# American Institute of Mathematical Sciences

2012, 9(3): 553-576. doi: 10.3934/mbe.2012.9.553

## 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, United States 2 Department of Mathematics, University of North Carolina, Chapel Hill, CB #3250, Chapel Hill, NC 27599, United States 3 Program in Applied Mathematics, University of Arizona, 617 N. Santa Rita Ave., PO Box 210089, Tucson, AZ 85721-0089, United States 4 Department of Mathematics, Morehouse College, 830 Westview Drive SW Unit 142133, Atlanta, GA 30314, United States 5 Department of Mathematics, North Carolina State University, Raleigh, NC 27695, United States 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, United States

Received  December 2009 Revised  April 2012 Published  July 2012

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. Mathematical Biosciences & Engineering, 2012, 9 (3) : 553-576. doi: 10.3934/mbe.2012.9.553
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