Abstract
Although the basic reproduction number, R 0, is useful for understanding the transmissibility of a disease and designing various intervention strategies, the classic threshold quantity theoretically assumes that the epidemic first occurs in a fully susceptible population, and hence, R 0 is essentially a mathematically defined quantity. In many instances, it is of practical importance to evaluate time-dependent variations in the transmission potential of infectious diseases. Explanation of the time course of an epidemic can be partly achieved by estimating the effective reproduction number, R(t), defined as the actual average number of secondary cases per primary case at calendar time t (for t >0). R(t) shows time-dependent variation due to the decline in susceptible individuals (intrinsic factors) and the implementation of control measures (extrinsic factors). If R(t)<1, it suggests that the epidemic is in decline and may be regarded as being under control at time t (vice versa, if R(t)>1). This chapter describes the primer of mathematics and statistics of R(t) and discusses other similar markers of transmissibility as a function of time.
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
Alho JM, Spencer BD (2005) Statistical Demography and Forecasting. Springer,New York.
Amundsen EJ, Stigum H, Rottingen JA, Aalen OO (2004) Definition and estimation of an actual reproduction number describing past infectious disease transmission: application to HIV epidemics among homosexual men in Denmark, Norway and Sweden. Epidemiology and Infection 132:1139–1149.
Anderson RM, May RM (1982) Directly transmitted infectious diseases: control by vaccination. Science 215:1053–1060.
Anderson RM, May RM (1991) Infectious Diseases of Humans: Dynamics and Control. Oxford University Press, Oxford.
Becker N (1977) Estimation for discrete time branching processes with application to epidemics. Biometrics 33:515–522.
Bettencourt LM, Ribeiro RM (2008) Real time bayesian estimation of the epidemic potential of emerging infectious diseases. PLoS ONE 3:e2185.
Bettencourt LMA, Ribeiro RM, Chowell G, Lant T, Castillo-Chavez C (2007) Towards real time epidemiology: data assimilation, modeling and anomaly detection of health surveillance data streams. in: Intelligence and security informatics: Biosurveillance. Proceedings of the 2nd NSF Workshop, Biosurveillance, 2007. Lecture Notes in Computer Science. eds F. Zeng et al. Springer-Verlag, Berlin pp. 79–90.
Cauchemez S, Boelle PY, Thomas G, Valleron AJ (2006) Estimating in real time the efficacy of measures to control emerging communicable diseases. American Journal of Epidemiology 164:591–597.
Cauchemez S, Boelle PY, Donnelly CA, Ferguson NM, Thomas G, Leung GM, Hedley AJ, Anderson RM, Valleron AJ (2006) Real-time estimates in early detection of SARS. Emerging Infectious Diseases 12:110–113.
Chowell G, Nishiura H, Bettencourt LM (2007) Comparative estimation of the reproduction number for pandemic influenza from daily case notification data. Journal of the Royal Society Interface 4:155–166.
Cowling BJ, Ho LM, Leung GM (2007) Effectiveness of control measures during the SARS epidemic in Beijing: a comparison of the Rt curve and the epidemic curve. Epidemiology and Infection 136:562–566.
Diekmann O (1977) Limiting behaviour in an epidemic model. Nonlinear Analysis, Theory, Methods and Applications 1:459–470.
Diekmann O, Heesterbeek JAP, Metz JAJ (1990) On the definition and the computation of the basic reproductive ratio R<Sub>0</Sub> in models for infectious diseases. Journal of Mathematical Biology 35:503–522.
Diekmann O, Heesterbeek JAP (2000) Mathematical Epidemiology of Infectious Diseases. Model Building, Analysis and Interpretation. John Wiley and Sons, New York.
Dietz K (1993) The estimation of the basic reproduction number for infectious diseases. Statistical Methods in Medical Research 2:23–41.
Dublin LI, Lotka AJ (1920) On the true rate of natural increase, as exemplified by the population of the United States, 1920. Journal of American Statistical Association 150:305–339.
Ferguson NM, Donnelly CA, Anderson RM (2001) Transmission intensity and impact of control policies on the foot and mouth epidemc in Great Britain. Nature 413:542–548.
Fine PE (1993) Herd immunity: history, theory, practice. Epidemiologic Reviews 15:265–302.
