Mathematical Biosciences and Engineering, 2009, 6(2): 261-282. doi: 10.3934/mbe.2009.6.261.

Primary: 62G05, 93E24, 49Q12, 37N25; Secondary: 62H12, 62N02.

Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

The estimation of the effective reproductive number from disease outbreak data

1. Center for Research in Scientific Computation, Center for Quantitative Sciences in Biomedicine, North Carolina State University, Raleigh, NC 27695
2. Department of Mathematics and Statistics, Arizona State University, P.O. Box 871804, Tempe, AZ 85287-1804
3. Theoretical Division, Mathematical Modeling and Analysis (T-7), Los Alamos National Laboratory, Mail Stop B284, Los Alamos, NM 87545
4. Center for Research in Scientific Computation, Biomathematics Graduate Program, Department of Mathematics, North Carolina State University, Raleigh, NC 27695
5. Center for Research in Scientific Computation, North Carolina State University, Raleigh, NC 27695-8212

We consider a single outbreak susceptible-infected-recovered (SIR) model and corresponding estimation procedures for the effective reproductive number $\mathcal{R}(t)$. We discuss the estimation of the underlying SIR parameters with a generalized least squares (GLS) estimation technique. We do this in the context of appropriate statistical models for the measurement process. We use asymptotic statistical theories to derive the mean and variance of the limiting (Gaussian) sampling distribution and to perform post statistical analysis of the inverse problems. We illustrate the ideas and pitfalls (e.g., large condition numbers on the corresponding Fisher information matrix) with both synthetic and influenza incidence data sets.
  Article Metrics

Keywords basic reproduction ratio; $\mathcal{R}$; reproduction number; parameter estimation; generalized least squares; $\mathcal{R}_0$; residual plots.; $\mathcal{R}(t)$; effective reproductive number

Citation: Ariel Cintrón-Arias, Carlos Castillo-Chávez, Luís M. A. Bettencourt, Alun L. Lloyd, H. T. Banks. The estimation of the effective reproductive number from disease outbreak data. Mathematical Biosciences and Engineering, 2009, 6(2): 261-282. doi: 10.3934/mbe.2009.6.261


