Search Advanced

Mathematical Biosciences and Engineering

2012, Volume 9, Issue 3: 553-576. doi: 10.3934/mbe.2012.9.553
Previous Article Next Article

Parameter estimation and uncertainty quantification for an epidemic model

  • 1.
    Center for Quantitative Sciences in Biomedicine and Department of Mathematics, North Carolina State University, Raleigh, NC 27695, and Department of Mathematics & Computer Science, Valparaiso University, 1900 Chapel Drive, Valparaiso, IN 46383
  • 2.
    Department of Mathematics, University of North Carolina, Chapel Hill, CB #3250, Chapel Hill, NC 27599
  • 3.
    Program in Applied Mathematics, University of Arizona, 617 N. Santa Rita Ave., PO Box 210089, Tucson, AZ 85721-0089
  • 4.
    Department of Mathematics, Morehouse College, 830 Westview Drive SW Unit 142133, Atlanta, GA 30314
  • 5.
    Department of Mathematics, North Carolina State University, Raleigh, NC 27695
  • 6.
    Biomathematics Graduate Program and Department of Mathematics, North Carolina State University, Raleigh NC, 27695, USA and Fogarty International Center, National Institutes of Health, Bethesda, MD 20892
  • Received: 01 December 2009 Accepted: 29 June 2018 Published: 01 July 2012

  • MSC : Primary: 92D30; Secondary: 62F99, 62P10, 65L09.

  • We examine estimation of the parameters of Susceptible-Infective-Recovered (SIR) models in the context of least squares. We review the use of asymptotic statistical theory and sensitivity analysis to obtain measures of uncertainty for estimates of the model parameters and the basic reproductive number ($R_0$)---an epidemiologically significant parameter grouping. We find that estimates of different parameters, such as the transmission parameter and recovery rate, are correlated, with the magnitude and sign of this correlation depending on the value of $R_0$. Situations are highlighted in which this correlation allows $R_0$ to be estimated with greater ease than its constituent parameters. Implications of correlation for parameter identifiability are discussed. Uncertainty estimates and sensitivity analysis are used to investigate how the frequency at which data is sampled affects the estimation process and how the accuracy and uncertainty of estimates improves as data is collected over the course of an outbreak. We assess the informativeness of individual data points in a given time series to determine when more frequent sampling (if possible) would prove to be most beneficial to the estimation process. This technique can be used to design data sampling schemes in more general contexts.

    Citation: Alex Capaldi, Samuel Behrend, Benjamin Berman, Jason Smith, Justin Wright, Alun L. Lloyd. Parameter estimation and uncertainty quantification for an epidemic model[J]. Mathematical Biosciences and Engineering, 2012, 9(3): 553-576. doi: 10.3934/mbe.2012.9.553

    Related Papers:

