# American Institute of Mathematical Sciences

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

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

Received  April 2008 Revised  August 2008 Published  March 2009

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