This article has been cited by

  • 1. Carlos Castillo-Chávez, Yun Kang, A simple epidemiological model for populations in the wild with Allee effects and disease-modified fitness, Discrete and Continuous Dynamical Systems - Series B, 2013, 19, 1, 89, 10.3934/dcdsb.2014.19.89
  • 2. Anne Cori, Neil M. Ferguson, Christophe Fraser, Simon Cauchemez, A New Framework and Software to Estimate Time-Varying Reproduction Numbers During Epidemics, American Journal of Epidemiology, 2013, 178, 9, 1505, 10.1093/aje/kwt133
  • 3. Alex Capaldi, Samuel Behrend, Benjamin Berman, Jason Smith, Justin Wright, Alun L. Lloyd, Parameter estimation and uncertainty quantification for an epidemic model, Mathematical Biosciences and Engineering, 2012, 9, 3, 553, 10.3934/mbe.2012.9.553
  • 4. Yves Emvudu, Danhrée Bongor, Rodoumta Koïna, Mathematical analysis of HIV/AIDS stochastic dynamic models, Applied Mathematical Modelling, 2016, 40, 21-22, 9131, 10.1016/j.apm.2016.05.007
  • 5. Mudassar Imran, Adnan Khan, Ali R. Ansari, Syed Touqeer Hussain Shah, Modeling transmission dynamics of Ebola virus disease, International Journal of Biomathematics, 2017, 10, 04, 1750057, 10.1142/S1793524517500577
  • 6. I. Takaidza, O. D. Makinde, O. K. Okosun, Computational Modelling and Optimal Control of Ebola Virus Disease with non-Linear Incidence Rate, Journal of Physics: Conference Series, 2017, 818, 012003, 10.1088/1742-6596/818/1/012003
  • 7. Jing Li, Daniel Blakeley, Robert J. Smith, The Failure ofR0, Computational and Mathematical Methods in Medicine, 2011, 2011, 1, 10.1155/2011/527610
  • 8. Naveen K. Vaidya, Libin Rong, Modeling Pharmacodynamics on HIV Latent Infection: Choice of Drugs is Key to Successful Cure via Early Therapy, SIAM Journal on Applied Mathematics, 2017, 77, 5, 1781, 10.1137/16M1092003
  • 9. Rahil Sachak-Patwa, Nabil T. Fadai, Robert A. Van Gorder, Understanding viral video dynamics through an epidemic modelling approach, Physica A: Statistical Mechanics and its Applications, 2018, 502, 416, 10.1016/j.physa.2018.02.083
  • 10. Matthew Biggerstaff, Simon Cauchemez, Carrie Reed, Manoj Gambhir, Lyn Finelli, Estimates of the reproduction number for seasonal, pandemic, and zoonotic influenza: a systematic review of the literature, BMC Infectious Diseases, 2014, 14, 1, 10.1186/1471-2334-14-480
  • 11. Eunha Shim, Gretchen B. Chapman, Jeffrey P. Townsend, Alison P. Galvani, The influence of altruism on influenza vaccination decisions, Journal of The Royal Society Interface, 2012, 9, 74, 2234, 10.1098/rsif.2012.0115
  • 12. Karen M. Campbell, C. D. Lin, Sopon Iamsirithaworn, Thomas W. Scott, The Complex Relationship between Weather and Dengue Virus Transmission in Thailand, The American Journal of Tropical Medicine and Hygiene, 2013, 89, 6, 1066, 10.4269/ajtmh.13-0321
  • 13. Cameron Browne, Hayriye Gulbudak, Glenn Webb, Modeling contact tracing in outbreaks with application to Ebola, Journal of Theoretical Biology, 2015, 384, 33, 10.1016/j.jtbi.2015.08.004
  • 14. , , Veterinary Epidemiology, 2018, 694, 10.1002/9781118280249.refs
  • 15. Jean-Baptiste Burie, Michel Langlais, Agnès Calonnec, Switching from a mechanistic model to a continuous model to study at different scales the effect of vine growth on the dynamic of a powdery mildew epidemic, Annals of Botany, 2011, 107, 5, 885, 10.1093/aob/mcq233
  • 16. Samira Akbari, Naveen K. Vaidya, Lindi M. Wahl, The Time Distribution of Sulfadoxine-Pyrimethamine Protection from Malaria, Bulletin of Mathematical Biology, 2012, 10.1007/s11538-012-9775-4
  • 17. Baojun Song, Zhilan Feng, Gerardo Chowell, From the guest editors, Mathematical Biosciences and Engineering, 2013, 10, 5/6, 10.3934/mbe.2013.10.5i
  • 18. Naveen K. Vaidya, Lindi M. Wahl, Avian Influenza Dynamics Under Periodic Environmental Conditions, SIAM Journal on Applied Mathematics, 2015, 75, 2, 443, 10.1137/140966642
  • 19. José Paulo Carvalho dos Santos, Lislaine Cristina Cardoso, Evandro Monteiro, Nelson H. T. Lemes, A Fractional-Order Epidemic Model for Bovine Babesiosis Disease and Tick Populations, Abstract and Applied Analysis, 2015, 2015, 1, 10.1155/2015/729894
  • 20. A. Cintrón-Arias, H. T. Banks, A. Capaldi, A. L. Lloyd, A sensitivity matrix based methodology for inverse problem formulation, Journal of Inverse and Ill-posed Problems, 2009, 17, 6, 10.1515/JIIP.2009.034
  • 21. Erin N. Bodine, Connor Cook, Mikayla Shorten, The potential impact of a prophylactic vaccine for Ebola in Sierra Leone, Mathematical Biosciences and Engineering, 2017, 15, 2, 337, 10.3934/mbe.2018015
  • 22. J.R. Artalejo, M.J. Lopez-Herrero, The SIS and SIR stochastic epidemic models: A maximum entropy approach, Theoretical Population Biology, 2011, 80, 4, 256, 10.1016/j.tpb.2011.09.005
