Influenza and Some Related Mathematical Models

  • Wei-min Liu
  • Simon A. Levin
Part of the Biomathematics book series (BIOMATHEMATICS, volume 18)


Despite advances in biology and medical science that have controlled many severe infectious diseases, influenza remains a recurrent problem, initiating new global pandemics because of its ability to change its form. In 1918–1919, an influenza pandemic (Spanish flu) killed about 20 million people and infected perhaps 2 billion. The special feature of this pandemic was a tendency towards bronchopneumonic complications fatal to previously healthy young adults. In Philadelphia, people were dying so quickly that bodies were stacked by the hundreds in temporary morgues, awaiting burial. Such horrible mortality caused tremendous social and economic disruption, and stimulated intensive research into the cause of the disease (Beveridge, 1977).


Influenza Virus Swine Influenza Virus Sustained Oscillation Trivial Equilibrium Epidemic Threshold 
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  1. Ackerman, E., Elveback, L.R., Fox, J.P. (1984) Simulation of Infectious Disease Epidemics. Thomas, Springfield, 111.Google Scholar
  2. Anderson, R.M., May, R.M. (1986) The invasion, persistence and spread of infectious diseases within animal and plant communities. Phil. Trans. Roy. Soc. Lond. B314, 533–570Google Scholar
  3. Bailey, N.T.J. (1975) The Mathematical Theory of Infectious Diseases and Its Applications, 2nd edn. Griffin, LondonMATHGoogle Scholar
  4. Baroyan, O.V., Rvachev, L.A., Basilevsky, U.V., Ermakov, V.V., Frank, K.D., Rvachev, M.A. and Shashkov, V.A. (1971) Computer modelling of influenza epidemics for the whole country (USSR). Adv. Appl. Prob. 3, 224–226CrossRefGoogle Scholar
  5. Baroyan, O.V., Rvachev, L.A. (1977) Mathematics and epidemiology. Moscow: Znanie (in Russian)Google Scholar
  6. Beveridge, W.I.B. (1977) Influenza: the last great plague. Heinemann, LondonGoogle Scholar
  7. Castillo-Chavez, C., Hethcote, H.W., Andreasen, V., Levin, S.A., Liu W. (1988a) Cross immunity in the dynamics of homogeneous and heterogeneous populations. Proceedings of the second autumn course on mathematical ecologyGoogle Scholar
  8. Castillo-Chavez, C., Hethcote, H.W., Andreasen, V., Levin, S.A., Liu W. (1988b) Epidemiological models with age structure and proportionate mixingGoogle Scholar
  9. Choi, K., Thacker, S.B. (1981 a) An evaluation of influenza mortality surveillance, 1962–1979 (I) Time series forecasts of expected pneumonia and influenza deaths. Amer. J. Epid. 113, 215–226Google Scholar
  10. Choi, K., Thacker, S.B. (1981 b) An evaluation of influenza mortality surveillance, 1962–1979(11) Percentage of pneumonia and influenza deaths as an indicator of influenza activity. Amer. J. Epid. 113, 227–235Google Scholar
  11. Couch, R.B., Kasel, J.A. (1983) Immunity to influenza in man. Ann. Rev. Microbiol. 37, 529–49CrossRefGoogle Scholar
  12. Dietz, K. (1979) Epidemiologic interference of virus populations. J. Math. Biol. 8, 291–300CrossRefMATHMathSciNetGoogle Scholar
  13. Dobson, A.P., May, R.M. (1986) Patterns of invasions by pathogens and parasites. In: Mooney, H.A., Drake, J.A. (ed.) Ecology of biological invasions of North America and Hawaii. Springer, Berlin Heidelberg New YorkGoogle Scholar
  14. Elveback, L.R., Fox, J.P., Varma, A. (1964) An extension of the Reed-Frost epidemic model for the study of competition between viral agents in the presence of interference. Amer. J. Epid. 80, 356–364Google Scholar
