American Institute of Mathematical Sciences

2010, 7(4): 851-869. doi: 10.3934/mbe.2010.7.851

Time variations in the generation time of an infectious disease: Implications for sampling to appropriately quantify transmission potential

 1 PRESTO, Japan Science and Technology Agency (JST), 4-1-8 Honcho Kawaguchi, Saitama 332-0012, Japan

Received  November 2009 Revised  June 2010 Published  October 2010

Although the generation time of an infectious disease plays a key role in estimating its transmission potential, the impact of the sampling time of generation times on the estimation procedure has yet to be clarified. The present study defines the period and cohort generation times, both of which are time-inhomogeneous, as a function of the infection time of secondary and primary cases, respectively. By means of analytical and numerical approaches, it is shown that the period generation time increases with calendar time, whereas the cohort generation time decreases as the incidence increases. The initial growth phase of an epidemic of Asian influenza A (H2N2) in the Netherlands in 1957 was reanalyzed, and estimates of the basic reproduction number, $R_0$, from the Lotka-Euler equation were examined. It was found that the sampling time of generation time during the course of the epidemic introduced a time-effect to the estimate of $R_0$. Other historical data of a primary pneumonic plague in Manchuria in 1911 were also examined to help illustrate the empirical evidence of the period generation time. If the serial intervals, which eventually determine the generation times, are sampled during the course of an epidemic, direct application of the sampled generation-time distribution to the Lotka-Euler equation leads to a biased estimate of $R_0$. An appropriate quantification of the transmission potential requires the estimation of the cohort generation time during the initial growth phase of an epidemic or adjustment of the time-effect (e.g., adjustment of the growth rate of the epidemic during the sampling time) on the period generation time. A similar issue also applies to the estimation of the effective reproduction number as a function of calendar time. Mathematical properties of the generation time distribution in a heterogeneously mixing population need to be clarified further.
Citation: Hiroshi Nishiura. Time variations in the generation time of an infectious disease: Implications for sampling to appropriately quantify transmission potential. Mathematical Biosciences & Engineering, 2010, 7 (4) : 851-869. doi: 10.3934/mbe.2010.7.851
References:
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Epidemiol., 164 (2006), 591. doi: doi:10.1093/aje/kwj274. Google Scholar [7] G. Chowell, H. Nishiura and L. M. Bettencourt, Comparative estimation of the reproduction number for pandemic influenza from daily case notification data,, J. R. Soc. Interface., 4 (2007), 155. doi: doi:10.1098/rsif.2006.0161. Google Scholar [8] O. Diekmann and J. A. P. Heesterbeek, "Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation,", Wiley, (2000). Google Scholar [9] O. Diekmann, J. A. P. Heesterbeek and J. A. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations,, J. Math. Biol., 28 (1990), 365. doi: doi:10.1007/BF00178324. Google Scholar [10] K. Dietz, The estimation of the basic reproduction number for infectious diseases,, Stat. Methods. Med. Res., 2 (1993), 23. doi: doi:10.1177/096228029300200103. Google Scholar [11] L. I. Dublin and A. J. Lotka, On the true rate of natural increase,, J. Am. Stat. Assoc., 151 (1925), 305. doi: doi:10.2307/2965517. Google Scholar [12] P. E. M. Fine, The interval between successive cases of an infectious disease,, Am. J. Epidemiol., 158 (2003), 755. doi: doi:10.1093/aje/kwg251. Google Scholar [13] C. Fraser, Estimating individual and household reproduction numbers in an emerging epidemic,, PLoS One, 2 (2007). doi: doi:10.1371/journal.pone.0000758. Google Scholar [14] C. Fraser, S. Riley, R. M. Anderson and N. M. Ferguson, Factors that make an infectious disease outbreak controllable,, Proc. Natl. Acad. Sci. U.S.A., 101 (2004), 6146. doi: doi:10.1073/pnas.0307506101. Google Scholar [15] T. Garske, P. Clarke and A. C. Ghani, The transmissibility of highly pathogenic avian influenza in commercial poultry in industrialised countries,, PLoS One, 2 (2007). doi: doi:10.1371/journal.pone.0000349. Google Scholar [16] N. C. Grassly and C. Fraser, Mathematical models of infectious disease transmission,, Nat. Rev. Microbiol., 6 (2008), 477. Google Scholar [17] D. T. Haydon, M. Chase-Topping, D. J. Shaw, L. Matthews, J. K. Friar, J. Wilesmith and M. E. Woolhouse, The construction and analysis of epidemic trees with reference to the 2001 UK foot-and-mouth outbreak,, Proc. R. Soc. Lond. Ser. B., 270 (2003), 121. doi: doi:10.1098/rspb.2002.2191. Google Scholar [18] R. E. Hope Simpson, The period of transmission in certain epidemic diseases: An observational method for its discovery,, Lancet, 2 (1948), 755. doi: doi:10.1016/S0140-6736(48)91328-2. Google Scholar [19] H. Inaba and H. Nishiura, The state-reproduction number for a multistate class age structured epidemic system and its application to the asymptomatic transmission model,, Math. Biosci., 216 (2008), 77. doi: doi:10.1016/j.mbs.2008.08.005. Google Scholar [20] J. D. Kalbfleisch and J. F. Lawless, Inference based on retrospective ascertainment: An analysis of the data on transfusion-related AIDS,, J. Am. Stat. Assoc., 84 (1989), 360. doi: doi:10.2307/2289919. Google Scholar [21] E. Kenah, M. Lipsitch and J. M. Robins, Generation interval contraction and epidemic data analysis,, Math. Biosci., 213 (2008), 71. doi: doi:10.1016/j.mbs.2008.02.007. Google Scholar [22] W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics. I.,, Proc. R. Soc. Ser. A., 115 (1927), 700. doi: doi:10.1016/S0092-8240(05)80040-0. Google Scholar [23] N. Keyfitz, "Applied Mathematical Demography,", John Wiley and Sons, (1977). Google Scholar [24] S. W. Lakagos, L. M. Barraj and V. De Gruttola, Nonparametric analysis of truncated survival data, with application to AIDS,, Biometrika, 75 (1988), 515. doi: doi:10.1093/biomet/75.3.515. Google Scholar [25] M. Lipsitch, T. Cohen, B. Cooper, J. M. Robins, S. Ma, L. James, G. Gopalakrishna, S. K. Chew, C. C. Tan, M. H. Samore, D. Fisman and M. Murray, Transmission dynamics and control of severe acute respiratory syndrome,, Science, 38 (2003), 115. Google Scholar [26] N. Masurel and W. M. Marine, Recycling of Asian and Hong Kong influenza A virus hemagglutinins in man,, Am. J. Epidemiol., 97 (1973), 44. Google Scholar [27] C. E. Mills, J. M. Robins and M. Lipsitch, Transmissibility of 1918 pandemic influenza,, Nature, 432 (2004), 904. doi: doi:10.1038/nature03063. Google Scholar [28] J. Mulder, N. Masurel and J. F. P. Hers, De Aziatische-influenza-pandemie van 1957 (Dutch),, Ned. Tijdschr. Geneeskd., 102 (1958), 1992. Google Scholar [29] H. Nishiura, Correcting the actual reproduction number: A simple method to estimate $R_0$ from early epidemic growth data,, Int. J. Environ. Res. Public. Health., 7 (2010), 291. doi: doi:10.3390/ijerph7010291. Google Scholar [30] H. Nishiura, Early efforts in modeling the incubation period of infectious diseases with an acute course of illness,, Emerg. Themes. Epidemiol., 4 (2007). doi: doi:10.1186/1742-7622-4-2. Google Scholar [31] H. Nishiura, Epidemiology of a primary pneumonic plague in Kantoshu, Manchuria, from 1910 to 1911: Statistical analysis of individual records collected by the Japanese Empire,, Int. J. Epidemiol., 35 (2006), 1059. doi: doi:10.1093/ije/dyl091. Google Scholar [32] H. Nishiura, Time variations in the transmissibility of pandemic influenza in Prussia, Germany, from 1918-19,, Theor. Biol. Med. Model., 4 (2007). doi: doi:10.1186/1742-4682-4-20. Google Scholar [33] H. Nishiura and G. Chowell, The effective reproduction number as a prelude to statistical estimtion of time-dependent epidemic trends,, in, (2009), 103. doi: doi:10.1007/978-90-481-2313-1_5. Google Scholar [34] H. Nishiura, G. Chowell, H. Heesterbeek and J. Wallinga, The ideal reporting interval for an epidemic to objectively interpret the epidemiological time course,, J. R. Soc. Interface., 7 (2010), 297. doi: doi:10.1098/rsif.2009.0153. Google Scholar [35] H. Nishiura, G. Chowell, M. Safan and C. Castillo-Chavez, Pros and cons of estimating the reproduction number from early epidemic growth rate of influenza A (H1N1) 2009,, Theor. Biol. Med. Model., 7 (2010). doi: doi:10.1186/1742-4682-7-1. Google Scholar [36] H. Nishiura, M. Schwehm, M. Kakehashi and M. Eichner, Transmission potential of primary pneumonic plague: Time inhomogeneous evaluation based on historical documents of the transmission network,, J. Epidemiol. Community. Health., 60 (2006), 640. doi: doi:10.1136/jech.2005.042424. Google Scholar [37] W. Pickles, "Epidemiology in Country Practice,", John Wright and Sons, (1939). Google Scholar [38] M. G. Roberts and J. A. Heesterbeek, Model-consistent estimation of the basic reproduction number from the incidence of an emerging infection,, J. Math. Biol., 55 (2007), 803. doi: doi:10.1007/s00285-007-0112-8. Google Scholar [39] G. Scalia Tomba, A. Svensson, T. Asikainen and J. Giesecke, Some model based considerations on observing generation times for communicable diseases,, Math. Biosci., 223 (2010), 24. doi: doi:10.1016/j.mbs.2009.10.004. Google Scholar [40] A. Svensson, A note on generation times in epidemic models,, Math. Biosci., 208 (2007), 300. doi: doi:10.1016/j.mbs.2006.10.010. Google Scholar [41] Temporary Quarantine Section, K. Totokufu, "Epidemic Record of Plague During 1910-1911" (Meiji 43,4-nen 'Pest' Ryuko-shi),, Manchurian Daily Press, (1912). Google Scholar [42] E. Vynnycky and W. J. Edmunds, Analyses of the 1957 (Asian) influenza pandemic in the United Kingdom and the impact of school closures,, Epidemiol. Infect., 136 (2008), 166. doi: doi:10.1017/S0950268807008369. Google Scholar [43] E. Vynnycky and P. E. Fine, Lifetime risks, incubation period, and serial interval of tuberculosis,, Am. J. Epidemiol., 152 (2000), 247. doi: doi:10.1093/aje/152.3.247. Google Scholar [44] E. Vynnycky, A. Trindall and P. Mangtani, Estimates of the reproduction numbers of Spanish influenza using morbidity data,, Int. J. Epidemiol., 36 (2007), 881. doi: doi:10.1093/ije/dym071. Google Scholar [45] J. Wallinga and M. Lipsitch, How generation intervals shape the relationship between growth rates and reproductive numbers,, Proc. R. Soc. Lond. Ser. B., 274 (2007), 599. doi: doi:10.1098/rspb.2006.3754. Google Scholar [46] J. Wallinga and P. Teunis, Different epidemic curves for severe acute respiratory syndrome reveal similar impacts of control measures,, Am. J. Epidemiol., 160 (2004), 509. doi: doi:10.1093/aje/kwh255. Google Scholar [47] L. F. White and M. Pagano, A likelihood-based method for real-time estimation of the serial interval and reproductive number of an epidemic,, Stat. Med., 27 (2007), 2999. doi: doi:10.1002/sim.3136. Google Scholar [48] P. Yan, 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,, J. Theor. Biol., 251 (2008), 238. doi: doi:10.1016/j.jtbi.2007.11.027. Google Scholar

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References:
 [1] J. M. Alho and B. D. Spencer, "Statistical Demography and Forecasting,", Springer, (2005). Google Scholar [2] R. M. Anderson and R. M. May, "Infectious Diseases of Humans: Dynamics and Control,", Oxford University Press, (1991). Google Scholar [3] N. T. J. Bailey, "The Mathematical Theory of Infectious Diseases and its Applications,", 2nd edition, (1975). Google Scholar [4] T. Burr and G. Chowell, The reproduction number $R(t)$ in structured and nonstructured populations,, Math. Biosci. Eng., 6 (2009), 239. doi: doi:10.3934/mbe.2009.6.239. Google Scholar [5] S. Cauchemez, P. Y. Boelle, C. A. Donnelly, N. M. Ferguson, G. Thomas, G. M. Leung, A. J. Hedley, R. M. Anderson and A. J. Valleron, Real-time estimates in early detection of SARS,, Emerg. Infect. Dis., 12 (2006), 110. Google Scholar [6] S. Cauchemez, P. Y. Boelle, G. Thomas and A. J. Valleron, Estimating in real time the efficacy of measures to control emerging communicable diseases,, Am. J. Epidemiol., 164 (2006), 591. doi: doi:10.1093/aje/kwj274. Google Scholar [7] G. Chowell, H. Nishiura and L. M. Bettencourt, Comparative estimation of the reproduction number for pandemic influenza from daily case notification data,, J. R. Soc. Interface., 4 (2007), 155. doi: doi:10.1098/rsif.2006.0161. Google Scholar [8] O. Diekmann and J. A. P. Heesterbeek, "Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation,", Wiley, (2000). Google Scholar [9] O. Diekmann, J. A. P. Heesterbeek and J. A. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations,, J. Math. Biol., 28 (1990), 365. doi: doi:10.1007/BF00178324. Google Scholar [10] K. Dietz, The estimation of the basic reproduction number for infectious diseases,, Stat. Methods. Med. Res., 2 (1993), 23. doi: doi:10.1177/096228029300200103. Google Scholar [11] L. I. Dublin and A. J. Lotka, On the true rate of natural increase,, J. Am. Stat. Assoc., 151 (1925), 305. doi: doi:10.2307/2965517. Google Scholar [12] P. E. M. Fine, The interval between successive cases of an infectious disease,, Am. J. Epidemiol., 158 (2003), 755. doi: doi:10.1093/aje/kwg251. Google Scholar [13] C. Fraser, Estimating individual and household reproduction numbers in an emerging epidemic,, PLoS One, 2 (2007). doi: doi:10.1371/journal.pone.0000758. Google Scholar [14] C. Fraser, S. Riley, R. M. Anderson and N. M. Ferguson, Factors that make an infectious disease outbreak controllable,, Proc. Natl. Acad. Sci. U.S.A., 101 (2004), 6146. doi: doi:10.1073/pnas.0307506101. Google Scholar [15] T. Garske, P. Clarke and A. C. Ghani, The transmissibility of highly pathogenic avian influenza in commercial poultry in industrialised countries,, PLoS One, 2 (2007). doi: doi:10.1371/journal.pone.0000349. Google Scholar [16] N. C. Grassly and C. Fraser, Mathematical models of infectious disease transmission,, Nat. Rev. Microbiol., 6 (2008), 477. Google Scholar [17] D. T. Haydon, M. Chase-Topping, D. J. Shaw, L. Matthews, J. K. Friar, J. Wilesmith and M. E. Woolhouse, The construction and analysis of epidemic trees with reference to the 2001 UK foot-and-mouth outbreak,, Proc. R. Soc. Lond. Ser. B., 270 (2003), 121. doi: doi:10.1098/rspb.2002.2191. Google Scholar [18] R. E. Hope Simpson, The period of transmission in certain epidemic diseases: An observational method for its discovery,, Lancet, 2 (1948), 755. doi: doi:10.1016/S0140-6736(48)91328-2. Google Scholar [19] H. Inaba and H. Nishiura, The state-reproduction number for a multistate class age structured epidemic system and its application to the asymptomatic transmission model,, Math. Biosci., 216 (2008), 77. doi: doi:10.1016/j.mbs.2008.08.005. Google Scholar [20] J. D. Kalbfleisch and J. F. Lawless, Inference based on retrospective ascertainment: An analysis of the data on transfusion-related AIDS,, J. Am. Stat. Assoc., 84 (1989), 360. doi: doi:10.2307/2289919. Google Scholar [21] E. Kenah, M. Lipsitch and J. M. Robins, Generation interval contraction and epidemic data analysis,, Math. Biosci., 213 (2008), 71. doi: doi:10.1016/j.mbs.2008.02.007. Google Scholar [22] W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics. I.,, Proc. R. Soc. Ser. A., 115 (1927), 700. doi: doi:10.1016/S0092-8240(05)80040-0. Google Scholar [23] N. Keyfitz, "Applied Mathematical Demography,", John Wiley and Sons, (1977). Google Scholar [24] S. W. Lakagos, L. M. Barraj and V. De Gruttola, Nonparametric analysis of truncated survival data, with application to AIDS,, Biometrika, 75 (1988), 515. doi: doi:10.1093/biomet/75.3.515. Google Scholar [25] M. Lipsitch, T. Cohen, B. Cooper, J. M. Robins, S. Ma, L. James, G. Gopalakrishna, S. K. Chew, C. C. Tan, M. H. Samore, D. Fisman and M. Murray, Transmission dynamics and control of severe acute respiratory syndrome,, Science, 38 (2003), 115. Google Scholar [26] N. Masurel and W. M. Marine, Recycling of Asian and Hong Kong influenza A virus hemagglutinins in man,, Am. J. Epidemiol., 97 (1973), 44. Google Scholar [27] C. E. Mills, J. M. Robins and M. Lipsitch, Transmissibility of 1918 pandemic influenza,, Nature, 432 (2004), 904. doi: doi:10.1038/nature03063. Google Scholar [28] J. Mulder, N. Masurel and J. F. P. Hers, De Aziatische-influenza-pandemie van 1957 (Dutch),, Ned. Tijdschr. Geneeskd., 102 (1958), 1992. Google Scholar [29] H. Nishiura, Correcting the actual reproduction number: A simple method to estimate $R_0$ from early epidemic growth data,, Int. J. Environ. Res. Public. Health., 7 (2010), 291. doi: doi:10.3390/ijerph7010291. Google Scholar [30] H. Nishiura, Early efforts in modeling the incubation period of infectious diseases with an acute course of illness,, Emerg. Themes. Epidemiol., 4 (2007). doi: doi:10.1186/1742-7622-4-2. Google Scholar [31] H. Nishiura, Epidemiology of a primary pneumonic plague in Kantoshu, Manchuria, from 1910 to 1911: Statistical analysis of individual records collected by the Japanese Empire,, Int. J. Epidemiol., 35 (2006), 1059. doi: doi:10.1093/ije/dyl091. Google Scholar [32] H. Nishiura, Time variations in the transmissibility of pandemic influenza in Prussia, Germany, from 1918-19,, Theor. Biol. Med. Model., 