Fine PE (2003) The interval between successive cases of an infectious disease. American Journal of Epidemiology 158:1039–1047.
Fraser C (2007) Estimating individual and household reproduction numbers in an emerging epidemic. PLoS ONE 2:e758.
Grassly NC, Fraser C (2008) Mathematical models of infectious disease transmission. Nature Review of Microbiology 6:477–487.
Haydon DT, Chase-Topping M, Shaw DJ, Matthews L, Friar JK, Wilesmith J, Woolhouse ME (2003) The construction and analysis of epidemic trees with reference to the 2001 UK foot-and-mouth outbreak. Proceedings of the Royal Society of London Series B 270:121–127.
Kendall DG (1956) Deterministic and stochastic epidemics in closed populations. In: Newman P (ed) Third Berkeley Symposium on Mathematical Statistics and Probability 4. University of California Press, New York, 1956, pp.149–165.
Kermack WO, McKendrick AG (1927) Contributions to the mathematical theory of epidemics – I. Proceedings of the Royal Society Series A 115:700–721 (reprinted in Bulletin of Mathematical Biology 53 (1991) 33–55).
Lloyd AL (2001a) Destabilization of epidemic models with the inclusion of realistic distributions of infectious periods. Proceedings of the Royal Society of London Series B 268:985–993.
Lloyd AL (2001b) Realistic distributions of infectious periods in epidemic models: Changing patterns of persistence and dynamics. Theoretical Population Biology 60:59–71.
Metz JAJ (1978) The epidemic in a closed population with all susceptibles equally vulnerable; some results for large susceptible populations and small initial infections. Acta Biotheoretica 27:75–123.
Nishiura H, Dietz K, Eichner M (2006) The earliest notes on the reproduction number in relation to herd immunity: theophil Lotz and smallpox vaccination. Journal of Theoretical Biology 241:964–967.
Nishiura H, Schwehm M, Kakehashi M, Eichner M (2006) Transmission potential of primary pneumonic plague: time inhomogeneous evaluation based on historical documents of the transmission network. Journal of Epidemiology and Community Health 60 (2006) 640–645.
Nishiura H (2007) Time variations in the transmissibility of pandemic influenza in Prussia, Germany, from 1918 to 1919. Theoretical Biology and Medical Modelling 4:20.
Peiper O (1920) Die Grippe-Epidemie in Preussen im Jahre 1918/19. Veroeffentlichungen aus dem Gebiete der Medizinalverwaltung 10:417–479 (in German).
Roberts MG, Heesterbeek JA (2007) Model-consistent estimation of the basic reproduction number from the incidence of an emerging infection. Journal of Mathematical Biology 55:803–816.
Smith CE(1964) Factors in the transmission of virus infections from animal to man. The Scientific Basis of Medicine Annual Reviews 125–150.
Svensson A (2007) A note on generation times in epidemic models. Mathematical Biosciences 208:300–311.
Wallinga J, Teunis P (2004) Different epidemic curves for severe acute respiratory syndrome reveal similar impacts of control measures. American Journal of Epidemiology 160:509–516.
Wallinga J, Lipsitch M (2007) How generation intervals shape the relationship between growth rates and reproductive numbers. Proceedings B 274:599–604.
Wearing HJ, Rohani P, Keeling MJ (2005) Appropriate models for the management of infectious diseases. PLoS Medicine 2:e174.
White PJ, Ward H, Garnett GP (2006) Is HIV out of control in the UK? An example of analysing patterns of HIV spreading using incidence-to-prevalence ratios. AIDS 20:1898–1901.
Yan P (2008) Separate roles of the latent and infectious periods in shaping the relation between the basic reproduction number and the intrinsic growth rate of infectious disease outbreaks. Journal of Theoretical Biology 251:238–252.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer Science+Business Media B.V.
About this chapter
Cite this chapter
Nishiura, H., Chowell, G. (2009). The Effective Reproduction Number as a Prelude to Statistical Estimation of Time-Dependent Epidemic Trends. In: Chowell, G., Hyman, J.M., Bettencourt, L.M.A., Castillo-Chavez, C. (eds) Mathematical and Statistical Estimation Approaches in Epidemiology. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-2313-1_5
Download citation
DOI: https://doi.org/10.1007/978-90-481-2313-1_5
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-2312-4
Online ISBN: 978-90-481-2313-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)