  15. Elveback, L.R., Fox, J.P., Ackerman, E., Langworthy, A., Boyd, M., Gatewood, L. (1976) An influenza simulation model for immunization studies. Amer. J. Epid. 103, 152–165Google Scholar
  16. Fine, P. (1982) Background paper: applications of mathematical models to the epidemiology of influenza: a critique. In: P. Selby (ed.) Influenza models: prospects for development and use. Sandoz Institute for Health and Socio-economic Studies, pp. 15–85Google Scholar
  17. Friedman, R.M. (1981) Interferon: a primer. Academic Press, New York London Toronto Sydney San FransciscoGoogle Scholar
  18. Garnick, E. (1986) A theoretical consideration of resource specialism vs. generalism in parasites and some related questions. Ph.D. Thesis, Cornell UniversityGoogle Scholar
  19. Hale, J.K. (1969) Ordinary differential equations. Wiley Interscience, New YorkMATHGoogle Scholar
  20. Hethcote, W.H. (1978) An immunization model for a heterogeneous population. Theor. Pop. Biol. 14, 338–349CrossRefMathSciNetGoogle Scholar
  21. Holt, R.D., Pickering, J. (1986) Infectious disease and species coexistence: a model of Lotka Volterra form. Amer. Nat.Google Scholar
  22. Kermack, W.O., McKendrick, A.G. (1927) Contributions to the mathematical theory of epidemics, pt. I. Proc. Roy. Soc. All5, 700–721Google Scholar
  23. Kilbourne, E.D., (ed.) (1975) The influenza viruses and influenza. Academic Press, New York London Toronto Sydney San FransciscoGoogle Scholar
  24. Liu, W. (1987) Dynamics of epidemiological models-recurrent outbreaks in autonomous systems. Ph.D. Thesis, Cornell UniversityGoogle Scholar
  25. London, W.E., Yorke, J.A. (1973) Recurrent outbreaks of measles, chicken pox and mumps, I. Seasonal variation in contact rates. Amer. J. Epid. 98, 453–468Google Scholar
  26. Longini, I.M., Fine, P.E.M., Thacker, S.B. (1983) Predicting the global spread of new infectious agents. Amer. J. Epid. 123, 383–391Google Scholar
  27. Palese, P., Young, J.F. (1982) Variation of influenza A, B, and C viruses. Science 215, 1468–1474CrossRefGoogle Scholar
  28. Rvachev, L.A., Longini, I.M., Jr. (1985) A mathematical model for the global spread of influenza. Math. Biosci. 75, 3–22CrossRefMATHMathSciNetGoogle Scholar
  29. Selby, P. (ed.) (1982) Influenza models: prospects for development and use. Sandoz Institute for Health and Socio-economic StudiesGoogle Scholar
  30. Serfling, R.E. (1963) Methods for current statistical analysis of excess pneumonia-influenza deaths. Publ. Hlth. Rep. 78, 494–506CrossRefGoogle Scholar
  31. Shope, R.E. (1936) The incidence of neutralizing antibodies for swine influenza virus in the sera of human beings of different ages. J. Exp. Med. 63, 669–684CrossRefGoogle Scholar
  32. Spicer, C.C. (1979) The mathematical modelling of influenza epidemics. Brit. Med. Bull. 35, 23–28Google Scholar
  33. Stuart-Harris, C.H. Schild, G.C. (1976) Influenza, the viruses and the disease. Publishing Sciences Group, Littleton, Mass.Google Scholar
  34. Thacker, S.B. (1986) The persistence of influenza A in human population. Epid. Rev. 8, 129–142.Google Scholar
  35. Webster, R.G., Laver, W.G., Air, G.M., Schild, G.C. (1982) Molecular mechanisms of variation in influenza viruses. Nature 296, 115–121CrossRefGoogle Scholar

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© Springer-verlag Berlin Heidelberg 1989

Authors and Affiliations

  • Wei-min Liu
  • Simon A. Levin

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