    [1] 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
    [2] H. T. Banks, Robert Baraldi, Karissa Cross, Kevin Flores, Christina McChesney, Laura Poag, Emma Thorpe . Uncertainty quantification in modeling HIV viral mechanics. Mathematical Biosciences and Engineering, 2015, 12(5): 937-964. doi: 10.3934/mbe.2015.12.937
    [3] H.T. Banks, Jimena L. Davis . Quantifying uncertainty in the estimation of probability distributions. Mathematical Biosciences and Engineering, 2008, 5(4): 647-667. doi: 10.3934/mbe.2008.5.647
    [4] Daniele Bernardo Panaro, Andrea Trucchia, Vincenzo Luongo, Maria Rosaria Mattei, Luigi Frunzo . Global sensitivity analysis and uncertainty quantification for a mathematical model of dry anaerobic digestion in plug-flow reactors. Mathematical Biosciences and Engineering, 2024, 21(9): 7139-7164. doi: 10.3934/mbe.2024316
    [5] Scott R. Pope, Laura M. Ellwein, Cheryl L. Zapata, Vera Novak, C. T. Kelley, Mette S. Olufsen . Estimation and identification of parameters in a lumped cerebrovascular model. Mathematical Biosciences and Engineering, 2009, 6(1): 93-115. doi: 10.3934/mbe.2009.6.93
    [6] H. T. Banks, D. Rubio, N. Saintier, M. I. Troparevsky . Optimal design for parameter estimation in EEG problems in a 3D multilayered domain. Mathematical Biosciences and Engineering, 2015, 12(4): 739-760. doi: 10.3934/mbe.2015.12.739
    [7] Jing Cai, Jianfeng Yang, Yongjin Zhang . Reliability analysis of s-out-of-k multicomponent stress-strength system with dependent strength elements based on copula function. Mathematical Biosciences and Engineering, 2023, 20(5): 9470-9488. doi: 10.3934/mbe.2023416
    [8] Robert G. McLeod, John F. Brewster, Abba B. Gumel, Dean A. Slonowsky . Sensitivity and uncertainty analyses for a SARS model with time-varying inputs and outputs. Mathematical Biosciences and Engineering, 2006, 3(3): 527-544. doi: 10.3934/mbe.2006.3.527
    [9] Li Cai, Jie Jiao, Pengfei Ma, Wenxian Xie, Yongheng Wang . Estimation of left ventricular parameters based on deep learning method. Mathematical Biosciences and Engineering, 2022, 19(7): 6638-6658. doi: 10.3934/mbe.2022312
    [10] Francisco Julian Ariza-Hernandez, Juan Carlos Najera-Tinoco, Martin Patricio Arciga-Alejandre, Eduardo Castañeda-Saucedo, Jorge Sanchez-Ortiz . Bayesian inverse problem for a fractional diffusion model of cell migration. Mathematical Biosciences and Engineering, 2024, 21(4): 5826-5837. doi: 10.3934/mbe.2024257
  • We examine estimation of the parameters of Susceptible-Infective-Recovered (SIR) models in the context of least squares. We review the use of asymptotic statistical theory and sensitivity analysis to obtain measures of uncertainty for estimates of the model parameters and the basic reproductive number ($R_0$)---an epidemiologically significant parameter grouping. We find that estimates of different parameters, such as the transmission parameter and recovery rate, are correlated, with the magnitude and sign of this correlation depending on the value of $R_0$. Situations are highlighted in which this correlation allows $R_0$ to be estimated with greater ease than its constituent parameters. Implications of correlation for parameter identifiability are discussed. Uncertainty estimates and sensitivity analysis are used to investigate how the frequency at which data is sampled affects the estimation process and how the accuracy and uncertainty of estimates improves as data is collected over the course of an outbreak. We assess the informativeness of individual data points in a given time series to determine when more frequent sampling (if possible) would prove to be most beneficial to the estimation process. This technique can be used to design data sampling schemes in more general contexts.


  • This article has been cited by:

