-
1
-
-
0016394337
-
-
R. M. May, Science 186, 645 (1974); T. Royama, Analytical Population Dynamics (Chapman and Hall, London, 1992); C. Zimmer, Science 284, 83 (1999).
-
(1974)
Science
, vol.186
, pp. 645
-
-
May, R.M.1
-
2
-
-
0016394337
-
-
Chapman and Hall, London
-
R. M. May, Science 186, 645 (1974); T. Royama, Analytical Population Dynamics (Chapman and Hall, London, 1992); C. Zimmer, Science 284, 83 (1999).
-
(1992)
Analytical Population Dynamics
-
-
Royama, T.1
-
3
-
-
0016394337
-
-
R. M. May, Science 186, 645 (1974); T. Royama, Analytical Population Dynamics (Chapman and Hall, London, 1992); C. Zimmer, Science 284, 83 (1999).
-
(1999)
Science
, vol.284
, pp. 83
-
-
Zimmer, C.1
-
4
-
-
0000065728
-
-
R. Levins, Bull. Entomol. Soc. Am. 15, 237 (1969); J. H. Brown and A. Kodric-Brown, Ecology 58, 445 (1977).
-
(1969)
Bull. Entomol. Soc. Am.
, vol.15
, pp. 237
-
-
Levins, R.1
-
5
-
-
0001011081
-
-
R. Levins, Bull. Entomol. Soc. Am. 15, 237 (1969); J. H. Brown and A. Kodric-Brown, Ecology 58, 445 (1977).
-
(1977)
Ecology
, vol.58
, pp. 445
-
-
Brown, J.H.1
Kodric-Brown, A.2
-
6
-
-
0030620649
-
-
R. F. Costantino, R. A. Desharnais, J. M. Cushing, B. Dennis, Science 275, 389 (1997); M. J. Keeling and B. T. Grenfell, Science 275, 65 (1997).
-
(1997)
Science
, vol.275
, pp. 389
-
-
Costantino, R.F.1
Desharnais, R.A.2
Cushing, J.M.3
Dennis, B.4
-
7
-
-
0031015249
-
-
R. F. Costantino, R. A. Desharnais, J. M. Cushing, B. Dennis, Science 275, 389 (1997); M. J. Keeling and B. T. Grenfell, Science 275, 65 (1997).
-
(1997)
Science
, vol.275
, pp. 65
-
-
Keeling, M.J.1
Grenfell, B.T.2
-
8
-
-
0032514553
-
-
B. T. Grenfell et al., Nature 394, 674 (1998); K. Higgins, A. Hastings, J. N. Sarvela, L. W. Botsford, Science 276, 1431 (1997).
-
(1998)
Nature
, vol.394
, pp. 674
-
-
Grenfell, B.T.1
-
9
-
-
0030616677
-
-
B. T. Grenfell et al., Nature 394, 674 (1998); K. Higgins, A. Hastings, J. N. Sarvela, L. W. Botsford, Science 276, 1431 (1997).
-
(1997)
Science
, vol.276
, pp. 1431
-
-
Higgins, K.1
Hastings, A.2
Sarvela, J.N.3
Botsford, L.W.4
-
10
-
-
0026268962
-
-
M. P. Hassell, H. N. Comins, R. M. May, Nature 353, 255 (1991); S. A. Levin, B. T. Grenfell, A. Hastings, A. S. Perelson, Science 275, 334 (1997); D. Grünbaum, Am. Nat. 151, 97 (1998); I. Hanski, Nature 396, 41 (1998); J. Swinton, J. Harwood, B. T. Grenfell, C. A. Gilligan, J. Anim. Evol. 67, 56 (1998).
-
(1991)
Nature
, vol.353
, pp. 255
-
-
Hassell, M.P.1
Comins, H.N.2
May, R.M.3
-
11
-
-
0030620646
-
-
M. P. Hassell, H. N. Comins, R. M. May, Nature 353, 255 (1991); S. A. Levin, B. T. Grenfell, A. Hastings, A. S. Perelson, Science 275, 334 (1997); D. Grünbaum, Am. Nat. 151, 97 (1998); I. Hanski, Nature 396, 41 (1998); J. Swinton, J. Harwood, B. T. Grenfell, C. A. Gilligan, J. Anim. Evol. 67, 56 (1998).
-
(1997)
Science
, vol.275
, pp. 334
-
-
Levin, S.A.1
Grenfell, B.T.2
Hastings, A.3
Perelson, A.S.4
-
12
-
-
0031913729
-
-
M. P. Hassell, H. N. Comins, R. M. May, Nature 353, 255 (1991); S. A. Levin, B. T. Grenfell, A. Hastings, A. S. Perelson, Science 275, 334 (1997); D. Grünbaum, Am. Nat. 151, 97 (1998); I. Hanski, Nature 396, 41 (1998); J. Swinton, J. Harwood, B. T. Grenfell, C. A. Gilligan, J. Anim. Evol. 67, 56 (1998).
-
(1998)
Am. Nat.
, vol.151
, pp. 97
-
-
Grünbaum, D.1
-
13
-
-
0032487859
-
-
M. P. Hassell, H. N. Comins, R. M. May, Nature 353, 255 (1991); S. A. Levin, B. T. Grenfell, A. Hastings, A. S. Perelson, Science 275, 334 (1997); D. Grünbaum, Am. Nat. 151, 97 (1998); I. Hanski, Nature 396, 41 (1998); J. Swinton, J. Harwood, B. T. Grenfell, C. A. Gilligan, J. Anim. Evol. 67, 56 (1998).
-
(1998)
Nature
, vol.396
, pp. 41
-
-
Hanski, I.1
-
14
-
-
0026268962
-
-
M. P. Hassell, H. N. Comins, R. M. May, Nature 353, 255 (1991); S. A. Levin, B. T. Grenfell, A. Hastings, A. S. Perelson, Science 275, 334 (1997); D. Grünbaum, Am. Nat. 151, 97 (1998); I. Hanski, Nature 396, 41 (1998); J. Swinton, J. Harwood, B. T. Grenfell, C. A. Gilligan, J. Anim. Evol. 67, 56 (1998).
-
(1998)
J. Anim. Evol.
, vol.67
, pp. 56
-
-
Swinton, J.1
Harwood, J.2
Grenfell, B.T.3
Gilligan, C.A.4
-
17
-
-
0000990735
-
-
M. P. Hassell, J. H. Lawton, R. M. May, J. Anim. Ecol. 46, 249 (1976); G. Sugihara and R. M. May, Nature 344, 734 (1990); S. P. Ellner and P. Turchin, Am. Nat. 145, 343 (1995).
-
(1976)
J. Anim. Ecol.
, vol.46
, pp. 249
-
-
Hassell, M.P.1
Lawton, J.H.2
May, R.M.3
-
18
-
-
0025199496
-
-
M. P. Hassell, J. H. Lawton, R. M. May, J. Anim. Ecol. 46, 249 (1976); G. Sugihara and R. M. May, Nature 344, 734 (1990); S. P. Ellner and P. Turchin, Am. Nat. 145, 343 (1995).
-
(1990)
Nature
, vol.344
, pp. 734
-
-
Sugihara, G.1
May, R.M.2
-
19
-
-
0028973866
-
-
M. P. Hassell, J. H. Lawton, R. M. May, J. Anim. Ecol. 46, 249 (1976); G. Sugihara and R. M. May, Nature 344, 734 (1990); S. P. Ellner and P. Turchin, Am. Nat. 145, 343 (1995).
-
(1995)
Am. Nat.
, vol.145
, pp. 343
-
-
Ellner, S.P.1
Turchin, P.2
-
21
-
-
0003022391
-
-
B. T. Grenfell and B. M. Bolker, Ecol. Lett. 1, 63 (1998); D. Schenzle, IMA (Inst. Math. Appl.) J. Math. Appl. Med. Biol. 1, 169 (1984). The weekly case reports of measles and whooping cough are published in the Registrar General's Weekly Returns. These data have been estimated to represent about 50 to 60% of the actual cases for measles and 5 to 25% for whooping cough [J. A. Clarkson and P. E. M. Fine, Int. J. Epidemiol. 14, 153 (1985)]. Although notification efficiency may affect the magnitudes of case reports, it will not create or alter the large-scale spatiotemporal trends observed in these data. Further, previous work has demonstrated very strong correlation between notified case reports, serological isolations of, for example, B. pertussis by the Public Health Laboratory Service, and independent notifications provided by the Royal College of General Practitioners General Practice Research Unit [(25); E. Miller, J. E. Vurdien, J. M. White, Communicable Dis. Rep. 2, R152 (1992)].
-
(1998)
Ecol. Lett.
, vol.1
, pp. 63
-
-
Grenfell, B.T.1
Bolker, B.M.2
-
22
-
-
77957185501
-
-
B. T. Grenfell and B. M. Bolker, Ecol. Lett. 1, 63 (1998); D. Schenzle, IMA (Inst. Math. Appl.) J. Math. Appl. Med. Biol. 1, 169 (1984). The weekly case reports of measles and whooping cough are published in the Registrar General's Weekly Returns. These data have been estimated to represent about 50 to 60% of the actual cases for measles and 5 to 25% for whooping cough [J. A. Clarkson and P. E. M. Fine, Int. J. Epidemiol. 14, 153 (1985)]. Although notification efficiency may affect the magnitudes of case reports, it will not create or alter the large-scale spatiotemporal trends observed in these data. Further, previous work has demonstrated very strong correlation between notified case reports, serological isolations of, for example, B. pertussis by the Public Health Laboratory Service, and independent notifications provided by the Royal College of General Practitioners General Practice Research Unit [(25); E. Miller, J. E. Vurdien, J. M. White, Communicable Dis. Rep. 2, R152 (1992)].
-
(1984)
IMA (Inst. Math. Appl.) J. Math. Appl. Med. Biol.
, vol.1
, pp. 169
-
-
Schenzle, D.1
-
23
-
-
0021958163
-
-
B. T. Grenfell and B. M. Bolker, Ecol. Lett. 1, 63 (1998); D. Schenzle, IMA (Inst. Math. Appl.) J. Math. Appl. Med. Biol. 1, 169 (1984). The weekly case reports of measles and whooping cough are published in the Registrar General's Weekly Returns. These data have been estimated to represent about 50 to 60% of the actual cases for measles and 5 to 25% for whooping cough [J. A. Clarkson and P. E. M. Fine, Int. J. Epidemiol. 14, 153 (1985)]. Although notification efficiency may affect the magnitudes of case reports, it will not create or alter the large-scale spatiotemporal trends observed in these data. Further, previous work has demonstrated very strong correlation between notified case reports, serological isolations of, for example, B. pertussis by the Public Health Laboratory Service, and independent notifications provided by the Royal College of General Practitioners General Practice Research Unit [(25); E. Miller, J. E. Vurdien, J. M. White, Communicable Dis. Rep. 2, R152 (1992)].
-
(1985)
Int. J. Epidemiol.
, vol.14
, pp. 153
-
-
Clarkson, J.A.1
Fine, P.E.M.2
-
24
-
-
0041113561
-
-
B. T. Grenfell and B. M. Bolker, Ecol. Lett. 1, 63 (1998); D. Schenzle, IMA (Inst. Math. Appl.) J. Math. Appl. Med. Biol. 1, 169 (1984). The weekly case reports of measles and whooping cough are published in the Registrar General's Weekly Returns. These data have been estimated to represent about 50 to 60% of the actual cases for measles and 5 to 25% for whooping cough [J. A. Clarkson and P. E. M. Fine, Int. J. Epidemiol. 14, 153 (1985)]. Although notification efficiency may affect the magnitudes of case reports, it will not create or alter the large-scale spatiotemporal trends observed in these data. Further, previous work has demonstrated very strong correlation between notified case reports, serological isolations of, for example, B. pertussis by the Public Health Laboratory Service, and independent notifications provided by the Royal College of General Practitioners General Practice Research Unit [(25); E. Miller, J. E. Vurdien, J. M. White, Communicable Dis. Rep. 2, R152 (1992)].
-
(1992)
Communicable Dis. Rep.
, vol.2
-
-
Miller, E.1
Vurdien, J.E.2
White, J.M.3
-
25
-
-
0032494698
-
-
P. Rohani, D. J. D. Earn, B. F. Finkenstädt, B. T. Grenfell, Proc. R. Soc. London Ser. B Biol. Sci. 265, 2033 (1998).
-
(1998)
Proc. R. Soc. London Ser. B Biol. Sci.
, vol.265
, pp. 2033
-
-
Rohani, P.1
Earn, D.J.D.2
Finkenstädt, B.F.3
Grenfell, B.T.4
-
28
-
-
0040519459
-
-
J. N. Perry and R. A. Smith, Eds. Academic Press, San Diego, CA
-
B. T. Grenfell, in Chaos from Real Data: The Analysis of Nonlinear Dynamics in Short Ecological Time Series, J. N. Perry and R. A. Smith, Eds. (Academic Press, San Diego, CA, 1999).
-
(1999)
Chaos from Real Data: the Analysis of Nonlinear Dynamics in Short Ecological Time Series
-
-
Grenfell, B.T.1
-
29
-
-
0344816239
-
-
note
-
The infectious period for whooping cough consists of two characteristically very distinct phases: a catarrhal phase (symptoms include a mild fever and a cough), which lasts about 2 weeks, and the paroxysmal phase (intense bouts of coughing punctuated by the "whoop"). A case would be successfully diagnosed as pertussis at the start of the whooping phase, and, hence, for epidemiological purposes, the infectious period effectively lasts 14 days (assuming susceptibles in the family would have been infected by this time). Our results are not qualitatively affected by this assumption.
-
-
-
-
30
-
-
0030867201
-
-
H. W. Hethcote, Math. Biosci. 145, 89 (1997); Can. Appl. Math. Q. 6, 61 (1998).
-
(1997)
Math. Biosci.
, vol.145
, pp. 89
-
-
Hethcote, H.W.1
-
31
-
-
0030867201
-
-
H. W. Hethcote, Math. Biosci. 145, 89 (1997); Can. Appl. Math. Q. 6, 61 (1998).
-
(1998)
Can. Appl. Math. Q
, vol.6
, pp. 61
-
-
-
32
-
-
0030758908
-
-
P. T. Scott, J. B. Clark, W. F. Miser, Am. Fam. Phys. 56, 1121 (1997); J. D. Cherry, Ann. Intern. Med. 128, 64 (1998).
-
(1997)
Am. Fam. Phys.
, vol.56
, pp. 1121
-
-
Scott, P.T.1
Clark, J.B.2
Miser, W.F.3
-
33
-
-
0031984398
-
-
P. T. Scott, J. B. Clark, W. F. Miser, Am. Fam. Phys. 56, 1121 (1997); J. D. Cherry, Ann. Intern. Med. 128, 64 (1998).
-
(1998)
Ann. Intern. Med.
, vol.128
, pp. 64
-
-
Cherry, J.D.1
-
40
-
-
0345247302
-
-
note
-
Environmental and demographic noise have little impact on the prevaccination biennial pattern of measles outbreaks, whereas they affect pertussis dynamics markedly (P. Rohani, M. J. Keeling, B. T. Grenfell, in preparation). This is caused by the differences in the local stability properties over the deterministic attractor for the two infections.
-
-
-
-
41
-
-
0345678986
-
-
note
-
For the unforced SEIR system, the natural (damping) period of oscillations is T ≈ 2π √AD, where A is the mean age at infection and D is the sum of infectious and incubation periods (9). For whooping cough before vaccination, the mean age at infection is 2 to 5 years and D = 0.055 years, giving a natural period of between 2.08 and 3.2 years. After vaccination, A would have risen to 5 to 7 years, giving a natural period of between 3.3 and 4.1 years. In (unforced) models with demographic or environmental stochasticity, resonance ensures that these oscillations are sustained, giving outbreaks with period T (assuming the population size is sufficiently large to avoid extinction of the pathogen) (9).
-
-
-
-
43
-
-
0027136227
-
-
Z. Agur, L. Cojocaru, G. Mazor, R. M. Anderson, Y. L. Danon, Proc. Natl. Acad. Sci. U.S.A. 90, 11698 (1993); D. J. Nokes and J. Swinton, IMA (Inst. Appl. Math.) J. Math. Appl. Med. Biol. 12, 29 (1997); B. Shulgin, L Stone, Z. Agur, Bull. Math. Biol. 60, 1123 (1998).
-
(1993)
Proc. Natl. Acad. Sci. U.S.A.
, vol.90
, pp. 11698
-
-
Agur, Z.1
Cojocaru, L.2
Mazor, G.3
Anderson, R.M.4
Danon, Y.L.5
-
44
-
-
0027136227
-
-
Z. Agur, L. Cojocaru, G. Mazor, R. M. Anderson, Y. L. Danon, Proc. Natl. Acad. Sci. U.S.A. 90, 11698 (1993); D. J. Nokes and J. Swinton, IMA (Inst. Appl. Math.) J. Math. Appl. Med. Biol. 12, 29 (1997); B. Shulgin, L Stone, Z. Agur, Bull. Math. Biol. 60, 1123 (1998).
-
(1997)
IMA (Inst. Appl. Math.) J. Math. Appl. Med. Biol.
, vol.12
, pp. 29
-
-
Nokes, D.J.1
Swinton, J.2
-
45
-
-
0032213158
-
-
Z. Agur, L. Cojocaru, G. Mazor, R. M. Anderson, Y. L. Danon, Proc. Natl. Acad. Sci. U.S.A. 90, 11698 (1993); D. J. Nokes and J. Swinton, IMA (Inst. Appl. Math.) J. Math. Appl. Med. Biol. 12, 29 (1997); B. Shulgin, L Stone, Z. Agur, Bull. Math. Biol. 60, 1123 (1998).
-
(1998)
Bull. Math. Biol.
, vol.60
, pp. 1123
-
-
Shulgin, B.1
Stone, L.2
Agur, Z.3
-
48
-
-
0345678985
-
-
note
-
We thank O. Bjørnstad, B. Bolker, B. Finkenstädt, R. Hamilton, M. Keeling, J. Swinton, and two anonymous reviewers for comments on this paper. P.R. was supported by a Natural Environment Research Council Postdoctoral Research Fellowship. D.J.D.E. was supported by a Wellcome Trust Postdoctoral Research Fellowship in Mathematical Biology. B.T.G. is supported by the Wellcome Trust.
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