# 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:
 [1] J. M. Alho and B. D. Spencer, "Statistical Demography and Forecasting," Springer, New York, 2005. [2] R. M. Anderson and R. M. May, "Infectious Diseases of Humans: Dynamics and Control," Oxford University Press, Oxford, 1991. [3] N. T. J. Bailey, "The Mathematical Theory of Infectious Diseases and its Applications," 2nd edition, Charles Griffin and Company Ltd., London, 1975. [4] T. Burr and G. Chowell, The reproduction number $R(t)$ in structured and nonstructured populations, Math. Biosci. Eng., 6 (2009), 239-259. doi: doi:10.3934/mbe.2009.6.239. [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), 110-113. [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), 591-597. doi: doi:10.1093/aje/kwj274. [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), 155-166. doi: doi:10.1098/rsif.2006.0161. [8] O. Diekmann and J. A. P. Heesterbeek, "Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation," Wiley, New York, 2000. [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), 365-382. doi: doi:10.1007/BF00178324. [10] K. Dietz, The estimation of the basic reproduction number for infectious diseases, Stat. Methods. Med. Res., 2 (1993), 23-41. doi: doi:10.1177/096228029300200103. [11] L. I. Dublin and A. J. Lotka, On the true rate of natural increase, J. Am. Stat. Assoc., 151 (1925), 305-339. doi: doi:10.2307/2965517. [12] P. E. M. Fine, The interval between successive cases of an infectious disease, Am. J. Epidemiol., 158 (2003), 755-760. doi: doi:10.1093/aje/kwg251. [13] C. Fraser, Estimating individual and household reproduction numbers in an emerging epidemic, PLoS One, 2 (2007), e758. doi: doi:10.1371/journal.pone.0000758. [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), 6146-6151. doi: doi:10.1073/pnas.0307506101. [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), e349. doi: doi:10.1371/journal.pone.0000349. [16] N. C. Grassly and C. Fraser, Mathematical models of infectious disease transmission, Nat. Rev. Microbiol., 6 (2008), 477-487. [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), 121-127. doi: doi:10.1098/rspb.2002.2191. [18] R. E. Hope Simpson, The period of transmission in certain epidemic diseases: An observational method for its discovery, Lancet, 2 (1948), 755-760. doi: doi:10.1016/S0140-6736(48)91328-2. [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), 77-89. doi: doi:10.1016/j.mbs.2008.08.005. [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), 360-372. doi: doi:10.2307/2289919. [21] E. Kenah, M. Lipsitch and J. M. Robins, Generation interval contraction and epidemic data analysis, Math. Biosci., 213 (2008), 71-79. doi: doi:10.1016/j.mbs.2008.02.007. [22] W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics. I., Proc. R. Soc. Ser. A., 115 (1927), 700-721, reprinted in Bull. Math. Biol., 53 (1991), 33-55. doi: doi:10.1016/S0092-8240(05)80040-0. [23] N. Keyfitz, "Applied Mathematical Demography," John Wiley and Sons, New York, 1977. [24] S. W. Lakagos, L. M. Barraj and V. De Gruttola, Nonparametric analysis of truncated survival data, with application to AIDS, Biometrika, 75 (1988), 515-523. doi: doi:10.1093/biomet/75.3.515. [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), 115-126. [26] N. Masurel and W. M. Marine, Recycling of Asian and Hong Kong influenza A virus hemagglutinins in man, Am. J. Epidemiol., 97 (1973), 44-49. [27] C. E. Mills, J. M. Robins and M. Lipsitch, Transmissibility of 1918 pandemic influenza, Nature, 432 (2004), 904-906. doi: doi:10.1038/nature03063. [28] J. Mulder, N. Masurel and J. F. P. Hers, De Aziatische-influenza-pandemie van 1957 (Dutch), Ned. Tijdschr. Geneeskd., 102 (1958), 1992-1999. [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), 291-302. doi: doi:10.3390/ijerph7010291. [30] H. Nishiura, Early efforts in modeling the incubation period of infectious diseases with an acute course of illness, Emerg. Themes. Epidemiol., 4 (2007), 2. doi: doi:10.1186/1742-7622-4-2. [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), 