The basic reproduction number of infectious diseases: Computation and estimation using compartmental epidemic models

Mathematical and Statistical Estimation Approaches in Epidemiology

Volumn , Issue , 2009, Pages 1-30

  Chowell, Gerardo ^a,b   Brauer, Fred ^c  

^a ARIZONA STATE UNIVERSITY   (United States)

^b NATIONAL INSTITUTES OF HEALTH   (United States)

^c UNIVERSITY OF BRITISH COLUMBIA   (Canada)

Author keywords

Basic reproduction number; Epidemiology; Influenza; Model; Pandemic

Indexed keywords

EID: 77954692449     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1007/978-90-481-2313-1_1     Document Type: Chapter

References (90)

1. Morbidity and Mortality Weekly Reports (MMWR). 121 US cities mortality surveillance (1968-1971) [http://www.cdc.gov/mmwr/].
2. Anderson RM and May RM (1982) Directly transmitted infectious diseases: Control by vaccination. Science 215:1053-1060.
3. Anderson RM and May RM (1991) Infectious Diseases of Humans: Dynamics and Control. Oxford University Press, Oxford.
4. Andreasen V, Viboud C and Simonsen L (2008) Epidemiologic characterization of the summer wave of the 1918 influenza pandemic in Copenhagen: Implications for pandemic control strategies. J. Infect. Dis. 197:270-278.
5. Arino J, Brauer F, van den Driessche P, Watmough J and Wu J (2007) A final size relation for epidemic models. Math. Biosc. Eng. 4:159-176.
6. Becker NG (1989) Analysis of Infectious Disease Data. Chapman and Hall, New York.
7. Berman A and Plemmons RJ (1994) Nonnegative Matrices in the Mathematical Sciences, Classics in Applied Mathematics 9, SIAM, Philadelphia.
8. Bickel P and Doksum KA (1977) Mathematical Statistics. Holden-Day, Oakland, California.
9. Blower SM and Mclean AR (1994) Prophylactic vaccines, risk behavior change, and the probability of eradicating HIV in San Francisco. Science 265: 1451.
10. Brauer F and Castillo-Chavez C (2000) Mathematical Models in Population Biology and Epidemiology. Springer-Verlag, New York.
11. Brauer F (2004) Backward bifurcations in simple vaccination models. J. Math. Anal. Appl. 298:418-431.
12. Brauer F (2005) The Kermack-McKendrick model revisited. Math. Biosc. 198: 119-131.
13. Brauer F (2008) Compartmental models in epidemiology. In: Brauer F, van den Driessche P, and Wu J (eds) Mathematical Epidemiology, Lecture Notes in Mathematics, Mathematical Biosciences Subseries 1945 Springer-Verlag, Berlin-Heidelberg: 19-79.
14. Cadham MFT (1919) The use of a vaccine in the recent epidemic of influenza. Can. Med. Assoc. J. 9:519-527.
15. Castillo-Chavez C and Thieme HR (1993) Asymptotically autonomous epidemic models. In: Arino O, Axelrod D, Kimmel M, Langlais M (eds) Mathematical Population Dynamics: Analysis of Heterogeneity, Vol. 1, Theory of Epidemics, Wuerz, Winnipeg: 33-50.
16. Castillo-Chavez C, Feng Z and Huang W (2002) On the computation of R0 and its role on global stability, in: Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction, IMA Volume 125. Springer-Veralg, Berlin pp. 229-250.
17. Carrat F, Vergu E, Ferguson NM, et al. (2008) Time lines of infection and disease in human influenza: A review of volunteer challenge studies. Am. J. Epidemiol. 167(7): 775-785.
18. Cauchemez S, Boelle PY, Thomas G and Valleron AJ (2006) Estimating in real time the efficacy of measures to control emerging communicable diseases. Am. J. Epidemiol. 164:591-597.
19. Chowell G, Ammon CE, Hengartner NW and Hyman JM (2006) Transmission dynamics of the great influenza pandemic of 1918 in Geneva, Switzerland: Assessing the effects of hypothetical interventions. J. Theor. Biol. 241:193-204.
20. Chowell G, Nishiura H and Bettencourt LM (2007) Comparative estimation of the reproduction number for pandemic influenza from daily case notification data. J. R. Soc. Interface 4:155-166.
