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Dynamics of influenza A drift: The linear three-strain model. (English) Zbl 0947.92017
Summary: We analyze an epidemiological model consisting of a linear chain of three cocirculating influenza $A$ strains that provide hosts exposed to a given strain with partial immune cross-protection against other strains. In the extreme case where infection with the middle strain prevents further infections from the other two strains, we reduce the model to a six-dimensional kernel capable of showing self-sustaining oscillations at relatively high levels of cross-protection. Dimensional reduction has been accomplished by a transformation of variables that preserves the eigenvalue responsible for the transition from damped oscillations to limit cycle solutions.

92C60Medical epidemiology
37N25Dynamical systems in biology
37G15Bifurcations of limit cycles and periodic orbits
Full Text: DOI
[1] Gupta, S.; Ferguson, N.; Anderson, R.: Chaos persistence and evolution of strain structure in antigenically diverse infectious agents. Science 280, 912 (1998)
[2] E.D. Kilbourne, Host determination of viral evolution: a variable tautology, in: S.S. Morse (Ed.), The evolutionary biology of viruses, Raven, New York, 1994, p. 253
[3] R.A. Lamb, Genes and proteins of the influenza viruses, in: R.M. Krug, H. Fraenkel-Conrat, R.R. Wagner (Eds.), The influenza viruses, Plenum, New York, 1989, p. 1
[4] P.E.M. Fine, Applications of mathematical models to the epidemiology of influenza: a critique, in: P. Selby (Ed.), Influenza models, MTP, Lancaster, 1982, p. 15
[5] Thacker, S. B.: The persistence of influenza in human populations. Epidemiol. rev. 8, 129 (1986)
[6] Palese, P.; Young, F. J.: Variation of influenza A, B and C viruses. Science 215, 1468 (1982)
[7] Webster, R. G.; Laver, W. G.; Air, G. M.; Schild, G. C.: Molecular mechanisms of variation in influenza viruses. Nature 296, 115 (1982)
[8] Webster, R. G.; Bean, W. J.; Gorman, O. T.; Chambers, T. M.; Kawaoka, Y.: Evolution and ecology of influenza A viruses. Microbiol. rev. 56, 152 (1992)
[9] Stuart-Harris, C.: The epidemiology and prevention of influenza. Amer. sci. 69, 166 (1981)
[10] F.L. Smith, P. Palese, Variation in influenza virus genes, in: R.M. Krug, H. Fraenkel-Conrat, R.R. Wagner (Eds.), The Influenza Viruses, Plenum, New York, 1989, p. 319
[11] Fine, P. E. M.: Herd immunity: history theory and practice. Epidemiol. rev. 15, 265 (1993)
[12] Hampson, A. W.: Surveillance for pandemic influenza. J. inf. Dis. 176, S8 (1997)
[13] Webster, R. G.: Predictions for future human influenza pandemics. J. inf. Dis. 176, S14 (1997)
[14] Fox, J. P.: Interference phenomena observed in the field. Recent prog. Microbiol. 8, 443 (1982)
[15] Frank, A. L.; Taber, L. H.; Wells, J. M.: Individuals infected with two sub-types of influenza A virus in the same season. J. inf. Dis. 147, 120 (1983)
[16] A.S. Monto, J.S. Koopman, I.M. Longini Jr, The Tecumseh study of illness. XII. Influenza infection and disease, 1976--1981. Am. J. Epidemiol. 121 (1985) 811
[17] Ackerman, E.; Jr, I. M. Longini; Seaholm, S. K.; Hedin, A. S.: Simulation of mechanisms of viral interference in influenza. Int. J. Epidemiol. 19, 444 (1990)
[18] Couch, R. B.; Kasel, J. A.: Immunity to influenza in man. Ann. rev. Microbiol. 37, 529 (1983)
[19] Davies, J. R.; Grilli, E. A.; Smith, A. J.: Influenza A: infection and reinfection. J. hyg. Camb. 92, 125 (1984)
[20] Davies, J. R.; Grilli, E. A.; Smith, A. J.: Infection with influenza a H1N1. 2. The effect of past experience on natural challenge. J. hyg. Camb. 96, 345 (1986)
[21] A.J. Levine, Viruses, W.H. Freeman, New York, 1992, p. 155
[22] Dietz, K.: Epidemiologic interference of virus populations. J. math. Biol. 8, 291 (1979) · Zbl 0412.92024
[23] Levin, S. A.; Pimentel, D.: Selection of intermediate rates of increase in parasite-hosts systems. Am. nat. 117, 308 (1981)
[24] Castillo-Chavez, C.; Hethcote, H. W.; Andreasen, V.; Levin, S. A.; Liu, W.: Epidemiological models with age structure, proportionate mixing, and cross-immunity. J. math. Biol. 27, 233 (1989) · Zbl 0715.92028
[25] V. Andreasen, Multiple time scales in the dynamics of infectious diseases, in: C. Castillo-Chavez, S.A. Levin, C.A. Shoemaker (Eds.), Mathematical Approaches to Problems in Resource Management and Epidemiology, Springer, Berlin, 1989, p. 142 · Zbl 0682.92010
[26] Gupta, S.; Swinton, J.; Anderson, R. M.: Theoretical studies of the effects of heterogeneity in the parasite population on the transmission dynamics of malaria. Proc. R. Soc. lond. B 256, 231 (1994)
[27] W. Liu, S. A. Levin, Influenza and some related mathematical models, in: S.A. Levin, T.G. Hallam, L.J. Gross (Eds.), Applied Mathematical Ecology, Springer, 1989, p. 235
[28] Thacker, S. B.; Stroup, D. F.: Persistence of influenza A by continuous close-contact transmission: the effect of non-random mixing. Int. J . Epidemiol. 19, 1078 (1990)
[29] Andreasen, V.; Lin, J.; Levin, S. A.: The dynamics of cocirculating influenza strains conferring partial cross-immunity. J. math. Biol. 35, 825 (1997) · Zbl 0874.92023
[30] Buonagurio, D. A.; Nakada, S.; Parvin, J. D.; Krystal, M.; Palese, P.; Fitch, W. M.: Evolution of human influenza A viruses over 50 years: rapid uniform rate of change in NS change. Science 232, 980 (1986)
[31] Fitch, W. M.; Leiter, J. M.; Li, X.; Palese, P.: Positive Darwinian evolution in human influenza A viruses. Proc. nat. Acad. sci. USA 88, 4270 (1991)
[32] Nichol, S. T.; Rowe, J. E.; Fitch, W. M.: Punctuated equilibrium and positive Darwinian evolution in vesicular stomatitis virus. Proc. nat. Acad. sci. USA 90, 10424 (1993)
[33] Gillespie, J. H.: Episodic evolution of RNA viruses. Proc. nat. Acad. sci. USA 90, 10411 (1993)
[34] Spicer, C. C.; Lawrence, C. J.: Epidemic influenza in greater London. J. hyg. Camb. 93, 105 (1984)
[35] W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes in C, second edition, p. 486ff. Cambridge University, New York, 1992 · Zbl 0778.65003
[36] Y.A. Kuznetsov, V.V. Levitin, CONTENT 1.5. Available via www.cwi.nl/ftp
[37] Ferguson, N.; Anderson, R.; Gupta, S.: The effect of antibody-dependent enhancement on the transmission dynamics and persistence of multiple-strain pathogens. Proc. nat. Acad. sci. USA 96, 790 (1999)
[38] Simon, H.: The science of artificial life. (1969)
[39] Levin, S. A.: Fragile dominion. (1999)
[40] Bremermann, H. J.; Thieme, H. R.: A competitive exclusion principle for pathogen virulence. J. math. Biol. 27, 179 (1989) · Zbl 0715.92027
[41] Pease, C. M.: An evolutionary epidemiological mechanism with application to type A influenza. Theor. pop. Biol. 31, 422 (1987) · Zbl 0614.92012
[42] Andreasen, V.; Levin, S. A.; Lin, J.: A model of influenza A drift evolution. Z. angew. Math. mech. 76, No. 2, 421 (1996) · Zbl 0886.92023
[43] Eigen, M.; Mccaskill, J.: P. schuster. The molecular quasispecies. J. phys. Chem. 92, 6881 (1988)