year":"2020","journal-title":"The Lancet Public Health"},{"key":"pcbi.1009472.ref049","first-page":"1","article-title":"Estimating the effects of non-pharmaceutical interventions on COVID-19 in Europe","author":"S Flaxman","year":"2020","journal-title":"Nature"},{"key":"pcbi.1009472.ref050","doi-asserted-by":"crossref","unstructured":"Radev ST, D\u2019Alessandro M, B\u00fcrkner PC, Mertens UK, Voss A, K\u00f6the U. Amortized Bayesian model comparison with evidential deep learning. arXiv:200410629. 2020;.","DOI":"10.1109\/TNNLS.2021.3124052"}],"updated-by":[{"DOI":"10.1371\/journal.pcbi.1009472","type":"new_version","label":"New version","source":"publisher","updated":{"date-parts":[[2021,11,11]],"date-time":"2021-11-11T00:00:00Z","timestamp":1636588800000}}],"container-title":["PLOS Computational Biology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dx.plos.org\/10.1371\/journal.pcbi.1009472","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,1,13]],"date-time":"2023-01-13T19:33:23Z","timestamp":1673638403000},"score":1,"resource":{"primary":{"URL":"https:\/\/dx.plos.org\/10.1371\/journal.pcbi.1009472"}},"subtitle":[],"editor":[{"given":"Mark M.","family":"Tanaka","sequence":"first","affiliation":[]}],"short-title":[],"issued":{"date-parts":[[2021,10,25]]},"references-count":50,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2021,10,25]]}},"URL":"https:\/\/doi.org\/10.1371\/journal.pcbi.1009472","relation":{"new_version":[{"id-type":"doi","id":"10.1371\/journal.pcbi.1009472","asserted-by":"object"}]},"ISSN":["1553-7358"],"issn-type":[{"value":"1553-7358","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,10,25]]}}}