The reproduction number R_t in structured and nonstructured populations

Previous Article
Mathematical modelling of tuberculosis epidemics
MBE Home
This Issue
Next Article
The estimation of the effective reproductive number from disease outbreak data

2009, 6(2): 239-259. doi: 10.3934/mbe.2009.6.239

The reproduction number $R_t$ in structured and nonstructured populations

1.	Statistical Sciences Group, Los Alamos National Laboratory, Los Alamos, NM 87545, United States
2.	School of Human Evolution and Social Change, Arizona State University, Box 872402, Tempe, AZ 85287, United States

Received November 2007 Revised September 2008 Published March 2009

Abstract
Related Papers

Using daily counts of newly infected individuals, Wallinga and Teunis (WT) introduced a conceptually simple method to estimate the number of secondary cases per primary case ($R_t$) for a given day. The method requires an estimate of the generation interval probability density function (pdf), which specifies the probabilities for the times between symptom onset in a primary case and symptom onset in a corresponding secondary case. Other methods to estimate $R_t$ are based on explicit models such as the SIR model; therefore, one might expect the WT method to be more robust to departures from SIR-type behavior. This paper uses simulated data to compare the quality of daily $R_t$ estimates based on a SIR model to those using the WT method for both structured (classical SIR assumptions are violated) and nonstructured (classical SIR assumptions hold) populations. By using detailed simulations that record the infection day of each new infection and the donor-recipient identities, the true $R_t$ and the generation interval pdf is known with negligible error. We find that the generation interval pdf is time dependent in all cases, which agrees with recent results reported elsewhere. We also find that the WT method performs essentially the same in the structured populations (except for a spatial network) as it does in the nonstructured population. And, the WT method does as well or better than a SIR-model based method in three of the four structured populations. Therefore, even if the contact patterns are heterogeneous as in the structured populations evaluated here, the WT method provides reasonable estimates of $R_t$, as does the SIR method.

Keywords: reproduction number; generation interval; structured population..

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.

Citation: 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

[1]	Hui Cao, Yicang Zhou. The basic reproduction number of discrete SIR and SEIS models with periodic parameters. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 37-56. doi: 10.3934/dcdsb.2013.18.37
[2]	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
[3]	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
[4]	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
[5]	Fred Brauer. A model for an SI disease in an age - structured population. Discrete & Continuous Dynamical Systems - B, 2002, 2 (2) : 257-264. doi: 10.3934/dcdsb.2002.2.257
[6]	Rinaldo M. Colombo, Mauro Garavello. Stability and optimization in structured population models on graphs. Mathematical Biosciences & Engineering, 2015, 12 (2) : 311-335. doi: 10.3934/mbe.2015.12.311
[7]	Jacek Banasiak, Wilson Lamb. Coagulation, fragmentation and growth processes in a size structured population. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 563-585. doi: 10.3934/dcdsb.2009.11.563
[8]	G. Buffoni, S. Pasquali, G. Gilioli. A stochastic model for the dynamics of a stage structured population. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 517-525. doi: 10.3934/dcdsb.2004.4.517
[9]	Ricardo Borges, Àngel Calsina, Sílvia Cuadrado. Equilibria of a cyclin structured cell population model. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 613-627. doi: 10.3934/dcdsb.2009.11.613
[10]	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
[11]	Dongxue Yan, Xianlong Fu. Asymptotic behavior of a hierarchical size-structured population model. Evolution Equations & Control Theory, 2018, 7 (2) : 293-316. doi: 10.3934/eect.2018015
[12]	Zhihua Liu, Hui Tang, Pierre Magal. Hopf bifurcation for a spatially and age structured population dynamics model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1735-1757. doi: 10.3934/dcdsb.2015.20.1735
[13]	Xianlong Fu, Dongmei Zhu. Stability analysis for a size-structured juvenile-adult population model. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 391-417. doi: 10.3934/dcdsb.2014.19.391
[14]	Bruno Buonomo, Deborah Lacitignola. On the stabilizing effect of cannibalism in stage-structured population models. Mathematical Biosciences & Engineering, 2006, 3 (4) : 717-731. doi: 10.3934/mbe.2006.3.717
[15]	Jacques Henry. For which objective is birth process an optimal feedback in age structured population dynamics?. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 107-114. doi: 10.3934/dcdsb.2007.8.107
[16]	L. M. Abia, O. Angulo, J.C. López-Marcos. Size-structured population dynamics models and their numerical solutions. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 1203-1222. doi: 10.3934/dcdsb.2004.4.1203
[17]	Gabriella Di Blasio, Alfredo Lorenzi. Direct and inverse problems in age--structured population diffusion. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 539-563. doi: 10.3934/dcdss.2011.4.539
[18]	Xianlong Fu, Dongmei Zhu. Stability results for a size-structured population model with delayed birth process. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 109-131. doi: 10.3934/dcdsb.2013.18.109
[19]	Z.-R. He, M.-S. Wang, Z.-E. Ma. Optimal birth control problems for nonlinear age-structured population dynamics. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 589-594. doi: 10.3934/dcdsb.2004.4.589
[20]	Inom Mirzaev, David M. Bortz. A numerical framework for computing steady states of structured population models and their stability. Mathematical Biosciences & Engineering, 2017, 14 (4) : 933-952. doi: 10.3934/mbe.2017049

2018 Impact Factor: 1.313

American Institute of Mathematical Sciences