1059-1065. doi: doi:10.1093/ije/dyl091. [32] H. Nishiura, Time variations in the transmissibility of pandemic influenza in Prussia, Germany, from 1918-19, Theor. Biol. Med. Model., 4 (2007), 20. doi: doi:10.1186/1742-4682-4-20. [33] H. Nishiura and G. Chowell, The effective reproduction number as a prelude to statistical estimtion of time-dependent epidemic trends, in "Mathematical and Statistical Estimation Approaches in Epidemiology" (eds. G. Chowell, J. M. Hyman, L. M. A. Bettencourt and C. Castillo-Chavez), Springer, New York, (2009), 103-121. doi: doi:10.1007/978-90-481-2313-1_5. [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), 297-307. doi: doi:10.1098/rsif.2009.0153. [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), 1. doi: doi:10.1186/1742-4682-7-1. [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), 640-645. doi: doi:10.1136/jech.2005.042424. [37] W. Pickles, "Epidemiology in Country Practice," John Wright and Sons, Bristol, 1939. [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), 803-816. doi: doi:10.1007/s00285-007-0112-8. [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), 24-31. doi: doi:10.1016/j.mbs.2009.10.004. [40] A. Svensson, A note on generation times in epidemic models, Math. Biosci., 208 (2007), 300-311. doi: doi:10.1016/j.mbs.2006.10.010. [41] Temporary Quarantine Section, K. Totokufu, "Epidemic Record of Plague During 1910-1911" (Meiji 43,4-nen 'Pest' Ryuko-shi), Manchurian Daily Press, Dalian, 1912. [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), 166-179. doi: doi:10.1017/S0950268807008369. [43] E. Vynnycky and P. E. Fine, Lifetime risks, incubation period, and serial interval of tuberculosis, Am. J. Epidemiol., 152 (2000), 247-263. doi: doi:10.1093/aje/152.3.247. [44] E. Vynnycky, A. Trindall and P. Mangtani, Estimates of the reproduction numbers of Spanish influenza using morbidity data, Int. J. Epidemiol., 36 (2007), 881-889. doi: doi:10.1093/ije/dym071. [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), 599-604. doi: doi:10.1098/rspb.2006.3754. [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), 509-516. doi: doi:10.1093/aje/kwh255. [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), 2999-3016. doi: doi:10.1002/sim.3136. [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), 238-252. doi: doi:10.1016/j.jtbi.2007.11.027.

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##### References:
 [1] J. M. Alho and B. D. Spencer, "Statistical Demography and Forecasting," Springer, New York, 2005. [2] R. M. Anderson and R. M. May, "Infectious Diseases of Humans: Dynamics and Control," Oxford University Press, Oxford, 1991. [3] N. T. J. Bailey, "The Mathematical Theory of Infectious Diseases and its Applications," 2nd edition, Charles Griffin and Company Ltd., London, 1975. [4] T. Burr and G. Chowell, The reproduction number $R(t)$ in structured and nonstructured populations, Math. Biosci. Eng., 6 (2009), 239-259. doi: doi:10.3934/mbe.2009.6.239. [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), 110-113. [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), 591-597. doi: doi:10.1093/aje/kwj274. [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), 155-166. doi: doi:10.1098/rsif.2006.0161. [8] O. Diekmann and J. A. P. Heesterbeek, "Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation," Wiley, New York, 2000. [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), 365-382. doi: doi:10.1007/BF00178324. [10] K. Dietz, The estimation of the basic reproduction number for infectious diseases, Stat. Methods. Med. Res., 2 (1993), 23-41. doi: doi:10.1177/096228029300200103. [11] L. I. Dublin and A. J. Lotka, On the true rate of natural increase, J. Am. Stat. Assoc., 151 (1925), 305-339. doi: doi:10.2307/2965517. [12] P. E. M. Fine, The interval between successive cases of an infectious disease, Am. J. Epidemiol., 158 (2003), 755-760. doi: doi:10.1093/aje/kwg251. [13] C. Fraser, Estimating individual and household reproduction numbers in an emerging epidemic, PLoS One, 2 (2007), e758. doi: doi:10.1371/journal.pone.0000758. [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), 6146-6151. doi: doi:10.1073/pnas.0307506101. [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), e349. doi: doi:10.1371/journal.pone.0000349. [16] N. C. Grassly and C. Fraser, Mathematical models of infectious disease transmission, Nat. Rev. Microbiol., 6 (2008), 477-487. [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), 121-127. doi: doi:10.1098/rspb.2002.2191. [18] R. E. Hope Simpson, The period of transmission in certain epidemic diseases: An observational method for its discovery, Lancet, 2 (1948), 755-760. doi: doi:10.1016/S0140-6736(48)91328-2. [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), 77-89. doi: doi:10.1016/j.mbs.2008.08.005. [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), 360-372. doi: doi:10.2307/2289919. [21] E. Kenah, M. Lipsitch and J. M. Robins, Generation interval contraction and epidemic data analysis, Math. Biosci., 213 (2008), 71-79. doi: doi:10.1016/j.mbs.2008.02.007. [22] W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics. I., Proc. R. Soc. Ser. A., 115 (1927), 700-721, reprinted in Bull. Math. Biol., 53 (1991), 33-55. doi: doi:10.1016/S0092-8240(05)80040-0. [23] N. Keyfitz, "Applied Mathematical Demography," John Wiley and Sons, New York, 1977. [24] S. W. Lakagos, L. M. Barraj and V. De Gruttola, Nonparametric analysis of truncated survival data, with application to AIDS, Biometrika, 75 (1988), 515-523. doi: doi:10.1093/biomet/75.3.515. [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), 115-126. [26] N. Masurel and W. M. Marine, Recycling of Asian and Hong Kong influenza A virus hemagglutinins in man, Am. J. Epidemiol., 97 (1973), 44-49. [27] C. E. Mills, J. M. Robins and M. Lipsitch, Transmissibility of 1918 pandemic influenza, Nature, 432 (2004), 904-906. doi: doi:10.1038/nature03063. [28] J. Mulder, N. Masurel and J. F. P. Hers, De Aziatische-influenza-pandemie van 1957 (Dutch), Ned. Tijdschr. Geneeskd., 102 (1958), 1992-1999. [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), 291-302. doi: doi:10.3390/ijerph7010291. [30] H. Nishiura, Early efforts in modeling the incubation period of infectious diseases with an acute course of illness, Emerg. Themes. Epidemiol., 4 (2007), 2. doi: doi:10.1186/1742-7622-4-2. [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), 1059-1065. doi: doi:10.1093/ije/dyl091. [32] H. Nishiura, Time variations in the transmissibility of pandemic influenza in Prussia, Germany, from 1918-19, Theor. Biol. Med. Model., 4 (2007), 20. doi: doi:10.1186/1742-4682-4-20. [33] H. Nishiura and G. Chowell, The effective reproduction number as a prelude to statistical estimtion of time-dependent epidemic trends, in "Mathematical and Statistical Estimation Approaches in Epidemiology" (eds. G. Chowell, J. M. Hyman, L. M. A. Bettencourt and C. Castillo-Chavez), Springer, New York, (2009), 103-121. doi: doi:10.1007/978-90-481-2313-1_5. [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), 297-307. doi: doi:10.1098/rsif.2009.0153. [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), 1. doi: doi:10.1186/1742-4682-7-1. [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), 640-645. doi: doi:10.1136/jech.2005.042424. [37] W. Pickles, "Epidemiology in Country Practice," John Wright and Sons, Bristol, 1939. [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), 803-816. doi: doi:10.1007/s00285-007-0112-8. [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), 24-31. doi: doi:10.1016/j.mbs.2009.10.004. [40] A. Svensson, A note on generation times in epidemic models, Math. 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B., 274 (2007), 599-604. doi: doi:10.1098/rspb.2006.3754. [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), 509-516. doi: doi:10.1093/aje/kwh255. [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), 2999-3016. doi: doi:10.1002/sim.3136. [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), 238-252. doi: doi:10.1016/j.jtbi.2007.11.027.
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