    1. Dave Osthus, Kyle S. Hickmann, Petruţa C. Caragea, Dave Higdon, Sara Y. Del Valle, Forecasting seasonal influenza with a state-space SIR model, 2017, 11, 1932-6157, 10.1214/16-AOAS1000
    2. Brett Matzuka, Jesper Mehlsen, Hien Tran, Mette Sofie Olufsen, Using Kalman Filtering to Predict Time-Varying Parameters in a Model Predicting Baroreflex Regulation During Head-Up Tilt, 2015, 62, 0018-9294, 1992, 10.1109/TBME.2015.2409211
    3. Hailay Weldegiorgis Berhe, Oluwole Daniel Makinde, David Mwangi Theuri, Parameter Estimation and Sensitivity Analysis of Dysentery Diarrhea Epidemic Model, 2019, 2019, 1110-757X, 1, 10.1155/2019/8465747
    4. Tulio Rodrigues, Otaviano Helene, Monte Carlo approach to model COVID-19 deaths and infections using Gompertz functions, 2020, 2, 2643-1564, 10.1103/PhysRevResearch.2.043381
    5. Necibe Tuncer, Hayriye Gulbudak, Vincent L. Cannataro, Maia Martcheva, Structural and Practical Identifiability Issues of Immuno-Epidemiological Vector–Host Models with Application to Rift Valley Fever, 2016, 78, 0092-8240, 1796, 10.1007/s11538-016-0200-2
    6. Dave Osthus, James Gattiker, Reid Priedhorsky, Sara Y. Del Valle, Dynamic Bayesian Influenza Forecasting in the United States with Hierarchical Discrepancy (with Discussion), 2019, 14, 1936-0975, 10.1214/18-BA1117
    7. Gerardo Chowell, Fitting dynamic models to epidemic outbreaks with quantified uncertainty: A primer for parameter uncertainty, identifiability, and forecasts, 2017, 2, 24680427, 379, 10.1016/j.idm.2017.08.001
    8. Chiara Piazzola, Lorenzo Tamellini, Raúl Tempone, A note on tools for prediction under uncertainty and identifiability of SIR-like dynamical systems for epidemiology, 2021, 332, 00255564, 108514, 10.1016/j.mbs.2020.108514
    9. Ming Liu, Xifen Xu, Jie Cao, Ding Zhang, Integrated planning for public health emergencies: A modified model for controlling H1N1 pandemic, 2020, 71, 0160-5682, 748, 10.1080/01605682.2019.1582589
    10. Ping Yan, Gerardo Chowell, 2019, Chapter 9, 978-3-030-21922-2, 317, 10.1007/978-3-030-21923-9_9
    11. Abdallah Alsayed, Hayder Sadir, Raja Kamil, Hasan Sari, Prediction of Epidemic Peak and Infected Cases for COVID-19 Disease in Malaysia, 2020, 2020, 17, 1660-4601, 4076, 10.3390/ijerph17114076
    12. Necibe Tuncer, Trang T. Le, Structural and practical identifiability analysis of outbreak models, 2018, 299, 00255564, 1, 10.1016/j.mbs.2018.02.004
    13. João N. C. Gonçalves, Helena Sofia Rodrigues, M. Teresa T. Monteiro, 2017, Chapter 96, 978-3-319-53479-4, 974, 10.1007/978-3-319-53480-0_96
    14. Andreas Widder, On the usefulness of set-membership estimation in the epidemiology of infectious diseases, 2017, 15, 1551-0018, 141, 10.3934/mbe.2018006
    15. Ming Liu, Jie Cao, Jing Liang, MingJun Chen, 2020, Chapter 9, 978-981-13-9352-5, 167, 10.1007/978-981-13-9353-2_9
    16. Punam R Thakare, S S Mathurkar, 2016, Modeling of epidemic spread by social interactions, 978-1-5090-0774-5, 1320, 10.1109/RTEICT.2016.7808045
    17. Mazair Raissi, Niloofar Ramezani, Padmanabhan Seshaiyer, On Parameter Estimation Approaches for Predicting Disease Transmission Through Optimization, Deep Learning and Statistical Inference Methods, 2019, 6, 23737867, 10.30707/LiB6.2Raissi
    18. Toshikazu Kuniya, Evaluation of the effect of the state of emergency for the first wave of COVID-19 in Japan, 2020, 5, 24680427, 580, 10.1016/j.idm.2020.08.004
    19. George Qian, Adam Mahdi, Sensitivity analysis methods in the biomedical sciences, 2020, 323, 00255564, 108306, 10.1016/j.mbs.2020.108306
    20. Yuxuan He, Nan Liu, Methodology of emergency medical logistics for public health emergencies, 2015, 79, 13665545, 178, 10.1016/j.tre.2015.04.007
    21. Siddhartha Paul, Jayendran Venkateswaran, Impact of drug supply chain on the dynamics of infectious diseases, 2017, 33, 0883-7066, 280, 10.1002/sdr.1592
