The basic reproduction number, R 0, a summary measure of the transmission potential of an infectious disease, is estimated from early epidemic growth rate, but a likelihood-based method for the estimation has yet to be developed. The present study corrects the concept of the actual reproduction number, offering a simple framework for estimating R 0 without assuming exponential growth of cases. The proposed method is applied to the HIV epidemic in European countries, yielding R 0 values ranging from 3.60 to 3.74, consistent with those based on the Euler-Lotka equation. The method also permits calculating the expected value of R 0 using a spreadsheet.
References
[1]
Diekmann, O; Heesterbeek, JA; Metz, JA. On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations. J. Math. Biol？1990, 28, 365–381. 2117040
[2]
Dietz, K. The estimation of the basic reproduction number for infectious diseases. Stat. Methods. Med. Res？1993, 2, 23–41, doi:10.1177/096228029300200103. 8261248
[3]
Becker, NG. Analysis of Infectious Disease Data; Chapman & Hall: Boca Raton, FL, USA, 1989.
[4]
Smith, CE. Factors in the transmission of virus infections from animal to man. Sci Basis Med Ann Rev？1964, 125–150.
[5]
Kendall, DG. Deterministic and stochastic epidemics in closed populations. In Proceedings of the 3rd Berkeley Symposium on Mathematical Statistics and Probability; University of California Press: Berkeley, CA, USA, 1956; pp. 149–165.
[6]
Ma, J; Earn, DJ. Generality of the final size formula for an epidemic of a newly invading infectious disease. Bull. Math. Biol？2006, 68, 679–702, doi:10.1007/s11538-005-9047-7. 16794950
[7]
Heffernan, JM; Wahl, LM. Improving estimates of the basic reproductive ratio: using both the mean and the dispersal of transition times. Theor. Pop. Biol？2006, 70, 135–145, doi:10.1016/j.tpb.2006.03.003.
[8]
Massad, E; Coutinho, FA; Burattini, MN; Amaku, M. Estimation of R0 from the initial phase of an outbreak of a vector-borne infection. Trop. Med. Int. Health？2009, 15, 120–126. 19891761
[9]
Wallinga, J; Lipsitch, M. How generation intervals shape the relationship between growth rates and reproductive numbers. Proc. Roy. Soc. Lon. Ser. B？2007, 274, 599–604, doi:10.1098/rspb.2006.3754.
[10]
Svensson, A. A note on generation times in epidemic models. Math. Biosci？2007, 208, 300–311, doi:10.1016/j.mbs.2006.10.010. 17174352
[11]
Diekmann, O; Heesterbeek, JAP. Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation; John Wiley & Son: Chichester, UK, 2000.
[12]
Garske, T; Clarke, P; Ghani, AC. The transmissibility of highly pathogenic avian influenza in commercial poultry in industrialised countries. PLoS ONE？2007, 2, e349, doi:10.1371/journal.pone.0000349. 17406673
[13]
White, LF; Pagano, M. A likelihood-based method for real-time estimation of the serial interval and reproductive number of an epidemic. Stat. Med？2008, 27, 2999–3016, doi:10.1002/sim.3136. 18058829
[14]
Amundsen, EJ; Stigum, H; Rottingen, JA; Aalen, OO. 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. Epidemiol. Infect？2004, 132, 1139–1149, doi:10.1017/S0950268804002997. 15635973
[15]
White, PJ; Ward, H; Garnett, GP. Is HIV out of control in the UK? An example of analysing patterns of HIV spreading using incidence-to-prevalence ratios. AIDS？2006, 20, 1898–1901, doi:10.1097/01.aids.0000244213.23574.fa. 16954735
[16]
Chowell, G; Nishiura, H. Quantifying the transmission potential of pandemic influenza. Phys. Life. Rev？2008, 5, 50–77, doi:10.1016/j.plrev.2007.12.001.
[17]
Fraser, C. Estimating individual and household reproduction numbers in an emerging epidemic. PLoS ONE？2007, 2, e758, doi:10.1371/journal.pone.0000758. 17712406
[18]
Nishiura, H; Chowell, G. The effective reproduction number as a prelude to statistical estimation of time-dependent epidemic trends. In Mathematical and Statistical Estimation Approaches in Epidemiology; Chowell, G, Hyman, JM, Bettencourt, LMA, Castillo-Chavez, C, Eds.; Springer: Dordrecht, Germany, 2009; pp. 103–121.