21. Chowell G, Miller MA and Viboud C (2008) Seasonal influenza in the United States, France, and Australia: Transmission and prospects for control. Epidemiol. Infect. 136:852-64.
22. Chowell G, Ammon CE, Hengartner NW and Hyman JM (2007) Estimating the reproduction number from the initial phase of the Spanish flu pandemic waves in Geneva, Switzerland. Math. Biosci. Eng. 4:457-470.
23. Chowell G, Bettencourt LMA, Johnson N, Alonso WJ and Viboud C (2008) The 1918-1919 influenza pandemic in England and Wales: Spatial patterns in transmissibility and mortality impact. Proc. R. Soc. B 275:501-509.
24. Chowell G, Bettencourt LMA, Johnson NPAS, Alonso WJ and Viboud C (2008) The 1918- 1919 influenza pandemic in England and Wales: Spatial patterns in transmissibility and mortality impact. Proc. Biol. Sci. 275:501-9.
25. Chowell G and Nishiura H (2008) Quantifying the transmission potential of pandemic influenza. Phys. Life Rev. 5, 50-77.
26. Cunha BA (2004) Influenza: Historical aspects of epidemics and pandemics. Infect. Dis. Clin. North Am. 18:141-155.
27. Davidian M and Giltinan DM (1995) Nonlinear Models for Repeated Measurement data. Monographs on Statistics and Applied Probability 62. Chapman and Hall, New York.
28. De Jong MC, Diekmann O and Heesterbeek JA (1994) The computation of R0 for discretetime epidemic models with dynamic heterogeneity. Math. Biosci. 119:97-114.
29. Diekmann O, Heesterbeek JAP and Metz JAJ (1990) On the definition and the computation of the basic reproductive ratioR0 in models for infectious diseases in heterogeneous populations. J. Math. Biol. 28:365-382.
30. Diekmann O and Heesterbeek JAP (2000) Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation. John Wiley and Sons, New York.
31. Dietz K (1988) Mathematical models for transmission and control of malaria. in: Malaria, Principles and Practice of Malariology, eds W.H. Wernsdorfer and I. McGregor. Churchill Livingstone, Edinburgh. pp.1091-1133.
32. Dietz K (1993) The estimation of the basic reproduction number for infectious diseases. Stat. Methods Med. Res. 2:23-41.
33. Dushoff J, Huang Wand, Castillo-Chavez C (1998) Backwards bifurcations and catastrophe in simple models of total diseases. J. Math. Biol. 36:227-248.
34. Efron B and Tibshirani RJ (1986) Bootstrap methods for standard errors, confidence intervals, and other measures of statistical accuracy. Stat. Sci. 1:54-75.
35. Ferguson NM, Donnelly CA and Anderson RM (2001) Transmission intensity and impact of control policies on the foot and mouth epidemc in Great Britain. Nature 413:542-548.
36. Ferguson NM, Cummings DAT, Cauchemez S, Fraser C, Riley S, Meeyai A, Iamsirithaworn S and Burke DS (2005) Strategies for containing an emerging influenza pandemic in Southeast Asia. Nature 437:209-214.
37. Fine PE (1993) Herd immunity: History, theory, practice. Epidemiol. Rev. 15:265-302.
38. Gani R, Hughes H, Fleming DM, Griffin T, Medlock J and Leach S (2005) Potential impact of antiviral drug use during influenza pandemic. Emerg. Infect. Dis. 11:1355-1362.
39. Hadeler KP and Castillo-Chavez C (1995) A core group model for disease transmission. Math Biosc. 128:41-55.
40. Hadeler KP and van den Driessche P (1997) Backward bifurcation in epidemic control. Math. Biosc. 146:15-35.
41. Halloran ME, Haber M, Longini IM and Struchiner CJ (1991) Direct and indirect effects in vaccine efficacy and effectiveness. Am. J. Epidemiol. 133:323-331.
42. Heesterbeek JAP (2002) A brief history of R0 and a recipe for its calculation. Acta Biotheor. 50:189-204.
43. Heffernan JM, Smith RJ and Wahl LM (2005) Perspectives on the basic reproductive ratio. J. R. Soc. Interface 2:281-293.
44. Heffernan JM and Wahl LM (2006) Improving estimates of the basic reproductive ratio: Using both the mean and the dispersal of transition times. Theor. Popul. Biol. 70:135-145.
45. Hethcote HW, Stech HW and van den Driessche P (1981) Periodicity and stability in epidemic models: A survey. In: Busenberg S and Cooke KL (eds.) Differential Equations and Applications in Ecology, Epidemics and Population Problems, Springer-Verlag, Berlin-Heidelberg: 65-82.
46. Hethcote HW and Levin SA (1989) Periodicity in epidemic models. In : Levin SA, Hallam TG, Gross LG (eds) Applied Mathematical Ecology. Biomathematics 18, Springer-Verlag, Berlin-Heidelberg-New York: 193-211.
47. Hethcote HW (2000) The mathematics of infectious diseases. SIAM Rev. 42:599-653.
48. Hyman J, Mand, Li J (2000) An intuitive formulation for the reproductive number for the spread of diseases in heterogeneous populations. Math. Biosci. 167:65-86.
49. Johnson NP and Mueller J (2002) Updating the accounts: Global mortality of the 1918-1920 "Spanish" influenza pandemic. Bull. Hist. Med. 76:105-115.
50. Jones EW (2005) Co-operarion in all human endeavour: Quarantine and immigrant disease vectors in the 1918-1919 influenza pandemic in winnipeg. Can. Bull. Med. Hist. 22:57-82.
51. Keeling MJ and Grenfell BT (2000) Individual-based perspectives on R(0). J. Theor. Biol. 203:51-61.
52. Keeling MJ and Eames KT (2005) Networks and epidemic models. J. R. Soc. Interface 2:295- 307.
53. Kendall DG (1956) Deterministic and stochastic epidemics in closed populations. in: Third Berkeley Symposium on Mathematical Statistics and Probability 4, ed P. Newman. University of California Press, New York. pp. 149-165.
54. Kermack WO and McKendrick AG (1927) Contributions to the mathematical theory of epidemics - I. Proc. R. Soc. A 115:700-721 (reprinted in Bulletin of Mathematical Biology 53 (1991) 33-55).
55. Kermack WO and McKendrick AG (1932) Contributions to the mathematical theory of epidemics, part. II. Proc. Roy. Soc. London 138:55-83.
56. Kermack WO and McKendrick AG (1933) Contributions to the mathematical theory of epidemics, part. III. Proc. Roy. Soc. London 141:94-112.
57. Kribs-Zaleta CM and Velasco-Hernandez JX (2000) A simple vaccination model with multiple endemic states. Math Biosc. 164:183-201.
58. Lipsitch M, Cohen T, Cooper B, Robins JM, Ma S, James L, Gopalakrishna G, Chew SK, Tan CC, Samore MH, Fisman D and Murray M (2003) Transmission dynamics and control of severe acute respiratory syndrome. Science 300:1966-1970.
59. Lloyd AL (2001) Destabilization of epidemic models with the inclusion of realistic distributions of infectious periods. Proc. R. Soc. London B 268:985-993.
60. Lloyd AL (2001) Realistic distributions of infectious periods in epidemic models: Changing patterns of persistence and dynamics. Theor. Popul. Biol. 60:59-71.
61. Ma J and Earn DJ (2006) Generality of the final size formula for an epidemic of a newly invading infectious disease. Bull. Math. Biol. 68:679-702.
62. MacKellar L (2007) Pandemic influenza: A review. Popul. Dev. Rev. 33:429-451.
63. Markel H, Lipman HB, Navarro JA, Sloan A, Michalsen JR, Stern AM and Cetron MS (2007) Nonpharmaceutical interventions implemented by US cities during the 1918-1919 influenza pandemic. JAMA 298:644-654.
64. Markus L (1956) Asymptotically autonomous differential systems. In: Lefschetz S (ed) Contributions to the Theory of Nonlinear Oscillations III. Annals of Mathematics Studies 36, Princeton University Press, Princeton, N.J.: 17-29.
65. Massad E, Burattini MN, Coutinho FA and Lopez LF (2007) The 1918 influenza A epidemic in the city of Sao Paulo, Brazil. Med. Hypotheses 68:442-445.
66. Mills CE, Robins JM and Lipsitch M (2004) Transmissibility of 1918 pandemic influenza. Nature 432:904-906.
67. Murray CJ, Lopez AD, Chin B, Feehan D and Hill KH (2006) Estimation of potential global pandemic influenza mortality on the basis of vital registry data from the 1918-1920 pandemic: A quantitative analysis. Lancet 368:2211-2218.
68. Nishiura H, Dietz K and Eichner M (2006) The earliest notes on the reproduction number in relation to herd immunity: Theophil Lotz and smallpox vaccination. J. Theor. Biol. 241:964- 967.
69. Nishiura H (2006) Mathematical and statistical analyses of the spread of dengue. Dengue Bull. 30:51-67.
70. Nishiura H (2007) Time variations in the transmissibility of pandemic influenza in Prussia, Germany, from 1918 to 1919. Theor. Biol. Med. Model. 4:20.
71. Nishiura H and Inaba H (2007) Discussion: Emergence of the concept of the basic reproduction number from mathematical demography. J. Theor. Biol. 244:357-364.
72. Patterson KD and Pyle GF (1991) The geography and mortality of the 1918 influenza pandemic. Bull. Hist. Med. 65:4-21.
73. RobertsMG and Heesterbeek JA (2007) Model-consistent estimation of the basic reproduction number from the incidence of an emerging infection. J. Math. Biol. 55:803-816.
74. Ross R (1911) The Prevention of Malaria. John Murray, London.
75. Rvachev LA, Longini IM (1985) A mathematical model for the global spread of influenza. Math. Biosci. 75:322.
76. Sattenspiel L and Herring DA (2003) Simulating the effect of quarantine on the spread of the 1918-1919 flu in central Canada. Bull. Math. Biol. 65:1-26.
77. Sertsou G, Wilson N, Baker M, Nelson P and Roberts MG (2006) Key transmission parameters of an institutional outbreak during the 1918 influenza pandemic estimated by mathematical modelling. Theor. Biol. Med. Model. 3:38.
78. Smith CE (1964) Factors in the transmission of virus infections from animal to man. Sci. Basis Med. Annu. Rev. 125-150.
79. Sydenstricker E (1921) Variations in case fatality during the influenza epidemic of 1918. Public Health Rep. 36:2201-2211.
80. Thieme HR and Castillo-Chavez C (1989) How may infection-age dependent infectivity affect the dynamics of HIV/AIDS? SIAM J. Appl. Math. 53:1447-1479.
81. Thieme HR (1994) Asymptotically autonomous differential equations in the plane. Rocky Mountain J. Math. 24:351-380.
82. van den Driessche P and Watmough J (2002) Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission. Math. Biosc. 180:29-48.
83. Viboud C, Tam T, Fleming D, Handel A, Miller MA and Simonsen L (2006) Transmissibility and mortality impact of epidemic and pandemic influenza, with emphasis on the unusually deadly 1951 epidemic. Vaccine 24:6701-6707.
84. Vynnycky E, Trindall A and Mangtani P (2007) Estimates of the reproduction numbers of Spanish influenza using morbidity data. Int. J. Epidemiol. 36:881-889.
85. Wallinga J and Teunis P (2004) Different epidemic curves for severe acute respiratory syndrome reveal similar impacts of control measures. Am. J. Epidemiol. 160:509-516.
86. Wallinga J, Lipsitch M(2007) How generation intervals shape the relationship between growth rates and reproductive numbers. Proc. Roy. Soc. B 274:599-604.
87. Wearing HJ, Rohani P, and Keeling MJ (2005) Appropriate models for the management of infectious diseases. PLOS Med. 2:621-627.
88. White LC and Pagano MA (2007) likelihood-based method for real-time estimation of the serial interval and reproductive number of an epidemic. Stat. Med. in press (doi: 10.1002/sim.3136).
89. 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. J. Theor. Biol. 251:238-252.
90. Yang CK and Brauer F (2008) Calculation of R0 for age-of-infection models. Math. Biosci. Eng. 5:585-599.