    22. Toshikazu Kuniya, Prediction of the Epidemic Peak of Coronavirus Disease in Japan, 2020, 2020, 9, 2077-0383, 789, 10.3390/jcm9030789
    23. Tianjian Zhou, Yuan Ji, Semiparametric Bayesian inference for the transmission dynamics of COVID-19 with a state-space model, 2020, 97, 15517144, 106146, 10.1016/j.cct.2020.106146
    24. T. Butler, L. Graham, S. Mattis, S. Walsh, A Measure-Theoretic Interpretation of Sample Based Numerical Integration with Applications to Inverse and Prediction Problems under Uncertainty, 2017, 39, 1064-8275, A2072, 10.1137/16M1063289
    25. Mattia Zanella, Chiara Bardelli, Mara Azzi, Silvia Deandrea, Pietro Perotti, Santino Silva, Ennio Cadum, Silvia Figini, Giuseppe Toscani, Social contacts, epidemic spreading and health system. Mathematical modeling and applications to COVID-19 infection, 2021, 18, 1551-0018, 3384, 10.3934/mbe.2021169
    26. Mattia Zanella, Chiara Bardelli, Giacomo Dimarco, Silvia Deandrea, Pietro Perotti, Mara Azzi, Silvia Figini, Giuseppe Toscani, A data-driven epidemic model with social structure for understanding the COVID-19 infection on a heavily affected Italian province, 2021, 31, 0218-2025, 2533, 10.1142/S021820252150055X
    27. Giacomo Albi, Giulia Bertaglia, Walter Boscheri, Giacomo Dimarco, Lorenzo Pareschi, Giuseppe Toscani, Mattia Zanella, 2022, Chapter 3, 978-3-030-96561-7, 43, 10.1007/978-3-030-96562-4_3
    28. Bin Hu, Guanhua Jiang, Xinyi Yao, Wei Chen, Tingyu Yue, Qitong Zhao, Zongliang Wen, Allocation of emergency medical resources for epidemic diseases considering the heterogeneity of epidemic areas, 2023, 11, 2296-2565, 10.3389/fpubh.2023.992197
    29. Yan Wang, Guichen Lu, Jiang Du, Calibration and prediction for the inexact SIR model, 2022, 19, 1551-0018, 2800, 10.3934/mbe.2022128
    30. Leah Mitchell, Andrea Arnold, Analyzing the effects of observation function selection in ensemble Kalman filtering for epidemic models, 2021, 339, 00255564, 108655, 10.1016/j.mbs.2021.108655
    31. Jiaji Pan, Zhongxiang Chen, Yixuan He, Tongliang Liu, Xi Cheng, Jun Xiao, Hao Feng, Why Controlling the Asymptomatic Infection Is Important: A Modelling Study with Stability and Sensitivity Analysis, 2022, 6, 2504-3110, 197, 10.3390/fractalfract6040197
    32. Giacomo Albi, Lorenzo Pareschi, Mattia Zanella, Modelling lockdown measures in epidemic outbreaks using selective socio-economic containment with uncertainty, 2021, 18, 1551-0018, 7161, 10.3934/mbe.2021355
    33. Collins Okoyo, Nelson Onyango, Idah Orowe, Charles Mwandawiro, Graham Medley, Sensitivity Analysis of a Transmission Interruption Model for the Soil-Transmitted Helminth Infections in Kenya, 2022, 10, 2296-2565, 10.3389/fpubh.2022.841883
    34. Hannah C. Lepper, Mark E. J. Woolhouse, Bram A. D. van Bunnik, The Role of the Environment in Dynamics of Antibiotic Resistance in Humans and Animals: A Modelling Study, 2022, 11, 2079-6382, 1361, 10.3390/antibiotics11101361
    35. Emmanuelle A. Dankwa, Andrew F. Brouwer, Christl A. Donnelly, Structural identifiability of compartmental models for infectious disease transmission is influenced by data type, 2022, 41, 17554365, 100643, 10.1016/j.epidem.2022.100643
    36. Giacomo Albi, Lorenzo Pareschi, Mattia Zanella, Control with uncertain data of socially structured compartmental epidemic models, 2021, 82, 0303-6812, 10.1007/s00285-021-01617-y
    37. Faraja Luhanda, Jacob I. Irunde, Dmitry Kuznetsov, Modeling cryptosporidiosis in humans and cattle: Deterministic and stochastic approaches, 2023, 21, 24056731, e00293, 10.1016/j.parepi.2023.e00293
    38. Divine Wanduku, The multilevel hierarchical data EM-algorithm. Applications to discrete-time Markov chain epidemic models, 2022, 8, 24058440, e12622, 10.1016/j.heliyon.2022.e12622
    39. G. Dimarco, B. Perthame, G. Toscani, M. Zanella, Kinetic models for epidemic dynamics with social heterogeneity, 2021, 83, 0303-6812, 10.1007/s00285-021-01630-1