4 (2007). doi: doi:10.1186/1742-4682-4-20. Google Scholar [33] H. Nishiura and G. Chowell, The effective reproduction number as a prelude to statistical estimtion of time-dependent epidemic trends,, in, (2009), 103. doi: doi:10.1007/978-90-481-2313-1_5. Google Scholar [34] H. Nishiura, G. Chowell, H. Heesterbeek and J. Wallinga, The ideal reporting interval for an epidemic to objectively interpret the epidemiological time course,, J. R. Soc. Interface., 7 (2010), 297. doi: doi:10.1098/rsif.2009.0153. Google Scholar [35] H. Nishiura, G. Chowell, M. Safan and C. Castillo-Chavez, Pros and cons of estimating the reproduction number from early epidemic growth rate of influenza A (H1N1) 2009,, Theor. Biol. Med. Model., 7 (2010). doi: doi:10.1186/1742-4682-7-1. Google Scholar [36] H. Nishiura, M. Schwehm, M. Kakehashi and M. Eichner, Transmission potential of primary pneumonic plague: Time inhomogeneous evaluation based on historical documents of the transmission network,, J. Epidemiol. Community. Health., 60 (2006), 640. doi: doi:10.1136/jech.2005.042424. Google Scholar [37] W. Pickles, "Epidemiology in Country Practice,", John Wright and Sons, (1939). Google Scholar [38] M. G. Roberts and J. A. Heesterbeek, Model-consistent estimation of the basic reproduction number from the incidence of an emerging infection,, J. Math. Biol., 55 (2007), 803. doi: doi:10.1007/s00285-007-0112-8. Google Scholar [39] G. Scalia Tomba, A. Svensson, T. Asikainen and J. Giesecke, Some model based considerations on observing generation times for communicable diseases,, Math. Biosci., 223 (2010), 24. doi: doi:10.1016/j.mbs.2009.10.004. Google Scholar [40] A. Svensson, A note on generation times in epidemic models,, Math. Biosci., 208 (2007), 300. doi: doi:10.1016/j.mbs.2006.10.010. Google Scholar [41] Temporary Quarantine Section, K. Totokufu, "Epidemic Record of Plague During 1910-1911" (Meiji 43,4-nen 'Pest' Ryuko-shi),, Manchurian Daily Press, (1912). Google Scholar [42] E. Vynnycky and W. J. Edmunds, Analyses of the 1957 (Asian) influenza pandemic in the United Kingdom and the impact of school closures,, Epidemiol. Infect., 136 (2008), 166. doi: doi:10.1017/S0950268807008369. Google Scholar [43] E. Vynnycky and P. E. Fine, Lifetime risks, incubation period, and serial interval of tuberculosis,, Am. J. Epidemiol., 152 (2000), 247. doi: doi:10.1093/aje/152.3.247. Google Scholar [44] E. Vynnycky, A. Trindall and P. Mangtani, Estimates of the reproduction numbers of Spanish influenza using morbidity data,, Int. J. Epidemiol., 36 (2007), 881. doi: doi:10.1093/ije/dym071. Google Scholar [45] J. Wallinga and M. Lipsitch, How generation intervals shape the relationship between growth rates and reproductive numbers,, Proc. R. Soc. Lond. Ser. B., 274 (2007), 599. doi: doi:10.1098/rspb.2006.3754. Google Scholar [46] J. Wallinga and P. Teunis, Different epidemic curves for severe acute respiratory syndrome reveal similar impacts of control measures,, Am. J. Epidemiol., 160 (2004), 509. doi: doi:10.1093/aje/kwh255. Google Scholar [47] L. F. White and M. Pagano, A likelihood-based method for real-time estimation of the serial interval and reproductive number of an epidemic,, Stat. Med., 27 (2007), 2999. doi: doi:10.1002/sim.3136. Google Scholar [48] P. Yan, 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,, J. Theor. Biol., 251 (2008), 238. doi: doi:10.1016/j.jtbi.2007.11.027. Google Scholar
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