American Institute of Mathematical Sciences

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

The estimation of the effective reproductive number from disease outbreak data

Ariel Cintrón-Arias 1, , Carlos Castillo-Chávez 2, , Luís M. A. Bettencourt 3, , Alun L. Lloyd 4, and  H. T. Banks 5,

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.
Keywords: basic reproduction ratio, $\mathcal{R}$, reproduction number, parameter estimation, generalized least squares, $\mathcal{R}_0$, residual plots., $\mathcal{R}(t)$, effective reproductive number.
Mathematics Subject Classification: Primary: 62G05, 93E24, 49Q12, 37N25; Secondary: 62H12, 62N0.
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
[1]

Nicolas Bacaër, Xamxinur Abdurahman, Jianli Ye, Pierre Auger. On the basic reproduction number $R_0$ in sexual activity models for HIV/AIDS epidemics: Example from Yunnan, China. Mathematical Biosciences & Engineering, 2007, 4 (4) : 595-607. doi: 10.3934/mbe.2007.4.595
[2]

Tom Burr, Gerardo Chowell. The reproduction number $R_t$ in structured and nonstructured populations. Mathematical Biosciences & Engineering, 2009, 6 (2) : 239-259. doi: 10.3934/mbe.2009.6.239
[3]

Gerardo Chowell, R. Fuentes, A. Olea, X. Aguilera, H. Nesse, J. M. Hyman. The basic reproduction number $R_0$ and effectiveness of reactive interventions during dengue epidemics: The 2002 dengue outbreak in Easter Island, Chile. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1455-1474. doi: 10.3934/mbe.2013.10.1455
[4]

Cameron J. Browne, Sergei S. Pilyugin. Minimizing $\mathcal R_0$ for in-host virus model with periodic combination antiviral therapy. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3315-3330. doi: 10.3934/dcdsb.2016099
[5]

Nitu Kumari, Sumit Kumar, Sandeep Sharma, Fateh Singh, Rana Parshad. Basic reproduction number estimation and forecasting of COVID-19: A case study of India, Brazil and Peru. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2021170
[6]

David Burguet. Examples of $\mathcal{C}^r$ interval map with large symbolic extension entropy. Discrete and Continuous Dynamical Systems, 2010, 26 (3) : 873-899. doi: 10.3934/dcds.2010.26.873
[7]

Hui Cao, Yicang Zhou. The basic reproduction number of discrete SIR and SEIS models with periodic parameters. Discrete and Continuous Dynamical Systems - B, 2013, 18 (1) : 37-56. doi: 10.3934/dcdsb.2013.18.37
[8]

Zhilan Feng, Qing Han, Zhipeng Qiu, Andrew N. Hill, John W. Glasser. Computation of $\mathcal R $ in age-structured epidemiological models with maternal and temporary immunity. Discrete and Continuous Dynamical Systems - B, 2016, 21 (2) : 399-415. doi: 10.3934/dcdsb.2016.21.399
[9]

Wenjun Liu, Yukun Xiao, Xiaoqing Yue. Classification of finite irreducible conformal modules over Lie conformal algebra $ \mathcal{W}(a, b, r) $. Electronic Research Archive, 2021, 29 (3) : 2445-2456. doi: 10.3934/era.2020123
[10]

Salvador Addas-Zanata, Fábio A. Tal. Support of maximizing measures for typical $\mathcal{C}^0$ dynamics on compact manifolds. Discrete and Continuous Dynamical Systems, 2010, 26 (3) : 795-804. doi: 10.3934/dcds.2010.26.795
[11]

Xiao-Wen Chang, David Titley-Peloquin. An improved algorithm for generalized least squares estimation. Numerical Algebra, Control and Optimization, 2020, 10 (4) : 451-461. doi: 10.3934/naco.2020044
[12]

Hisashi Inaba. The Malthusian parameter and $R_0$ for heterogeneous populations in periodic environments. Mathematical Biosciences & Engineering, 2012, 9 (2) : 313-346. doi: 10.3934/mbe.2012.9.313
[13]

Tianhui Yang, Ammar Qarariyah, Qigui Yang. The effect of spatial variables on the basic reproduction ratio for a reaction-diffusion epidemic model. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3005-3017. doi: 10.3934/dcdsb.2021170
[14]

Timothy C. Reluga, Jan Medlock, Alison Galvani. The discounted reproductive number for epidemiology. Mathematical Biosciences & Engineering, 2009, 6 (2) : 377-393. doi: 10.3934/mbe.2009.6.377
[15]

Alonso sepúlveda Castellanos. Generalized Hamming weights of codes over the $\mathcal{GH}$ curve. Advances in Mathematics of Communications, 2017, 11 (1) : 115-122. doi: 10.3934/amc.2017006
[16]

Qiang Li. A kind of generalized transversality theorem for $C^r$ mapping with parameter. Discrete and Continuous Dynamical Systems - S, 2017, 10 (5) : 1043-1050. doi: 10.3934/dcdss.2017055
[17]

Gerardo Chowell, Catherine E. Ammon, Nicolas W. Hengartner, James M. Hyman. Estimating the reproduction number from the initial phase of the Spanish flu pandemic waves in Geneva, Switzerland. Mathematical Biosciences & Engineering, 2007, 4 (3) : 457-470. doi: 10.3934/mbe.2007.4.457
[18]

Ling Xue, Caterina Scoglio. Network-level reproduction number and extinction threshold for vector-borne diseases. Mathematical Biosciences & Engineering, 2015, 12 (3) : 565-584. doi: 10.3934/mbe.2015.12.565
[19]

Yu-Ming Chu, Saima Rashid, Fahd Jarad, Muhammad Aslam Noor, Humaira Kalsoom. More new results on integral inequalities for generalized $ \mathcal{K} $-fractional conformable Integral operators. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2119-2135. doi: 10.3934/dcdss.2021063
[20]

Agnieszka Badeńska. No entire function with real multipliers in class $\mathcal{S}$. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3321-3327. doi: 10.3934/dcds.2013.33.3321

2018 Impact Factor: 1.313

Tools

Metrics

  • PDF downloads (426)
  • HTML views (0)
  • Cited by (48)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy