Noise, nonlinearity and seasonality: The epidemics of whooping cough revisited

Journal of the Royal Society Interface

Volumn 5, Issue 21, 2008, Pages 403-413

  Nguyen, Hanh T H ^a   Rohani, Pejman ^a  

^a UNIVERSITY OF GEORGIA   (United States)

Author keywords

Epidemiology; Mathematical model; Population dynamics

Indexed keywords

CLIMATE CHANGE; DISEASES; DISTRIBUTION FUNCTIONS; MATHEMATICAL MODELS; NOISE POLLUTION;

CHILDHOOD INFECTIOUS; REALISTIC DISTRIBUTIONS; WHOOPING COUGH REVISITED;

EPIDEMIOLOGY;

ARTICLE; DISEASE PREDISPOSITION; ECOLOGY; EPIDEMIC; HUMAN; INCIDENCE; MATHEMATICAL COMPUTING; NOISE; NONLINEAR SYSTEM; OSCILLATION; PERTUSSIS; QUALITATIVE ANALYSIS; SEASONAL VARIATION;

EID: 39149109571     PISSN: 17425689     EISSN: 17425662     Source Type: Journal    
DOI: 10.1098/rsif.2007.1168     Document Type: Article

References (97)

1. Alonso, D., McKane, A. J. & Pascual, M. 2007 Stochastic amplification in epidemics. J. R. Soc. Interface 4, 575-582. (doi:10.1098/rsif.2006.0192)
2. Anderson, R. M. & May, R. M. 1979 Population biology of infectious diseases 1. Nature 280, 361-367. (doi:10.1038/280361a0)
3. Anderson, R. M. & May, R. M. 1982 Directly transmitted infectious diseases: control by vaccination. Science 215, 1053-1060. (doi:10.1126/science.7063839)
4. Bailey, N. T. J. 1954 A statistical method of estimating the periods of incubation and infection of an infectious disease. Nature 174, 139-140. (doi:10.1038/174139a0)
5. Bailey, N. T. J. 1975 The mathematical theory of infectious diseases, 2nd edn. London, UK: Griffin.
6. Bartlett, M. S. 1957 Measles periodicity and community size. J. R. Stat. Soc. 1, 48-59.
7. Bauch, C. T. & Earn, D. J. D. 2003 Transients and attractors in epidemics. Proc. R. Soc. B 270, 1573-1578. (doi:10.1098/rspb.2003. 2410)
8. Benton, T. G., Plaistow, S. J. & Coulson, T. N. 2006 Complex population dynamics and complex causation: devils, details and demography. Proc. R. Soc. B 273, 1173-1181. (doi:10.1098/rspb.2006.3495)
9. Bjørnstad, O. N. & Grenfell, B. T. 2001 Noisy clockwork: time series analysis of population fluctuations in animals. Science 293, 638-643. (doi:10.1126/science.1062226)
10. Bjørnstad, O. N., Grenfell, B. T. & Finkenstädt, B. F. 2002 Dynamics of measles epidemics: estimating scaling of transmission rates using a time series SIR model. Ecol. Monogr. 72, 169-184. (doi:10.2307/3100023)
11. Blythe, S. P. & Anderson, R. M. 1988 Distributed incubation and infectious periods in models of the transmission dynamics of the human immunodeficiency virus (HIV). IMA J. Math. Appl. Med. Biol. 5, 1-19. (doi:10.1093/imammb/5.1.1)
12. Bolker, B. M. & Grenfell, B. T. 1993 Chaos and biological complexity in measles dynamics. Proc. R. Soc. B 251, 75-81. (doi:10.1098/rspb.1993.0011)
13. Broutin, H., Rohani, P., Guégan, J. F., Grenfell, B. T. & Simondon, F. 2004 Loss of immunity to pertussis in a rural community in Senegal. Vaccine 22, 595-597. (doi:10.1016/j.vaccine.2003.07.018)
14. Broutin, H., Guégan, J. F., Elguero, E., Simondon, F. & Cazelles, B. 2005a Large-scale comparative analysis of pertussis population dynamics: periodicity, synchrony, and impact of vaccination. Am. J. Epidemiol. 161, 1159-1167. (doi:10.1093/aje/kwi141)
15. Broutin, H., Mantilla-Beniers, N. B., Simondon, F., Aaby, P., Grenfell, B. T., Guégan, J. F. & Rohani, P. 2005b Epidemiological impact of vaccination on the dynamics of two childhood diseases in rural Senegal. Microbes Infect. 7, 593-599. (doi:10.1016/j.micinf.2004.12.018)
16. Caswell, H. & Neubert, M. G. 2005 Reactivity and transient dynamics of discrete-time ecological systems. J. Differ. Equ. Appl. 11, 295-310. (doi:10.1080/10236190412331335382)
17. Choisy, M., Guegan, J. F. & Rohani, P. 2006 Resonance effects and the dynamics of infectious diseases. Physica D 223, 26-35. (doi:10.1016/j.physd.2006.08.006)
18. Clarkson, J. A. & Fine, P. E. M. 1985 The efficiency of measles and pertussis notification in England and Wales. Int. J. Epidemiol. 14, 153-168. (doi:10.1093/ije/14.1.153)
19. Coulson, T., Catchpole, E. A., Albon, S. D., Morgan, B. J. T., Pemberton, J. M., Clutton-Brock, T. H., Crawley, M. J. & Grenfell, B. T. 2001 Age, sex, density, winter weather, and population crashes in soay sheep. Science 292, 1528-1531. (doi:10.1126/science.292.5521.1528)
20. Coulson, T., Rohani, P. & Pascual, M. 2004 Skeletons, noise and the population growth: the end of an old debate? Trends Ecol. Evol. 19, 359-364. (doi:10.1016/j.tree.2004.05.008)
21. Cox, D. R. & Miller, H. D. 1965 The theory of stochastic processes. London, UK: Chapman and Hall.
22. Crowcroft, N. S. & Pebody, R. G. 2006 Recent developments in pertussis. Lancet 367, 1926-1936. (doi:10.1016/S0140-6736(06) 68848-X)
23. Cushing, J. M., Dennis, B., Desharnais, R. A. & Costantino, R. F. 1998 Moving toward an unstable equilibrium: saddle nodes in population systems. J. Anim. Ecol. 67, 298-306. (doi:10.1046/j.1365-2656.1998.00194.x)
24. Daszak, P., Cunningham, A. A. & Hyatt, A. D. 2000 Emerging infectious diseases of wildlife - threats to biodiversity and human health. Science 287, 443-449. (doi:10.1126/science.287.5452.443)
25. Dietz, K. 1976 The incidence of infectious diseases under the influence of seasonal fluctuations. Lecture Notes Biomath. 11, 1-15.
26. Dwyer, G., Dushoff, J. & Yee, S. H. 2004 The combined effects of pathogens and predators on insect outbreaks. Nature 430, 341-345. (doi:10.1038/nature02569)
27. Dye, C. & Williams, B. G. 2000 Criteria for the control of drug-resistant tuberculosis. Proc. Natl Acad. Sci. USA 97, 8180-8185. (doi:10.1073/pnas.140102797)
28. Earn, D. J. D., Rohani, P. & Grenfell, B. T. 1998 Persistence, chaos and synchrony in ecology and epidemiology. Proc. R. Soc. B 265, 7-10. (doi:10.1098/rspb.1998.0562)
29. Earn, D. J. D., Rohani, P., Bolker, B. M. & Grenfell, B. T. 2000 A simple model for complex dynamical transitions in epidemics. Science 287, 667-670. (doi:10.1126/science.287.5453.667)
30. Edmunds, W. J., Van de Heijden, O. G., Eerola, M. & Gay, N. J. 2000 Modelling rubella in Europe. Epidemiol. Infect. 125, 617-634. (doi:10.1017/S0950268800004660)
31. Ferguson, N. M., Nokes, D. J. & Anderson, R. M. 1996 Dynamical complexity in age-structured models of the transmission of measles virus: Epidemiological implications at high levels of vaccine uptake. Math. Biosci. 138, 101-130. (doi:10.1016/S0025-5564(96)00127-7)
32. Fine, P. E. M. & Clarkson, J. A. 1982 The recurrence of whooping cough: possible implications for assessment of vaccine efficacy. Lancet 1, 666-669. (doi:10.1016/S0140-6736(82)92214-0)
33. Fine, P. E. M. & Clarkson, J. A. 1986 Seasonal influences on pertussis. Int. J. Epidemiol. 15, 237-247. (doi:10.1093/ije/15.2. 237)
34. Frankish, H. 2003 Death tolls continue to climb in Congo Ebola outbreak. Lancet 361, 1020. (doi:10.1016/S0140-6736(03)12846-2)
35. Gillespie, D. T. 1977 Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem. 81, 2340-2361. (doi:10.1021/j100540a008)
36. Glass, K., Xia, Y. & Grenfell, B. T. 2003 Interpreting time-series analyses for continuous-time biological models - measles as a case study. J. Theor. Biol. 223, 19-25. (doi:10.1016/S0022-5193(03)00031-6)
37. Grassly, N. C. & Fraser, C. 2006 Seasonal infectious disease epidemiology. Proc. R. Soc. B 273, 2541-2550. (doi:10.1098/rspb. 2006.3604)
38. Gomes, M. C., Gomes, J. J. & Paulo, A. C. 1999 Diptheria, pertussis, and measles in Portugal before and after mass vaccination: a time series analysis. Eur. J. Epidemiol. 15, 791-798. (doi:10.1023/A: 1007615513441)
39. Gough, K. J. 1977 The estimation of latent and infectious periods. Biometrika 64, 559-565. (doi:10.1093/biomet/64.3.559)
40. Greenman, J., Kamo, M. & Boots, M. 2004 External forcing of ecological and epidemiological systems: a resonance approach. Phys. D Nonlinear Phenomena 190, 136-151. (doi:10.1016/j.physd.2003.08.008)
41. Grenfell, B. T. 1992 Chance and chaos in measles dynamics. J. R. Stat. Soc. 54, 383-398.
42. Grenfell, B. T. & Anderson, R. M. 1989 Pertussis in England and Wales: an investigation of transmission dynamics and control by mass vaccination. Proc. R. Soc. Lond. Biol. Sci. B 236, 213-252. (doi:10.1098/rspb.1989.0022)
43. Grenfell, B. T., Wilson, K., Finkenstädt, B. F., Coulson, T. N., Murray, S., Albon, S. D., Pemberton, J. M., Clutton-Brock, T. H. & Crawley, M. J. 1998 Noise and determinism in synchronised sheep dynamics. Nature 394, 674-677. (doi:10.1038/29291)
44. Grenfell, B. T., Bjørnstad, O. N. & Kappey, J. 2001 Travelling waves and spatial hierarchies in measles epidemics. Nature 414, 716-723. (doi:10.1038/414716a)
45. Güris, D., Strebel, P. M., Bardenheier, B., Brennan, M., Tachdjian, R., Finch, E., Wharton, M. & Livengood, J. R. 1999 Changing epidemiology of pertussis in the United States: increasing reported incidence among adolescents and adults, 1990-1996. Clin. Infect. Dis. 28, 1230-1237.
46. Guzmán, M. & Kouri, G. 2002 Dengue: an update. Lancet Infect. Dis. 2, 33-42. (doi:10.1016/S1473-3099(01)00171-2)
47. Hamer, W. H. 1897 Age-incidence in relation with cycles of disease-prevalence. Trans. Epidemiol. Soc. Lond. XVI, 64-77.
48. Hastings, A. 1977 Spatial heterogeneity and the stability of predator-prey systems. Theor. Popul. Biol. 12, 37-48. (doi:10.1016/0040-5809(77)90034-X)
49. Hastings, A. 2004 Transients: the key to long-term ecological understanding? Trends Ecol. Evol. 19, 39-45. (doi:10.1016/j.tree. 2003.09.007)
50. Hastings, A. & Higgins, K. 1994 Persistence of transients in spatially structured ecological models. Science 263, 1133-1136. (doi:10.1126/science.263.5150.1133)
51. Heffernan, J. M. & Wahl, L. M. 2006 Improving estimates of the basic reproductive ratio: using both the mean and the dispersal of transition times. Theor. Popul. Biol. 70, 135-145. (doi:10.1016/j.tpb.2006.03.003)
52. Heininger, U., Cherry, J. D., Stehr, K., Schmitt-Grohe, S., Úberall, M., Laussucq, S., Eckhardt, T., Meyer, M. & Gornbein, J. 1998 Comparative efficacy of the Lederle/ Takeda acellular pertussis component DTP (DTaP) vaccine and Lederle whole-cell component DTP vaccine in German children after household exposure. Pediatrics 102, 546-553. (doi:10.1542/peds.102.3.546)
53. Hethcote, H. W. 1998 Oscillations in an endemic model for pertussis. Can. Appl. Math. Quart. 6, 61-88.
54. Higgins, K., Hastings, A., Sarvela, J. N. & Botsford, L. W. 1997 Stochastic dynamics and deterministic skeletons: population behavior of Dungeness crab. Science 276, 1431-1435. (doi:10.1126/science.276. 5317.1431)
55. Ionides, E. L., Breto, C. & King, A. A. 2006 Inference for nonlinear dynamical systems. Proc. Natl Acad. Sci. USA 103, 18 438-18 443. (doi:10.1073/pnas.0603181103)
56. Keeling, M. J. & Grenfell, B. T. 1997 Disease extinction and community size: modeling the persistence of measles. Science 275, 65-67. (doi:10.1126/science.275.5296.65)
57. Keeling, M. J. & Grenfell, B. 2002 Understanding the persistence of measles: reconciling theory, simulation and observation. Proc. R. Soc. B 269, 335-343. (doi:10.1098/rspb.2001.1898)
58. Keeling, M. J. & Rohani, P. 2007 Modelling infectious diseases: in humans and animals. Princeton, NJ: Princeton University Press.
59. Keeling, M. J., Rohani, P. & Grenfell, B. T. 2001 Seasonally forced disease dynamics explained by switching between attractors. Physica D 148, 317-335. (doi:10.1016/S0167-2789(00)00187-1)
60. Kuske, R., Gordillo, L. F. & Greenwood, G. 2007 Sustained oscillations via coherence resonance in SIR. J. Theor. Biol. 245, 459-469. (doi:10.1016/j.jtbi.2006.10.029)
61. Lipsitch, M. et al. 2003 Transmission dynamics and control of severe acute respiratory syndrome. Science 300, 1966-1970. (doi:10.1126/science.1086616)
62. Lloyd, A. L. 2001a Destabilization of epidemic models with the inclusion of realistic distributions of infectious periods. Proc. R. Soc. B 268, 985-993. (doi:10.1098/rspb.2001.1599)
63. Lloyd, A. L. 2001b Realistic distributions of infectious periods in epidemic models: changing patterns of persistence and dynamics. Theor. Popul. Biol. 60, 59-71. (doi:10.1006/tpbi.2001.1525)
64. Lloyd, A. L. & Jansen, V. A. A. 2004 Spatiotemporal dynamics of epidemics: synchrony in metapopulation models. Math. Biosci. 188, 1-16. (doi:10.1016/j.mbs.2003.09.003)
65. Lloyd-Smith, J. O., Schreiber, S. J., Kopp, P. E. & Getz, W. M. 2005 Superspreading and the effect of individual variation on disease emergence. Nature 438, 355-359. (doi:10.1038/nature04153)
66. London, W. P. & Yorke, J. A. 1973 Recurrent outbreaks of measles, chickenpox and mumps. I. Seasonal variations in contact rates. Am. J. Epidemiol. 98, 453-468.
67. MacDonald, N. 1978 Time lags in biological models. Berlin, Germany: Springer.
68. May, R. M. & Anderson, R. M. 1984 Spatial heterogeneity and the design of immunization programs. Math. Biosci. 72, 83-111. (doi:10.1016/0025-5564(84)90063-4)
69. Millonas, M. 1995 Fluctuations and order: the new synthesis. Berlin, Germany: Springer.
70. Morens, D. M., Folkers, G. K. & Fauci, A. S. 2004 The challenge of emerging and re-emerging infectious diseases. Nature 430, 242-249. (doi:10.1038/nature02759)
71. Murdoch, W. W., Briggs, C. J. & Nisbet, R. M. 2003 Consumer-resource dynamics. Princeton, NJ: Princeton University Press.
72. Nasell, I. 2005 A new look at the critical community size for childhood infections. Theor. Popul. Biol. 67, 203-216. (doi:10.1016/j.tpb. 2005.01.002)
73. Noonburg, E. G. & Abrams, P. A. 2005 Transient dynamics limit the effectiveness of keystone predation in bringing about coexistence. Am. Nat. 165, 322-335. (doi:10.1086/428068)
74. Obenauer, J. C. et al. 2006 Large-scale sequence analysis of avian influenza isolates. Science 311, 1576-1580. (doi:10.1126/science. 1121586)
75. Olsen, B., Munster, V. J., Wallensten, A., Waldenstrom, J., Osterhaus, A. D. M. E. & Fouchier, R. A. M. 2006 Global patterns of influenza A virus in wild birds. Science 312, 384-388. (doi:10.1126/science.1122438)
76. Quinn, T. C. & Overbaugh, J. 2005 HIV/AIDS in women: an expanding epidemic. Science 308, 1582-1583. (doi:10.1126/science.1112489)
77. Rand, D. A. & Wilson, H. B. 1991 Chaotic stochasticity: a ubiquitous source of unpredictability in epidemics. Proc. R. Soc. B 246, 179-184. (doi:10.1098/rspb.1991.0142)
78. Ransome, A. 1880 On epidemic cycles. Proc. Manchester Lit. Phil. Soc. 19, 75-96.
79. Rand, D. A. & Wilson, H. B. 1991 Chaotic stochasticity: a ubiquitous source of unpredictability in epidemics. Proc. R. Soc. B 246, 179-184. (doi:10.1098/rspb.1991.0142)
80. Reuman, D. C., Desharnais, R. A., Costantino, R. F., Ahmad, O. S. & Cohen, J. E. 2006 Power spectra reveal the influence of stochasticity on nonlinear population dynamics. Proc. Natl Acad. Sci. USA 103, 18 860-18 865. (doi:10.1073/pnas.0608571103)
81. Riley, S. et al. 2003 Transmission dynamics of the etiological agent of SARS in Hong Kong: impact of public health interventions. Science 300, 1961-1966. (doi:10.1126/science.1086478)
82. Rohani, P., Earn, D. J. D., Finkenstadt, B. F. & Grenfell, B. T. 1998 Population dynamic interference among childhood diseases. Proc. R. Soc. B 265, 2033-2041. (doi:10.1098/rspb.1998.0537)
83. Rohani, P., Earn, D. J. D. & Grenfell, B. T. 1999 Opposite patterns of synchrony in sympatric disease metapopulations. Science 286, 968-971. (doi:10.1126/science.286.5441.968)
84. Rohani, P., Earn, D. J. D. & Grenfell, B. T. 2000 Impact of immunisation on pertussis transmission in England and Wales. Lancet 355, 285-286. (doi:10.1016/S0140-6736(99)04482-7)
85. Rohani, P., Keeling, M. J. & Grenfell, B. T. 2002 The interplay between noise and determinism in childhood diseases. Am. Nat. 159, 469-481. (doi:10.1086/339467)
86. Sartwell, P. E. 1950 The distribution of incubation periods of infectious disease. Am. J. Hyg. 51, 310-318.
87. Schenzle, D. 1984 An age-structured model of pre- and post-vaccination measles transmission. IMA J. Math. Appl. Med. Biol. 1, 169-191. (doi:10.1093/imammb/1.2.169)
88. Schwartz, I. B. & Smith, H. L. 1983 Infinite subharmonic bifurcation in an SEIR epidemic model. J. Math. Biol. 18, 233-253. (doi:10.1007/BF00276090)
89. Simpson, R. E. H. 1952 Infectiousness of communicable diseases in the household (measles, chickenpox, and mumps). Lancet 263, 549-554. (doi:10.1016/S0140-6736(52)91357-3)
90. Soper, H. E. 1929 The interpretation of periodicity in disease prevalence. J. R. Stat. Soc. A 92, 34-61. (doi:10.2307/2341437)
91. Sugihara, G. & May, R. M. 1990 Nonlinear forecasting as a way of distinguishing chaos from measurement error in time series. Nature 344, 734-741. (doi:10.1038/344734a0)
92. Torrence, C. & Compo, G. P. 1998 A practical guide to wavelet analysis. Bull. Am. Meteorol. Soc. 79, 61-78. (doi:10.1175/1520- 0477(1998)079〈0061:APGTWA〉2.0.CO;2)
93. von König, C. H. W., Postelsmultani, S., Bock, H. L. & Schmitt, H. J. 1995 Pertussis in adults: frequency of transmission after household exposure. Lancet 346, 1326-1329. (doi:10.1016/S0140-6736(95)92343- 8)
94. Walker, P. R., Worobey, M., Rambaut, A., Holmes, E. C. & Pybus, O. G. 2003 Epidemiology: sexual transmission of HIV in Africa. Nature 422, 679. (doi:10.1038/422679a)
95. Wearing, H. J. & Rohani, P. Submitted. Dynamical implications of waning immunity for pertussis epidemiology.
96. Wearing, H. J., Rohani, P. & Keeling, M. J. 2005 Appropriate models for the management of infectious diseases. PLoS Med. 2, e174. (doi:10.1371/journal.pmed.0020174)
97. Xia, Y., Bjørnstad, O. N. & Grenfell, B. T. 2004 Measles metapopulation dynamics: a gravity model for epidemiological coupling and dynamics. Am Nat. 164, 267-281. (doi:10.1086/422341)