Parameter estimation and uncertainty quantification for an epidemic model
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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
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2.
Department of Mathematics, University of North Carolina, Chapel Hill, CB #3250, Chapel Hill, NC 27599
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3.
Program in Applied Mathematics, University of Arizona, 617 N. Santa Rita Ave., PO Box 210089, Tucson, AZ 85721-0089
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4.
Department of Mathematics, Morehouse College, 830 Westview Drive SW Unit 142133, Atlanta, GA 30314
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Department of Mathematics, North Carolina State University, Raleigh, NC 27695
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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
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Received:
01 December 2009
Accepted:
29 June 2018
Published:
01 July 2012
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MSC :
Primary: 92D30; Secondary: 62F99, 62P10, 65L09.
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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
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Abstract
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.
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