Applied Mathematical Ecology pp 193-211

Part of the Biomathematics book series (BIOMATHEMATICS, volume 18)

Periodicity in Epidemiological Models

  • Herbert W. Hethcote
  • Simon A. Levin
Chapter

Abstract

Various epidemiological mechanisms have been shown to lead to periodic solutions. The most direct way in which periodicity arises is through extrinsic forcing by a parameter such as the contact rate, but periodicity can also arise autonomously. Cyclic models of SIRS or SEIRS type can have periodic solutions if there is a large time delay in the removed class. Epidemiological models with nonlinear incidence of certain general forms can have periodic solutions. Some models with variable population size and disease-related deaths have periodic solutions; most of these are host-parasite models where the parasite lifetime is much shorter than that of the host. Recently, periodic solutions have been found numerically in age structured models with cross immunity between two viral strains.

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References

  1. Anderson, R.M. (1982) Directly transmitted viral and bacterial infections of man. In: Anderson, R.M. (ed.) Population Dynamics of Infectious Diseases. Theory and Applications. Chapman and Hall, New York, pp. 1–37Google Scholar
  2. Anderson, R.M., Jackson, H.C., May, R.M., Smith, A.D.M. (1981) Populations dynamics of fox rabies in Europe. Nature 289, 765–777CrossRefGoogle Scholar
  3. Anderson, R.M., May, R.M. (1979) Population biology of infectious diseases I. Nature 280, 361–367CrossRefGoogle Scholar
  4. Anderson, R.M., May, R.M. (1981) The population dynamics of microparasites and their invertebrate hosts. Phil. Trans. Roy. Soc. London B291, 451–524Google Scholar
  5. Anderson, R.M., May, R.M. (1982) Directly transmitted infectious diseases: control by vaccination. Science 215, 1053–1060CrossRefMATHMathSciNetGoogle Scholar
  6. Anderson, R.M., May, R.M. (1983) Vaccination against rubella and measles: quantitative investigations-of different policies. J. Hyg. Camb. 90, 259–325CrossRefGoogle Scholar
  7. Andreasen, V. (1987) Dynamical behaviour of epidemiological models. Preprint Aron, J.L., Schwartz, I.B. (1984a) Seasonality and period-doubling bifurcations in an epidemic model. J. Theor. Biol. 110, 665–679Google Scholar
  8. Aron, J.L., Schwartz, I.B. (1984b) Some new directions for research in epidemic models, IMA J. Math. Appl. Med. Biol. 267–276Google Scholar
  9. Aronsson, G., Mellander, I. (1980) A deterministic model in biomathematics: asymptotic behavior and threshold conditions, Math. Biosci. 49, 207–222CrossRefMATHMathSciNetGoogle Scholar
  10. Bailey, N.T.J. (1975) The Mathematical Theory of Infectious Diseases, Second Edition, Hafner, New YorkMATHGoogle Scholar
  11. Bartlett, M.S. (1956) Deterministic and stochastic models for recurrent epidemics, Proc. Third Berkeley Symp. Math. Stat. Prob. 4, 81–109MathSciNetGoogle Scholar
  12. Bartlett, M.S. (1960) Stochastic Population Models in Ecology and Epidemiology, Methuen, LondonMATHGoogle Scholar
  13. Boland, W.R., Powers, M.W. (1977) A numerical technique for obtaining approximate solutions of certain functional equations arising in the theory of epidemics, Math. Biosci. 33, 297–319CrossRefMATHMathSciNetGoogle Scholar
  14. Busenberg, S.N., Cooke, K.L. (1978a) Periodic solutions of delay differential equations arising in some models of epidemics, in Proceedings of the Applied Nonlinear Analysis Conference, Univ. of Texas, Arlington, Academic Press, New YorkGoogle Scholar
  15. Busenberg, S.N., Cooke, K.L. (1978b) Periodic solutions of a periodic nonlinear delay differential equation, SIAM J. on Applied Math. 35, 704–721CrossRefMATHMathSciNetGoogle Scholar
  16. Busenberg, S.N., Cooke, K.L. (1980) The effect of integral conditions in certain equations modeling epidemics and population growth, J. Math. Biol. 10, 13–32CrossRefMATHMathSciNetGoogle Scholar
  17. Capasso, V., Serio, G. (1978) A generalization of the Kermack-McKendrick deterministic epidemic model, Math. Biosci. 42, 43–61CrossRefMATHMathSciNetGoogle Scholar
  18. Castillo-Chavez, C., Hethcote, H.W., Andreasen, V., Levin, S.A., Liu, W.M. (1988) Cross-immunity in the dynamics of homogeneous and heterogeneous populations. In Hallam, T.G., Gross, L. and Levin, S.A. (eds.) Mathematical Ecology, World Scientific Publishing Co., Singapore, 303–316Google Scholar
  19. Castillo-Chavez, C., Hethcote, H.W., Andreasen, V., Levin, S.A., Liu, W.M. (1989) Epidemiological models with age-structure and proportionate mixing, J. Math. Biology (to appear) Cooke, K.L. (1982) Models for endemic infections with asymptomatic cases: one group, Math. Modelling 3, 1–15MathSciNetGoogle Scholar
  20. Cooke, K., Kaplan, J. (1976) A periodicity threshold theorem for epidemics and population growth, Math. Biosci. 31, 87–104CrossRefMATHMathSciNetGoogle Scholar
  21. Cunningham, J. (1979) A deterministic model for measles, Z. Naturforsch 34c, 647–648Google Scholar
  22. Diekmann, O., Montijn, R. (1982) Prelude to Hopf bifurcation in an epidemic model: analysis of the characteristic equation associated with a nonlinear Volterra integral equation, J. Math. Biol. 14, 117–127CrossRefMATHMathSciNetGoogle Scholar
  23. Dietz, K. (1975) Transmission and control of arbovirus diseases, in Epidemiology, SIMS 1974 Utah Conference Proceedings, SIAM, Philadelphia, pp. 104–121Google Scholar
  24. Dietz, K. (1976) The incidence of infectious diseases under the influence of season fluctuations, in Mathematical Models in Medicine, Lecture Notes in Biomathematics, No. 11, Springer-Verlag, New York, pp. 1–15Google Scholar
  25. Dietz, K. (1981) The evaluation of rubella vaccination strategies. In: Hiorns, R.W., Cooke. D. (eds.) The Mathematical Theory of the Dynamics of Biological Populations, vol. II. Academic Press, London, pp. 81–87Google Scholar
  26. Dietz, K., Schenzle, D. (1985a) Mathematical models for infectious disease statistics. In: Atikinson, A.C., Fienberg, S.E. (eds.) A Celebration of Statistics. Springer-Verlag, New York, pp. 167–204CrossRefGoogle Scholar
  27. Dietz, K., Schenzle, D. (1985b) Proportionate mixing models for age-dependent infection transmission, J. Math. Biol. 22, 117–120CrossRefMATHMathSciNetGoogle Scholar
  28. El-Doma, M. (1987) Analysis of nonlinear integro-difTerential equations arising in age-dependent epidemic models, Nonlinear Anal. TMA 11, 913–937CrossRefMATHMathSciNetGoogle Scholar
  29. Enderle, J.D. (1980) A stochastic communicable disease model with age-specific states and applications to measles, Ph.D. dissertation, Rensselaer Polytechnic InstituteGoogle Scholar
  30. Fine, P.E.M., Clarkson, J.A. (1982) Measles in England and Wales. I. An analysis of factors underlying seasonal patterns, Int. J. Epidem. 11, 5–14CrossRefGoogle Scholar
  31. Fine, P.E.M., Clarkson, J.A. (1983) Measles in England and Wales. III. Assessing published predictions of the impact of vaccination on incidence, Int. J. Epidem. 12, 332–339CrossRefGoogle Scholar
  32. Gabriel, J.P., Hanisch, H., Hirsch, W.M. (1981) Dynamic equilibria of helminthic infections. In: Chapman, D.G., Gallucci, V.F. (eds.) Quantitative Population Dynamics. Intern. Cooperative Publ. House, Maryland, Stat. Ecology Series 13, 83–104Google Scholar
  33. Gani, J. (1978) Some problems in epidemic theory, J. Roy. Statist. Soc. Ser. A140, 323–347CrossRefMathSciNetGoogle Scholar
  34. Green, D. (1978) Self-oscillations for epidemic models, Math. Biosci. 38, 91–111CrossRefMATHMathSciNetGoogle Scholar
  35. Gripenberg, G. (1980) Periodic solutions of an epidemic model, J. Math. Biol. 10, 271–280CrossRefMATHMathSciNetGoogle Scholar
  36. Grossman, Z. (1980) Oscillatory phenomena in a model of infectious diseases, Theor. Pop. Biol. 18, 204–243CrossRefMATHMathSciNetGoogle Scholar
  37. Grossman, Z., Gumowski, I., Dietz, K. (1977) The incidence of infectious diseases under the influence of seasonal fluctuations-analytic approach, in Nonlinear Systems and Applications to Life Sciences, Academic Press, New York, pp. 525–546Google Scholar
  38. Hethcote, H.W. (1973) Asymptotic behavior in a deterministic epidemic model, Bull. Math. Biology 35, 607–614MATHGoogle Scholar
  39. Hethcote, H.W. (1976) Qualitative analysis for communicable disease models, Math. Biosci. 28,335–356CrossRefMATHMathSciNetGoogle Scholar
  40. Hethcote, H.W., (1983) Measles and rubella in the United States, Am. J. Epidemiol. 117, 2–13Google Scholar
  41. Hethcote, H.W. (1986) Choosing a strategy for rubella vaccination, in Proceedings of the Third International Colloquium on Theoretical Biology and Medicine: Models in Epidemiology, Fontevraud, France, SeptemberGoogle Scholar
  42. Hethcote, H.W. (1989) Three basic epidemiological models. In: Levin, S.A., Hallam, T.G. and Gross, L. (eds.) Applied Mathematical Ecology, Biomathematics vol. 18, Springer-Verlag, Berlin, Heidelberg, New York, 119–144Google Scholar
  43. Hethcote, H.W., Lewis, M.A., van den Driessche, P. (1989) An epidemiological model with a delay and a nonlinear incidence rate, J. Math. Biol. 27, 49–64CrossRefMATHMathSciNetGoogle Scholar
  44. Hethcote, H.W., Stech, H.W., van den Driessche, P. (1981a) Nonlinear oscillations in epidemic models. SIAM J. Appl. Math. 40, 1–9CrossRefMATHMathSciNetGoogle Scholar
  45. Hethcote, H.W., Stech, H.W., van den Driessche, P. (1981b) Stability analysis for models of diseases without immunity. J. Math. Biology 13, 185–198CrossRefMATHGoogle Scholar
  46. Hethcote, H.W., Stech, H.W., van den Driessche, P. (1981c) Periodicity and stability in epidemic models: a survey, In: Busenberg, S., Cooke, K.L. (eds.) Differential Equations and Applications in Ecology, Epidemics and Populations Problems. Academic Press, New York, pp. 65–82Google Scholar
  47. Hethcote, H.W., Tudor, D.W. (1980) Integral equation models for endemic infectious diseases, J. Math. Biol. 9, 37–47CrossRefMATHMathSciNetGoogle Scholar
  48. Hethcote, H.W., Van Ark, J.W. (1987) Epidemiological models for heterogeneous populations: proportionate mixing, parameter estimation and immunization programs, Math. Biosci. 84, 85–118CrossRefMATHMathSciNetGoogle Scholar
  49. Hethcote, H.W., Yorke, J.A. (1984) Gonorrhea Transmission Dynamics and Control. Lecture Notes in Biomathematics, vol. 56. Springer, Berlin Heidelberg New YorkGoogle Scholar
  50. Hirsch, M.W. (1984) The differential equations approach to dynamical systems, Bull. Amer. Math. Soc. 11, 1–64CrossRefMATHMathSciNetGoogle Scholar
  51. Hoppensteadt, F. (1975) Mathematical Theories of Populations: Demographics, Genetics and Epidemics, SIAM, PhiladelphiaGoogle Scholar
  52. Hoppensteadt, F., Waltman, P. (1971) A problem in the theory of epidemics II, Math. Biosci. 12,133–145CrossRefMATHMathSciNetGoogle Scholar
  53. Kermack, W.O., Mckendrick, A.G. (1927) A contribution to the mathematical theory of epidemics, Proc. Roy. Soc. AU5, 700–721Google Scholar
  54. Knox, E.G. (1980) Strategy for rubella vaccination, Int. J. Epidemiol. 9, 13–23CrossRefGoogle Scholar
  55. Lajmanovich, A., Yorke, J.A. (1976) A deterministic model for gonorrhea in a nonhomogeneous population, Math. Biosci. 28, 221–236CrossRefMATHMathSciNetGoogle Scholar
  56. Liu, W.M. (1987) Dynamics of epidemiological models-recurrent outbreaks in autonomous systems, Ph.D. Thesis. Cornell UniversityGoogle Scholar
  57. Liu, W.M., Levin, S.A., Iwasa, Y. (1986) Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J. Math. Biol. 23, 187–204CrossRefMATHMathSciNetGoogle Scholar
  58. Liu, W.M., Hethcote, H.W., Levin, S.A. (1987) Dynamical behavior of epidemiological models with nonlinear incidence rates. 25, 359–380MATHMathSciNetGoogle Scholar
  59. London, W.P., Yorke, J.A. (1973) Recurrent outbreaks of measles, chickenpox and mumps, I, Am. J. Epid. 98, 453–468Google Scholar
  60. Longini, Jr., I.M., Ackerman, E., Elveback, L.R. (1978) An optimization model for influenza A epidemics, Math. Biosci. 38, 141–157CrossRefGoogle Scholar
  61. May, R.M., Anderson, R.M. (1978) Regulation and stability of hostparasite population interactions. II. Destabilizing processes, J. Animal Ecology 47, 249–267CrossRefGoogle Scholar
  62. May, R.M., Anderson, R.M. (1979) Population biology of infectious diseases II, Nature 280, 455–461CrossRefGoogle Scholar
  63. Mena, J. (1988) Periodicity and stability in epidemiological models with disease-related deaths, Ph.D. Thesis in Mathematics, University of IowaGoogle Scholar
  64. Mosevich, J. (1975) A numerical method for approximating solutions to the functional equation arising in the epidemic model of Hoppensteadt and Waltman, Math. Biosci. 24, 333–344CrossRefMATHMathSciNetGoogle Scholar
  65. Nussbaum, R. (1977) Periodic solutions of some integral equations from the theory of epidemics, in Nonlinear Systems and Applications to Life Sciences. Academic Press, New York, pp. 235–255Google Scholar
  66. Nussbaum, R. (1978) A periodicity threshold theorem for some nonlinear integral equations, SIAM J. Math. Anal. 9, 356–376CrossRefMATHMathSciNetGoogle Scholar
  67. Ross, R. (1916) An application of the theory of probabilities to the study of a priori pathometry, Part I, Proc. Roy. Soc. A92, 204–230Google Scholar
  68. Ross, R., Hudson, H.P. (1917) An application of the theory of probabilities to the study of a priori pathometry-Part III, Proc. Roy. Soc. A93, 225–240Google Scholar
  69. Schaffer, W.M. (1985) Can nonlinear dynamics help us infer mechanisms in ecology and epidemiology?, IMA J. Math. Appl. Biol. Med. 2, 221–252CrossRefMATHMathSciNetGoogle Scholar
  70. Schaffer, W.M., Kot, M. (1985) Nearly one dimensional dynamics in an epidemic, J. Theor. Biol. 112, 403–427CrossRefMathSciNetGoogle Scholar
  71. Schenzle, D. (1984) An age structured model of pre and post-vaccination measles transmission, IMAJ. Math. Appl. Biol. Med. 1, 169–191CrossRefMATHMathSciNetGoogle Scholar
  72. Schwartz, I.B. (1983) Estimating regions of existence of unstable periodic orbits using computer-based techniques, SIAM J. Num. Anal. 20, 106–120CrossRefMATHGoogle Scholar
  73. Schwartz, I.B. (1985) Multiple recurrent outbreaks and predictability in seasonally forced nonlinear epidemic models, J. Math. Biol. 21, 347–361CrossRefMATHMathSciNetGoogle Scholar
  74. Schwartz, I.B. (1988) Nonlinear dynamics of seasonally driven epidemic models, PreprintGoogle Scholar
  75. Schwartz, I.B., Smith, H.L. (1983) Infinite subharmonic bifurcations in an SEIR model, J. Math. Biol.18, 233–253CrossRefMATHMathSciNetGoogle Scholar
  76. Severo, N.C. (1969) Generalizations of some stochastic epidemic models, Math. Biosci. 4, 395–402CrossRefMATHMathSciNetGoogle Scholar
  77. Smith, H.L. (1977) On periodic solutions of a delay integral equation modeling epidemics, J. Math. Biol. 4, 69–80CrossRefMATHMathSciNetGoogle Scholar
  78. Smith, H.L. (1978) Periodic solutions for a class of epidemic equations, J. Math. Anal, and Applic. 64, 467–479CrossRefMATHGoogle Scholar
  79. Smith, H.L. (1979) Periodic solutions for an epidemic model with a threshold, Rocky Mountain J. of Math. 9, 131–142CrossRefMATHGoogle Scholar
  80. Smith, H.L. (1983) Subharmonic bifurcation in an SIR epidemic model, J. Math. Biology 17, 163–177CrossRefMATHGoogle Scholar
  81. Smith, H.L. (1983) Multiple stable subharmonics for a periodic epidemic model, J. Math. Biology 17, 179–190CrossRefMATHGoogle Scholar
  82. Smith, H.L. (1983) Hopf bifurcation in a system of functional equations modeling the spread of an infectious disease, SIAM J. Appi. Math. 43, 370–385CrossRefMATHGoogle Scholar
  83. Smith, H.L. (1986) Cooperative systems of differential equations with concave nonlinearities, J. Nonlin. Anal. T.M.A. 10, 1037–1052CrossRefMATHGoogle Scholar
  84. Stech, H.W., Williams, M. (1981) Stability for a class of cyclic epidemic models with delay, J. Math. Biol. 11, 95–103CrossRefMATHMathSciNetGoogle Scholar
  85. Stirzaker, D.R. (1975) A perturbation method for the stochastic recurrent epidemic, J. Inst. Maths Applies 15, 135–160CrossRefMATHMathSciNetGoogle Scholar
  86. Takens, F. (1981) Detecting strange attractors in turbulence, in Dynamical Systems and Turbulence. In: Rand, D.A., Young, L.S. (eds.) Warwick, 1980, Springer-Verlag, New York, pp. 366–381Google Scholar
  87. Tudor, D.W. (1985) An age dependent epidemic model with application to measles, Math. Biosci. 73, 131–147CrossRefMATHMathSciNetGoogle Scholar
  88. van den Driessche, P. (1981) An SIRS model with constant temporary immunity and constant births and deaths. In: Freedman, H.I., Strobeck, D. (eds.) Population Biology. Lecture Notes in Biomathematics, vol. 52. Springer, Berlin Heidelberg New York, pp. 433–440Google Scholar
  89. Wang, F.J.S. (1978) Asymptotic behavior of some deterministic epidemic models, SIAM J. Math. Anal. 9, 529–534CrossRefMATHMathSciNetGoogle Scholar
  90. Wilson, E.B., Worcester, J. (1945) The law of mass action in epidemiology, Proc. N.A.S. 31, 24–34CrossRefMathSciNetGoogle Scholar
  91. Wilson, E.B., Worcester, J. (1945) The law of mass action in epidemiology, II., Proc. N.A.S. 31,109–116CrossRefMathSciNetGoogle Scholar
  92. Yorke, J.A., London, W.P. (1973) Recurrent outbreaks of measles, chickenpox and mumps II, Am. J. Epid. 98, 469–482Google Scholar

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© Springer-Verlag Berlin Heidelberg 1989

Authors and Affiliations

  • Herbert W. Hethcote
  • Simon A. Levin

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