[19]
Shiboski, SC; Jewell, NP. Statistical analysis of the time dependence of HIV infectivity based on partner study data. J. Amer. Statist. Assn？1991, 87, 360–372.
[20]
Hollingsworth, TD; Anderson, RM; Fraser, C. HIV-1 transmission, by stage of infection. Int. J. Infect. Dis？2008, 198, 687–693, doi:10.1086/590501.
[21]
Wawer, MJ; Gray, RH; Sewankambo, NK; Serwadda, D; Li, X; Laeyendecker, O; Kiwanuka, N; Kigozi, G; Kiddugavu, M; Lutalo, T; Nalugoda, F; Wabwire-Mangen, F; Meehan, MP; Quinn, TC. Rates of HIV-1 transmission per coital act, by stage of HIV-1 infection, in Rakai, Uganda. Int. J. Infect. Dis？2005, 191, 1403–1409, doi:10.1086/429411.
[22]
Abu-Raddad, LJ; Longini, IM. No HIV stage is dominant in driving the HIV epidemic in sub-Saharan Africa. AIDS？2008, 22, 1055–1061, doi:10.1097/QAD.0b013e3282f8af84. 18520349
[23]
Artzrouni, M. Back-calculation and projection of the HIV/AIDS epidemic among homosexual/bisexual men in three European countries: Evaluation of past projections and updates allowing for treatment effects. Eur. J. Epidemiol？2004, 19, 171–179. 15074573
[24]
Brookmeyer, R; Gail, MH. AIDS Epidemiology: A Quantitative Approach (Monographs in Epidemiology and Biostatistics); Oxford University Press: New York, NY, USA, 1994.
Nishiura, H. Lessons from previous predictions of HIV/AIDS in the United States and Japan: epidemiologic models and policy formulation. Epidemiol. Perspect. Innov？2007, 4, 3, doi:10.1186/1742-5573-4-3. 17567907
[27]
Bailey, NTJ. The Elements of Stochastic Processes with Applications to the Natural Sciences; Wiley: New York, NY, USA, 1964.
[28]
Nishiura, H; Castillo-Chavez, C; Safan, M; Chowell, G. Transmission potential of the new influenza A(H1N1) virus and its age-specificity in Japan. Euro. Surveill？2009, 14, 19227. 19497256
[29]
Wallinga, J; Teunis, P. Different epidemic curves for severe acute respiratory syndrome reveal similar impacts of control measures. Amer. J. Epidemiol？2004, 160, 509–516, doi:10.1093/aje/kwh255.
[30]
Haydon, DT; Chase-Topping, M; Shaw, DJ; Matthews, L; Friar, JK; Wilesmith, J; Woolhouse, ME. The construction and analysis of epidemic trees with reference to the 2001 UK foot-and-mouth outbreak. Proc. Roy. Soc. Lon. Ser. B？2003, 270, 121–127, doi:10.1098/rspb.2002.2191.
[31]
Nishiura, H; Schwehm, M; Kakehashi, M; Eichner, M. Transmission potential of primary pneumonic plague: time inhomogeneous evaluation based on historical documents of the transmission network. J. Epidemiol. Community Health？2006, 60, 640–645, doi:10.1136/jech.2005.042424. 16790838
[32]
Jacquez, JA; Simon, CP; Koopman, JS. The reproduction number in deterministic models of contagious diseases. Comments Theor. Biol？1991, 2, 159–209.
[33]
Grassly, NC; Fraser, C. Mathematical models of infectious disease transmission. Nat. Rev. Microbiol？2008, 6, 477–487. 18533288
[34]
Nishiura, H; Chowell, G; Heesterbeek, H; Wallinga, J. The ideal reporting interval for an epidemic to objectively interpret the epidemiological time course. J. R. Soc. Interface？2010, 7, 297–307, doi:10.1098/rsif.2009.0153. 19570792
[35]
Nishiura, H; Chowell, G; Safan, M; Castillo-Chavez, C. Pros and cons of estimating the reproduction number from early epidemic growth rate of influenza A (H1N1) 2009. Theor. Biol. Med. Model？2010, 7, 1, doi:10.1186/1742-4682-7-1. 20056004
[36]
Hethcote, HW; Yorke, JA. Gonorrhea Transmission Dynamics and Control (Lecture Notes in Biomathematics, 56); Springer-Verlag: Berlin, Germany, 1980.
[37]
Muller, H; Bauch, C. When do sexual partnerships need to be accounted for in transmission models of human papilloma virus. Int J Environ Res Public Health？2010. 20195428