  • 23. Elissa J. Schwartz, Boseung Choi, Grzegorz A. Rempala, Estimating epidemic parameters: Application to H1N1 pandemic data, Mathematical Biosciences, 2015, 270, 198, 10.1016/j.mbs.2015.03.007
  • 24. Manoj Gambhir, Catherine Bozio, Justin J. O'Hagan, Amra Uzicanin, Lucinda E. Johnson, Matthew Biggerstaff, David L. Swerdlow, Infectious Disease Modeling Methods as Tools for Informing Response to Novel Influenza Viruses of Unknown Pandemic Potential, Clinical Infectious Diseases, 2015, 60, suppl_1, S11, 10.1093/cid/civ083
  • 25. Mudassar Imran, Tufail Malik, Ali R Ansari, Adnan Khan, Mathematical analysis of swine influenza epidemic model with optimal control, Japan Journal of Industrial and Applied Mathematics, 2016, 33, 1, 269, 10.1007/s13160-016-0210-3
  • 26. Zain Ul Abadin Zafar, Kashif Rehan, M Mushtaq, Fractional-order scheme for bovine babesiosis disease and tick populations, Advances in Difference Equations, 2017, 2017, 1, 10.1186/s13662-017-1133-2
  • 27. Saraya Tavornpanich, Hildegunn Viljugrein, Anne Stene, Edgar Brun, Estimation of the reproduction number of salmon pancreas disease virus subtype 3 in homogeneously mixed populations of Norwegian farmed Atlantic salmon, Preventive Veterinary Medicine, 2013, 111, 3-4, 329, 10.1016/j.prevetmed.2013.05.012
  • 28. Junjing Lin, Michael Ludkovski, Sequential Bayesian inference in hidden Markov stochastic kinetic models with application to detection and response to seasonal epidemics, Statistics and Computing, 2014, 24, 6, 1047, 10.1007/s11222-013-9419-z
  • 29. Vanja Dukic, Hedibert F. Lopes, Nicholas G. Polson, Tracking Epidemics With Google Flu Trends Data and a State-Space SEIR Model, Journal of the American Statistical Association, 2012, 107, 500, 1410, 10.1080/01621459.2012.713876
  • 30. Paul J. Hurtado, Spencer R. Hall, Stephen P. Ellner, Infectious disease in consumer populations: dynamic consequences of resource-mediated transmission and infectiousness, Theoretical Ecology, 2014, 7, 2, 163, 10.1007/s12080-013-0208-2
  • 31. R. Luo, E. F. Cardozo, M. J. Piovoso, H. Wu, M. J. Buzon, J. Martinez-Picado, R. Zurakowski, Modelling HIV-1 2-LTR dynamics following raltegravir intensification, Journal of The Royal Society Interface, 2013, 10, 84, 20130186, 10.1098/rsif.2013.0186
  • 32. Erida Gjini, M. Gabriela M. Gomes, Expanding vaccine efficacy estimation with dynamic models fitted to cross-sectional prevalence data post-licensure, Epidemics, 2016, 14, 71, 10.1016/j.epidem.2015.11.001
  • 33. Marco Tulio Angulo, Jorge X. Velasco-Hernandez, Robust qualitative estimation of time-varying contact rates in uncertain epidemics, Epidemics, 2018, 10.1016/j.epidem.2018.03.001
  • 34. K. Glass, G. N. Mercer, H. Nishiura, E. S. McBryde, N. G. Becker, Estimating reproduction numbers for adults and children from case data, Journal of The Royal Society Interface, 2011, 8, 62, 1248, 10.1098/rsif.2010.0679
  • 35. Adnan Khan, Muhammad Waleed, Mudassar Imran, Mathematical analysis of an influenza epidemic model, formulation of different controlling strategies using optimal control and estimation of basic reproduction number, Mathematical and Computer Modelling of Dynamical Systems, 2015, 21, 5, 432, 10.1080/13873954.2015.1016975
  • 36. A. Calonnec, J-B. Burie, M. Langlais, S. Guyader, S. Saint-Jean, I. Sache, B. Tivoli, Impacts of plant growth and architecture on pathogen processes and their consequences for epidemic behaviour, European Journal of Plant Pathology, 2013, 135, 3, 479, 10.1007/s10658-012-0111-5
  • 37. Quan-Hui Liu, Marco Ajelli, Alberto Aleta, Stefano Merler, Yamir Moreno, Alessandro Vespignani, Measurability of the epidemic reproduction number in data-driven contact networks, Proceedings of the National Academy of Sciences, 2018, 201811115, 10.1073/pnas.1811115115
  • 38. Rahil Sachak-Patwa, Nabil T. Fadai, Robert A. Van Gorder, Modeling multi-group dynamics of related viral videos with delay differential equations, Physica A: Statistical Mechanics and its Applications, 2019, 521, 197, 10.1016/j.physa.2019.01.052
  • 39. Carlos Castillo-Chavez, Sunmi Lee, , Encyclopedia of Applied and Computational Mathematics, 2015, Chapter 85, 427, 10.1007/978-3-540-70529-1_85
  • 40. H. T. Banks, Ariel Cintrón-Arias, Franz Kappel, , Mathematical Modeling and Validation in Physiology, 2013, Chapter 3, 43, 10.1007/978-3-642-32882-4_3
  • 41. George Qian, Adam Mahdi, Sensitivity analysis methods in the biomedical sciences, Mathematical Biosciences, 2020, 108306, 10.1016/j.mbs.2020.108306

Reader Comments

your name: *   your email: *  

Copyright Info: 2009, , licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved