Species Coexistence and Periodicity in Host-Host-Pathogen Models



Models for the transmission of an infectious disease in one and two host populations with and without self-regulation are analyzed. Many unusual behaviors such as multiple positive equilibria and periodic solutions occur in previous models that use the mass-action (density-dependent) incidence. In contrast, the models formulated using the frequency-dependent (standard) incidence have the behavior of a classic endemic model, since below the threshold, the disease dies out, and above the threshold, the disease persists and the infectious fractions approach an endemic equilibrium. The results given here reinforce previous examples in which there are major differences in behavior between models using mass-action and frequency-dependent incidences.

Keywords or Pharses

Coexistence Regulation Host-pathogen Infectious disease Dynamics 


  1. 1.
    Anderson, R. M., Jackson, H. C., May, R. M., Smith, A. D. M.: Population dynamics of fox rabies in Europe. Nature 289, 765–777 (1981)Google Scholar
  2. 2.
    Anderson, R. M., May, R. M.: Regulation and stability of host-parasite population interactions I: Regulatory processes. J. Animal Ecology 47, 219–247 (1978)Google Scholar
  3. 3.
    Anderson, R. M., May, R. M.: Population biology of infectious diseases: Part I. Nature 280, 361–367 (1979)Google Scholar
  4. 4.
    Anderson, R. M., May, R. M. (Eds.): Population Biology of Infectious Diseases. Springer-Verlag, Berlin, Heidelberg, New York, 1982Google Scholar
  5. 5.
    Anderson, R. M., May, R. M.: The invasion, persistence, and spread of infectious diseases within animal and plant communities. Trans. R. Soc. Lond. B 314, 533–570 (1986)Google Scholar
  6. 6.
    Anderson, R. M., May, R. M.: Infectious Diseases of Humans: Dynamics and Control. Oxford University Press, Oxford, 1991Google Scholar
  7. 7.
    Becker, N., Angelo, J.: On estimating the contagiousness of the disease transmitted from person to person. Math. Biosci. 54, 137–154 (1981)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Begon, M., Bennett, M., Bowers, R. G., French, N. P., Hazel, S. M., Turner, J.: A clarification of transmission terms in host-microparasite models: numbers, densities and areas. Epidemiol. Infect. 129, 147–153 (2002)CrossRefGoogle Scholar
  9. 9.
    Begon, M., Bowers, R. G.: Beyond host-pathogen dynamics. In: Grenfell, B. T., Dobson, A. P. (Eds.), Ecology of Disease in Natural Populations, Cambridge University Press, Cambridge, 1995, pp. 479–509Google Scholar
  10. 10.
    Begon, M., Bowers, R. G., Kadianakis, N., Hodgkinson, D. E.: Disease and community structure: The importance of host self-regulation in a host host-pathogen model. Am. Nat. 139, 1131–1150 (1992)CrossRefGoogle Scholar
  11. 11.
    Begon, M., Hazel, S. M., Telfer, S., Bown, K., Cavanagh, R., Chantrey, J., Jones, T., Bennett, M.: Transmission dynamics of a zoonotic pathogen within and between wildlife host species. Proc. R. Soc. Lond. B 266, 1939–1945 (1999)Google Scholar
  12. 12.
    Begon, M., Hazel, S. M., Baxby, D., Bown, K., Carslake, D., Cavanagh, R., Chantrey, J., Jones, T., Bennett, M.: Rodents, cowpox virus and islands: Densities, numbers and thresholds. J. Animal Ecol. 72, 343–355 (2003)Google Scholar
  13. 13.
    Bonsall, M. B., Hassell, M. P.: Apparent competition structures ecological assemblages, Nature 388, 371–373 (1997)Google Scholar
  14. 14.
    Bouma, A., de Jong, M. C. M., Kimman, T. G.: Transmission of pseudorabies virus within pig populations is independent of the size of the population. Prev. Vet. Med. 23, 163–172 (1995)CrossRefGoogle Scholar
  15. 15.
    De Jong, M. C. M., Diekmann, O., Heesterbeek, J. A. P.: How does transmission depend on population size? In: Mollison, D. (Ed.), Epidemic Models: Their Structure and Relation to Data, Cambridge University Press, Cambridge, 1995, pp. 84–94Google Scholar
  16. 16.
    Diekmann, O., De Jong, M. C. M., De Koeijer, A. A., Reijnders, P.: The force of infection in populations of varying size: A modelling problem. J. Biol. Syst. 3, 519–529 (1995)Google Scholar
  17. 17.
    Diekmann, O., Heesterbeek, J. A. P.: Mathematical Epidemiology of Infectious Diseases. John Wiley and Sons, Chichester, England, 2000Google Scholar
  18. 18.
    Diekmann, O., Heesterbeek, J. A. P., Metz, J. A. J.: On the definition and the computation of the basic reproduction ratio in models for infectious diseases in heterogeneous populations. J. Math. Biol. 28, 365–382 (1990)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Dietz, K.: Overall population patterns in the transmission cycle of infectious disease agents. In: Population Biology of Infectious Diseases, Anderson, R. M., May, R. M. (eds.), Springer-Verlag, Berlin, Heidelberg, New York, 1982, pp. 87–102Google Scholar
  20. 20.
    Dobson, A. P., Meagher, M.: The population dynamics of brucellosis in the Yellowstone National Park. Ecology 77, 1026–1036 (1996)Google Scholar
  21. 21.
    Fromont, E., Artois, M., Langlais, M., Courchamp, F., Pontier, D.: Modelling the Feline Leukemia Virus (FeLV) in natural populations of cats (Felis catus). Theor. Pop. Biol. 52, 60–70 (1997)CrossRefMATHGoogle Scholar
  22. 22.
    Gao, L. Q., Hethcote, H. W.: Disease transmission models with density-dependent demographics. J. Math. Biol. 30, 717–731 (1992)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Gao, L. Q., Mena-Lorca, J., Hethcote, H. W.: Four SEI endemic models with periodicity and separatrices. Math. Biosci. 128, 157–184 (1995)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Gao, L. Q., Mena-Lorca, J., Hethcote, H. W.: Variations on a theme of SEI endemic models. In: Martelli, M., Cooke, K., Cumberbatch, E., Tang, B., Thieme, H. (eds.), Differential Equations and Applications to Biology and to Industry, World Scientific Publishing, Singapore, 1996, pp. 191–207Google Scholar
  25. 25.
    Greenman, J. V., Hudson, P. J.: Infected coexistence instability with and without density-dependent regulation. J. Theor. Biol. 185, 345–356 (1997)CrossRefGoogle Scholar
  26. 26.
    Greenwood, M., Bradford Hill, A. T., Topley, W. W. C., Wilson, J.: Experimental Epidemiology. MRC Special Report Series 209, 1936Google Scholar
  27. 27.
    Grenfell, B. T., Dobson, A. P. (Eds.): Ecology of Disease in Natural Populations. Cambridge University Press, Cambridge, 1995Google Scholar
  28. 28.
    Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer-Verlag, New York, 1983Google Scholar
  29. 29.
    Hale, J. K.: Ordinary Differential Equations. John Wiley, New York, 1969Google Scholar
  30. 30.
    Heesterbeek, J. A. P., Metz, J. A. J.: The saturating contact rate in marriage- and epidemic models. J. Math. Biol. 31, 529–539 (1993)CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    Hethcote, H. W.: Qualitative analyses of communicable disease models. Mathematical Biosciences 28, 335–356 (1976)CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    Hethcote, H. W.: An immunization model for a heterogeneous population. Theor. Popul. Biol. 14, 338–349 (1978)MATHMathSciNetGoogle Scholar
  33. 33.
    Hethcote, H. W.: A thousand and one epidemic models. In: Levin, S. A. (Ed.), Frontiers in Theoretical Biology, Volume 100 of Lecture Notes in Biomathematics, Springer-Verlag, Berlin, 1994, pp. 504–515Google Scholar
  34. 34.
    Hethcote, H. W.: Modeling heterogenous mixing in infectious disease models. In: Isham, V., Medley, G. (Eds.), Models for Infectious Human Diseases, Cambridge University Press, Cambridge, 1996, pp. 215–238Google Scholar
  35. 35.
    Hethcote, H. W.: The mathematics of infectious diseases. SIAM Review 42, 599–653 (2000)CrossRefMATHMathSciNetGoogle Scholar
  36. 36.
    Hethcote, H. W., Levin, S. A.: Periodicity in epidemiological models. In: Gross, L., Hallam, T. G., Levin, S. A. (eds.), Applied Mathematical Ecology, Springer-Verlag, Berlin, 1989, pp. 193–211Google Scholar
  37. 37.
    Hethcote, H. W., Van Ark, J. W.: Epidemiological models with heterogeneous populations: proportional mixing, parameter estimation and immunization programs. Math. Biosci. 84, 85–118 (1987)CrossRefMATHMathSciNetGoogle Scholar
  38. 38.
    Hethcote, H. W., Wang, W., Han, L., Ma, Z.: A predator-prey model with infected prey. Theor. Pop. Biol. 66, 259–268 (2004)CrossRefGoogle Scholar
  39. 39.
    Hirsch, W. M., Smith, H. L., Zhao, X.-Q.: Chain transitivity, attractivity, and strong repellors for semidynamical systems. J. Dynam. Diff. Equ. 13, 107–131 (2001)MATHMathSciNetGoogle Scholar
  40. 40.
    Holt, R. D.: Predation, apparent comptetition and the structure of prey communities. Theor. Popul. Biol. 12, 197–229 (1977)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Holt, R. D., Pickering, J.: Infectious disease and species coexistence: A model of Lotka-Volterra form. Am. Nat. 126, 196–211 (1985)CrossRefGoogle Scholar
  42. 42.
    Horn, R. A., Johnson, C. R.: Matrix Analysis. Cambridge University Press, Cambridge, 1990Google Scholar
  43. 43.
    Lloyd-Smith, J. O., Getz, W. M., Westerhoff, H. V.: frequency-dependent incidence in models of sexually transmitted diseases: Portrayal of pair-based transmission and effects of illness on contact behaviour. Proc. R. Soc. Lond. B 271, 625–635 (2004)CrossRefGoogle Scholar
  44. 44.
    May, R. M., Anderson, R. M.: Regulation and stability of host-parasite population interactions II: Destabilizing processes. J. Animal Ecology 47, 249–267 (1978)MathSciNetGoogle Scholar
  45. 45.
    McCallum, H., Barlow, N., Hone, J.: How should pathogen transmission be modelled? Trends Ecol. Evol. 16, 295–300 (2001)Google Scholar
  46. 46.
    Mena-Lorca, J., Hethcote, H. W.: Dynamic models of infectious diseases as regulators of population sizes. J. Math. Biol. 30, 693–716 (1992)MATHMathSciNetGoogle Scholar
  47. 47.
    Roberts, M. G.: The dynamics of bovine tuberculosis in possum populations, and its eradication or control by culling or vaccination. J. Anim. Ecol. 65, 451–464 (1996)Google Scholar
  48. 48.
    Shen, J., Jing, Z.-J.: A new detecting method for conditions of existence of Hopf bifurcation. Acta Mathematica Applicatae Sinica 11(1), 79–93 (1993)Google Scholar
  49. 49.
    Smith, H. L.: Monotone Dynamical Systems. American Mathematical Society, Providence, Rhode Island, 1995Google Scholar
  50. 50.
    Strogatz, S. H.: Nonlinear Dynamics and Chaos. Addison-Wesley, Reading, Massachusetts, 1994Google Scholar
  51. 51.
    Thieme, H. R.: Convergence results and a Poincare-Bendixson trichotomy for asymptotically autonomous differential equations. J. Math. Biol. 30, 755–763 (1992)CrossRefMATHMathSciNetGoogle Scholar
  52. 52.
    Turner, J., Begon, M., Bowers, R. G.: Modelling pathogen transmission: The interrelationship between local and global approaches. Proc. R. Soc. Lond. B 270, 105–112 (2003)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of IowaIowa CityUSA
  2. 2.Department of MathematicsHunan Normal UniversityHunanP.R. China
  3. 3.University of IowaIowa CityUSA
  4. 4.Department of MathematicsSouthwest Normal UniversityChongqingP. R. China

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