Sustained oscillations in an evolutionary epidemiological model of influenza A drift

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

Volumn 466, Issue 2116, 2010, Pages 965-992

  Magal, Pierre ^a   Ruan, Shigui ^b  

^a UNIVERSITÉ DE BORDEAUX   (France)

^b Department of Biology   (United States)

Author keywords

Endemic equilibrium; Epidemic model; Hopf bifurcation; Influenza A drift; Stability

Indexed keywords

EPIDEMIOLOGY; VIRUSES;

ENDEMIC EQUILIBRIUM; EPIDEMIC MODEL; EPIDEMIC MODELS; INFLUENZA A; INFLUENZA A DRIFT;

HOPF BIFURCATION;

EID: 77950669100     PISSN: 13645021     EISSN: 14712946     Source Type: Journal    
DOI: 10.1098/rspa.2009.0435     Document Type: Article

References (51)

1. Andreasen, V. 2003 Dynamics of annual influenza A epidemics with immune-selection. J. Math. Biol. 46, 504-536. (doi:10.1007/s00285-002-0186-2) (Pubitemid 38355890)
2. Andreasen, V., Levin, S. A. & Lin, J. 1996 A model of influenza A drift evolution. Z. Angew. Math. Mech. 76, 421-424.
3. Andreasen, V., Lin, J. & Levin, S. A. 1997 The dynamics of circulating influenza strains conferring partial cross-immunity. J. Math. Biol. 35, 825-842. (doi:10.1007/s002850050079)
4. Arendt, W. 1987 Resolvent positive operators. Proc. Lond. Math. Soc. 54, 321-349. (doi:10.1112/ plms/s3-54.2.321)
5. Arendt, W., Batty, C. J. K., Hieber, M. & Neubrander, F. 2001 Vector-valued Laplace transforms and Cauchy problems. Basel, Switzerland: Birkhäuser.
6. Buonagurio, D. A., Nakada, S., Parvin, J. D., Krystal, M., Palese, P. & Fitch, W. M. 1986 Evolution of human influenza A viruses over 50 years: rapid, uniform rate of change in NS gene. Science 232, 980-982. (doi:10.1126/science.2939560)
7. Castillo-Chavez, C., Hethcote, H. W., Andreasen, V., Levin, S. A. & Liu, W. M. 1989 Epidemiological models with age structure, proportionate mixing, and cross-immunity. J. Math. Biol. 27, 233-258. (doi:10.1007/BF00275810)
8. CDC. 2009 Seasonal flu. See http://www.cdc.gov/flu/.
9. Cox, N. J. & Subbarao, K. 2000 Global epidemiology of influenza: past and present. Annu. Rev. Med. 51, 407-421. (doi:10.1146/annurev.med.51.1.407) (Pubitemid 30216046)
10. Ducrot, A., Liu, Z. & Magal, P. 2008 Essential growth rate for bounded linear perturbation of non-densely defined Cauchy problems. J. Math. Anal. Appl. 341, 501-518. (doi:10.1016/ j.jmaa.2007.09.074)
11. Dushoff, J., Plotkin, J. B., Levin, S. A. & Earn, D. J. D. 2004 Dynamical response can account for seasonality of influenza epidemics. Proc. Natl Acad. Sci. USA 101, 16 915-16 916. (doi:10.1073/ pnas.0407293101)
12. Earn, D. J. D., Dushoff, J. & Levin, S. A. 2002 Ecology and evolution of the flu. Trends Ecol. Evol. 17, 334-340. (doi:10.1016/S0169-5347(02)02502-8) (Pubitemid 34628660)
13. Engel, K.-J. & Nagel, R. 2000 One parameter semigroups for linear evolution equations. New York, NY: Springer-Verlag.
14. Fine, P. 1982 Applications of mathematical models to the epidemiology of influenza: a critique. In Influenza models (ed. P. Selby), pp. 15-85. Boston, MA: MTP Press.
15. Fitch, W. M., Bush, R. M., Bender, C. A. & Cox, N. J. 1997 Long term trends in the evolution of H(3) HA1 human influenza type A. Proc. Natl Acad. Sci. USA 94, 7712-7718. (doi:10.1073/ pnas.94.15.7712)
16. Hassard, B. D., Kazarinoff, N. D. & Wan, Y. H. 1981 Theory and applications of Hopf Birfurcation. Cambridge, UK: Cambridge University Press.
17. Hethcote, H. W. & Levin, S. A. 1989 Periodicity in epidemiological models. In Applied mathematical ecology. Biomathematics texts, vol. 18 (eds S. A. Levin, T. G. Hallam & L. J. Gross), pp 193-211. New York, NY: Springer-Verlag.
18. Hethcote, H. W., Stech, H. W. & van den Driessche, P. 1981 Periodicity and stability in epidemic models: a survey. In Differential equations and applications in ecology, epidemics and population problems (eds S. N. Busenberg & K. L. Cook), pp. 65-82. New York, NY: Academic Press.
19. Iannelli, M. 1994 Mathematical theory of age-structured population dynamics. Pisa, Italy: Giadini Editori e Stampatori.
20. Ina, Y. & Gojobori, T. 1994 Statistical analysis of nucleotide-sequences of the hemagglutinin gene of human influenza A viruses. Proc. Natl Acad. Sci. USA 91, 8388-8392. (doi:10.1073/ pnas.91.18.8388)
21. Inaba, H. 1998 Mathematical analysis for an evolutionary epidemic model. In Mathematical methods in medical and health sciences (ed. M. A. Horn, G. Simonett & G.Webb), pp. 213-236. Nashville, TN: Vanderbilt University Press.
22. Inaba, H. 2002 Epidemic threshold and stability in an evolutionary epidemic model. In Mathematical approaches for emerging and reemerging infectious diseases: models, methods, and theory. IMA Volumes in Mathematics and its Applications, vol. 126 (eds C. Castillo-Chavez, S. Blower, P. van d. Driessche, D. Kirscher & A.-A. Yakubu), pp. 337-359. New York, NY: Springer.
23. Kellermann, H. & Hieber, M. 1989 Integrated semigroups. J. Funct. Anal. 84, 160-180. (doi:10.1016/0022-1236(89)90116-X)
24. Kermack, W. O. & McKendrick, A. G. 1927 A contribution to the mathematical theory of epidemics. Proc. R. Soc. 115, 700-721. (doi:10.1098/rspa.1927.0118)
25. Kermack, W. O. & McKendrick, A. G. 1932 Contributions to the mathematical theory of epidemics. II. The problem of endemicity. Proc. R. Soc. 138, 55-83. (doi:10.1098/rspa.1932.0171)
26. Kermack, W. O. & McKendrick, A. G. 1933 Contributions to the mathematical theory of epidemics-III. Further studies of the problem of endemicity. Proc. R. Soc. 141A, 94-122.
27. Koelle, K., Cobey, S., Grenfell, B. & Pascual, M. 2006 Epochal evolution shapes the phylodynamics of interpandemic influenza A (H3N2) in humans. Science 314, 1898-1903. (doi:10.1126/ science.1132745)
28. Levin, S. A., Dushoff, J. & Plotkin, J. B. 2004 Evolution and persistence of influenza A and other diseases. Math. Biosci. 188, 17-28. (doi:10.1016/j.mbs.2003.08.010) (Pubitemid 38177302)
29. Lin, J., Andreasen, V. & Levin, S. A. 1999 Dynamics of influenza A drift: the linear three-strain model. Math. Biosci. 162, 33-51. (doi:10.1016/S0025-5564(99)00042-5)
30. Lin, J., Andreasen, V., Casagrandi, R. & Levin, S. A. 2003 Traveling waves in a model of influenza A drift. J. Theoret. Biol. 222, 437-445.
31. Magal, P. 2001 Compact attractors for time periodic age-structured population models. Electr. J. Diff. Equations 2001, 1-35.
32. p spaces. Diff. Integr. Equations 20, 197-239.
33. Magal, P. & Ruan, S. 2009a Center manifolds for semilinear equations with non-dense domain and applications on Hopf bifurcation in age structured models. Mem. Amer. Math. Soc. 202, no. 951. (doi:10.1090/S0065-9266-09-00568-7)
34. Magal, P. & Ruan, S. 2009b On semilinear Cauchy problems with non-densely defined domain. Adv. Diff. Equations 14, 1041-1084.
35. Neubrander, F. 1988 Integrated semigroups and their application to the abstract Cauchy problem. Pacific J. Math. 135, 111-155.
36. Nuño, M., Feng, Z., Martcheva, M. &. Castillo-Chavez, C. 2005 Dynamics of two-strain influenza with isolation and partial cross-immunity. SIAM J. Appl. Math. 65, 964-982. (doi:10.1137/ S003613990343882X)
37. Pease, C. M. 1987 An evolutionary epidemiological mechanism with application to type A influenza. Theoret. Pop. Biol. 31, 422-452. (doi:10.1016/0040-5809(87)90014-1)
38. Ruan, S. & Wang, W. 2003 Dynamical behaviour of an epidemic model with a nonlinear incidence rate. J. Diff. Equations 188, 135-163. (doi:10.1016/S0022-0396(02)00089-X) (Pubitemid 36233149)
39. Schweiger, B., Zadow, I. & Heckler, R. 2002 Antigenic drfit and variability of influenza viruses. Med. Microbiol. Immunol. 191, 133-138. (doi:10.1007/s00430-002-0132-3) (Pubitemid 35435754)
40. Shih, A. C.-C., Hsiao, T.-C., Ho, M.-S. & Li, W.-H. 2007 Simultaneous amino acid substitutions at antigenic sites drive influenza A hemagglutinin evolutiion. Proc. Natl Acad. Sci. USA 104, 6283-6288. (doi:10.1073/pnas. 0701396104) (Pubitemid 47186086)
41. Smith, D. J., Lapedes, A. S., de Jong, J. C., Bestebroer, T. M. Rimmelzwaan, G. F. Osterhaus, A. D. M. E. & Fouchier, R. A. M. 2004 Mapping the antigenic and genetic evolution of influenza virus. Science 305, 371-376. (doi:10.1126/science.1097211) (Pubitemid 38938148)
42. Tang, Y., Huang, D., Ruan, S. & Zhang, W. 2008 Coexistence of limit cycles and homoclinic loops in an SIRS model with a nonlinear incidence rate. SIAM J. Appl. Math. 69, 621-639. (doi:10.1137/ 070700966)
43. Thieme, H. R. 1990a Semiflows generated by Lipschitz perturbations of non-densely defined operators. Diff. Integr. Equations 3, 1035-1066.
44. Thieme, H. R. 1990b 'Integrated semigroupś and integrated solutions to abstract Cauchy problems. J. Math. Anal. Appl. 152, 416-447. (doi:10.1016/0022-247X(90)90074-P)
45. Thieme, H. R. 1997 Quasi-compact semigroups via bounded perturbation. In Advances in mathematical population dynamics-molecules, cells and man (eds O. Arino, D. Axelrod & M. Kimmel), pp. 691-713. River Edge, NJ: World Scientific Publishers.
46. Thieme, H. R. & Yang, J. 2002 An endemic model with variable re-infection rate and application to influenza. Math. Biosci. 180, 207-235. (doi:10.1016/S0025-5564(02)00102-5) (Pubitemid 35304354)
47. Treanor, J. 2004 Influenza vaccine - outmaneuvering antigenic schift and drift. N. Engl. J. Med. 350, 218-220. (doi:10.1056/NEJMp038238) (Pubitemid 38067634)
48. Viboud, C., Bjørnstad, O. N., Smith, D. L., Simonsen, L., Miller, M. A. & Grenfell, B. T. 2006 Synchrony, waves, and spatial hierarchies in the spread of influenza. Science 312, 447-451. (doi:10.1126/science.1125237)
49. Webb, G. F. 1985 Theory of nonlinear age-dependent population dynamics. New York, NY: Marcel Dekker.
50. Webb, G. F. 1987 An operator-theoretic formulation of asynchronous exponential growth. Trans. Am. Math. Soc. 303, 155-164. (doi:10.2307/2000695)
51. WHO. 2009 Influenza. See http://www.who.int/mediacentre/factsheets/fs211/ en/index.html.