    40. Linjie Wen, Jinglai Li, Affine-mapping based variational ensemble Kalman filter, 2022, 32, 0960-3174, 10.1007/s11222-022-10156-5
    41. Miriam R. Ferrández, Benjamin Ivorra, Juana L. Redondo, Ángel M. Ramos, Pilar M. Ortigosa, A multi-objective approach to identify parameters of compartmental epidemiological models—Application to Ebola Virus Disease epidemics, 2023, 120, 10075704, 107165, 10.1016/j.cnsns.2023.107165
    42. G. Dimarco, G. Toscani, M. Zanella, Optimal control of epidemic spreading in the presence of social heterogeneity, 2022, 380, 1364-503X, 10.1098/rsta.2021.0160
    43. Albert Orwa Akuno, L. Leticia Ramírez-Ramírez, Jesús F. Espinoza, Inference on a Multi-Patch Epidemic Model with Partial Mobility, Residency, and Demography: Case of the 2020 COVID-19 Outbreak in Hermosillo, Mexico, 2023, 25, 1099-4300, 968, 10.3390/e25070968
    44. Yuqing Sun, Zhonghua Zhang, Yulin Sun, Calculation Method and Application of Time-Varying Transmission Rate via Data-Driven Approach, 2023, 11, 2227-7390, 2955, 10.3390/math11132955
    45. Alvan Caleb Arulandu, Padmanabhan Seshaiyer, PHYSICS-INFORMED NEURAL NETWORKS FOR INFORMED VACCINE DISTRIBUTION INMETA-POPULATIONS , 2023, 4, 2689-3967, 83, 10.1615/JMachLearnModelComput.2023047642
    46. Jacques Hermes, Marcus Rosenblatt, Christian Tönsing, Jens Timmer, 2023, 2872, 0094-243X, 030006, 10.1063/5.0163819
    47. Shan Qiao, Mingke He, Jing Wang, Jianping Cai, Jie Zheng, Robust optimization for a dynamic emergency materials supply chain network under major infectious disease epidemics, 2023, 1367-5567, 1, 10.1080/13675567.2023.2269101
    48. Chih-Li Sung, Ying Hung, Efficient calibration for imperfect epidemic models with applications to the analysis of COVID-19, 2023, 0035-9254, 10.1093/jrsssc/qlad083
    49. B. K. M. Case, Jean-Gabriel Young, Laurent Hébert-Dufresne, Accurately summarizing an outbreak using epidemiological models takes time, 2023, 10, 2054-5703, 10.1098/rsos.230634
    50. Jacques Hermes, Marcus Rosenblatt, Christian Tönsing, Jens Timmer, Non-Parametric Model-Based Estimation of the Effective Reproduction Number for SARS-CoV-2, 2023, 16, 1999-4893, 533, 10.3390/a16120533
    51. D. Bichara, A. Iggidr, M. Oumoun, A. Rapaport, G. Sallet, Identifiability and Observability via decoupled variables: Application to a malaria intra-host model, 2023, 56, 24058963, 576, 10.1016/j.ifacol.2023.10.1629
    52. Martina Cendoya, Ana Navarro-Quiles, Antonio López-Quílez, Antonio Vicent, David Conesa, An Individual-Based Spatial Epidemiological Model for the Spread of Plant Diseases, 2024, 1085-7117, 10.1007/s13253-024-00604-2
    53. Praachi Das, Morganne Igoe, Alexanderia Lacy, Trevor Farthing, Archana Timsina, Cristina Lanzas, Suzanne Lenhart, Agricola Odoi, Alun L. Lloyd, Modeling county level COVID-19 transmission in the greater St. Louis area: Challenges of uncertainty and identifiability when fitting mechanistic models to time-varying processes, 2024, 00255564, 109181, 10.1016/j.mbs.2024.109181
    54. Nik Cunniffe, Frédéric Hamelin, Abderrahman Iggidr, Alain Rapaport, Gauthier Sallet, 2024, Chapter 5, 978-981-97-2538-0, 59, 10.1007/978-981-97-2539-7_5
    55. Hannah Kravitz, Christina Durón, Moysey Brio, A Coupled Spatial-Network Model: A Mathematical Framework for Applications in Epidemiology, 2024, 86, 0092-8240, 10.1007/s11538-024-01364-3
    56. Mahmudul Bari Hridoy, An exploration of modeling approaches for capturing seasonal transmission in stochastic epidemic models, 2025, 22, 1551-0018, 324, 10.3934/mbe.2025013
  • Reader Comments
  • © 2012 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
Mathematical Biosciences and Engineering
3.9

Metrics

Article views(3888) PDF downloads(876) Cited by(56)

Preview PDF
Download XML
Export Citation
Article outline
Abstract

Other Articles By Authors

Related pages

Tools

Copyright © AIMS Press

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog