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Endemic Disease Models

  • Fred Brauer
  • Carlos Castillo-Chavez
  • Zhilan Feng
Chapter
Part of the Texts in Applied Mathematics book series (TAM, volume 69)

Abstract

In this chapter, we consider models for disease that may be endemic. In the preceding chapter we studied SIS models with and without demographics and SIR models with demographics. In each model, the basic reproduction number \(\mathcal {R}_0\) determined a threshold. If \(\mathcal {R}_0 < 1\) the disease dies out, while if \(\mathcal {R}_0 > 1\) the disease becomes endemic. The analysis in each case involves determination of equilibria and determining the asymptotic stability of each equilibrium by linearization about the equilibrium. In each of the cases studied in the preceding chapter the disease-free equilibrium was asymptotically stable if and only if \(\mathcal {R}_0 < 1\) and if \(\mathcal {R}_0 > 1\) there was a unique endemic equilibrium that was asymptotically stable. In this chapter, we will see that these properties continue to hold for many more general models, but there are situations in which there may be an asymptotically stable endemic equilibrium when \(\mathcal {R}_0 < 1\), and other situations in which there is an endemic equilibrium that is unstable for some values of \(\mathcal {R}_0 > 1\).

References

  1. 1.
    Aparicio, J.P., A. Capurro, and C. Castillo-Chavez (2000) Markers of disease evolution: the case of tuberculosis, J. Theor. Biol., 215: 227–238.MathSciNetCrossRefGoogle Scholar
  2. 2.
    Aparicio J. P.,A.F. Capurro, and C. Castillo-Chavez (2000) On the fall and rise of tuberculosis. Department of Biometrics, Cornell University, Technical Report Series, BU-1477-M.Google Scholar
  3. 3.
    Aparicio J.P., A. Capurro, and C. Castillo-Chavez (2002) Frequency Dependent Risk of Infection and the Spread of Infectious Diseases. In: Mathematical Approaches for Emerging and Reemerging Infectious Diseases : Models, Methods and Theory, Edited by Castillo-Chavez, C. with S. Blower, P. van den Driessche, D. Kirschner, and A.A. Yakubu, Springer-Verlag (2002), pp. 341–350.Google Scholar
  4. 4.
    Aparicio, J.P., A. Capurro, and C. Castillo-Chavez (2002) On the long-term dynamics and re-emergence of tuberculosis. In: Mathematical Approaches for Emerging and Reemerging Infectious Diseases: Models, Methods and Theory, Edited by Castillo-Chavez, C. with S. Blower, P. van den Driessche, D. Kirschner, and A.A. Yakubu, Springer-Verlag, (2002) pp. 351–360.Google Scholar
  5. 5.
    Bellman, R.E. and K.L. Cooke (1963) Differential-Difference Equations, Academic Press, New York.zbMATHGoogle Scholar
  6. 6.
    Blower, S.M. & A.R. Mclean (1994) Prophylactic vaccines, risk behavior change, and the probability of eradicating HIV in San Francisco, Science 265: 1451–1454.CrossRefGoogle Scholar
  7. 7.
    Boldin, B. and O. Diekmann (2008) Superinfections can induce evolutionarily stable coexistence of pathogens, J. Math. Biol. 56: 635–672.MathSciNetCrossRefGoogle Scholar
  8. 8.
    Busenberg, S. and K.L. Cooke (1993) Vertically Transmitted Diseases: Models and Dynamics, Biomathematics 23, Springer-Verlag, Berlin-Heidelberg-New York.CrossRefGoogle Scholar
  9. 9.
    Castillo-Chavez, C., and Z. Feng (1997) To treat or not to treat; the case of tuberculosis, J. Math. Biol, 35: 629–656.MathSciNetCrossRefGoogle Scholar
  10. 10.
    Castillo-Chavez, C., and Z. Feng (1998) Global stability of an age-structure model for TB and its applications to optimal vaccination strategies, Math. Biosc. 151: 135–154.CrossRefGoogle Scholar
  11. 11.
    Castillo-Chavez, C. and Z. Feng (1998) Mathematical models for the disease dynamics of tuberculosis, Advances in Mathematical Population Dynamics - Molecules, Cells, and Man (O. Arino, D. Axelrod, M. Kimmel, (eds)): 629–656, World Scientific Press, Singapore.Google Scholar
  12. 12.
    Cen, X., Z. Feng and Y. Zhao (2014) Emerging disease dynamics in a model coupling within-host and between-host systems, J. Theor. Biol. 361: 141–151.MathSciNetCrossRefGoogle Scholar
  13. 13.
    Conlan, A.J.K. and B.T. Grenfell (2007) Seasonality and the persistence and invasion of measles, Proc. Roy. Soc. B. 274: 1133–1141.CrossRefGoogle Scholar
  14. 14.
    Dushoff, J., W. Huang, & C. Castillo-Chavez (1998) Backwards bifurcations and catastrophe in simple models of fatal diseases, J. Math. Biol. 36: 227–248.MathSciNetCrossRefGoogle Scholar
  15. 15.
    Feng, Z., C. Castillo-Chavez, and A. Capurro (2000) A model for TB with exogenous re-infection, J. Theor. Biol. 5: 235–247.CrossRefGoogle Scholar
  16. 16.
    Feng, Z., W. Huang and C. Castillo-Chavez (2001) On the role of variable latent periods in mathematical models for tuberculosis, J. Dynamics and Differential Equations 13: 425–452.MathSciNetCrossRefGoogle Scholar
  17. 17.
    Feng, Z., J. Velasco-Hernandez and B. Tapia-Santos (2013) A mathematical model for coupling within-host and between-host dynamics in an environmentally-driven infectious disease, Math. Biosc. 241: 49–55.MathSciNetCrossRefGoogle Scholar
  18. 18.
    Feng, Z., D. Xu, & H. Zhao (2007) Epidemiological models with non-exponentially distributed disease stages and applications to disease control, Bull. Math. biol. 69: 1511–1536.MathSciNetCrossRefGoogle Scholar
  19. 19.
    Gilchrist, M.A. and D. Coombs (2006) Evolution of virulence: interdependence, constraints, and selection using nested models, Theor. Pop. Biol. 69: 145–153.CrossRefGoogle Scholar
  20. 20.
    Hadeler, K.P. & C. Castillo-Chavez (1995) A core group model for disease transmission, Math Biosc. 128: 41–55.CrossRefGoogle Scholar
  21. 21.
    Hadeler, K.P. and P. van den Driessche (1997) Backward bifurcation in epidemic control, Math. Biosc. 146: 15–35.MathSciNetCrossRefGoogle Scholar
  22. 22.
    Hethcote, H.W. (1978) An immunization model for a heterogeneous population, Theor. Pop. Biol. 14: 338–349.MathSciNetCrossRefGoogle Scholar
  23. 23.
    Hethcote, H.W. and D.W. Tudor (1980) Integral equation models for endemic infectious diseases, J. Math. Biol. 9: 37–47.MathSciNetCrossRefGoogle Scholar
  24. 24.
    Hethcote, H.W., H.W. Stech, and P. van den Driessche (1981) Periodicity and stability in epidemic models: A survey, in Differential Equations and Applications in Ecology, Epidemics, and Population Problems, S. Busenberg and K.L. Cooke (eds.), Academic Press, New York, pages 65–82.CrossRefGoogle Scholar
  25. 25.
    Hopf, E. (1942) Abzweigung einer periodischen Lösungen von einer stationaren Lösung eines Differentialsystems, Berlin Math-Phys. Sachsiche Akademie der Wissenschaften, Leipzig, 94: 1–22.Google Scholar
  26. 26.
    Hurwitz, A. (1895) Über die Bedingungen unter welchen eine Gleichung nur Wurzeln mit negativen reellen Teilen bezizt, Math. Annalen 46: 273–284.CrossRefGoogle Scholar
  27. 27.
    Keeling, M.J. and B.T. Grenfell (1997) Disease extinction and community size: Modeling the persistence of measles, Science 275: 65–67.CrossRefGoogle Scholar
  28. 28.
    Kermack, W.O. and A.G. McKendrick (1932) Contributions to the mathematical theory of epidemics, part. II, Proc. Roy. Soc. London, 138: 55–83.CrossRefGoogle Scholar
  29. 29.
    Kribs-Zaleta, C.M. and J.X. Velasco-Hernandez (2000) A simple vaccination model with multiple endemic states, Math Biosc. 164: 183–201.CrossRefGoogle Scholar
  30. 30.
    Lloyd, A. L. (2001) Destabilization of epidemic models with the inclusion of realistic distributions of infectious periods, Proc. Royal Soc. London. Series B: Biological Sciences 268: 985–993.Google Scholar
  31. 31.
    Lloyd, A. L. (2001b) Realistic distributions of infectious periods in epidemic models: changing patterns of persistence and 60 (1), 59–71.Google Scholar
  32. 32.
    MacDonald, N. (1978) Time lags in biological models (Vol. 27), Heidelberg, Springer-Verlag.Google Scholar
  33. 33.
    McNeill, W.H. (1976) Plagues and Peoples, Doubleday, New York.Google Scholar
  34. 34.
    Nowak, M.A., S. Bonhoeffer, G. M. Shaw and R. M. May (1997) Anti-viral drug treatment: dynamics of resistance in free virus and infected cell populations, J. Theor. Biol. 184: 203–217.CrossRefGoogle Scholar
  35. 35.
    Perelson, A.S., D. E. Kirschner and R. De Boer (1993) Dynamics of HIV infection of CD4+ T cells, Math. Biosc. 114: 81–125.CrossRefGoogle Scholar
  36. 36.
    Perelson, A.S. and P. W. Nelson (2002) Modelling viral and immune system dynamics, Nature Rev. Immunol. 2: 28–36.Google Scholar
  37. 37.
    Routh, E.J. (1877) A Treatise on the Stability of a Given State of Motion: Particularly Steady Motion, MacMillan.Google Scholar
  38. 38.
    U. S. Bureau of the Census (1975) Historical statistics of the United States: colonial times to 1970, Washington, D. C. Government Printing Office.Google Scholar
  39. 39.
    U.S. Bureau of the Census (1980) Statistical Abstracts of the United States, 101st edition.Google Scholar
  40. 40.
    U.S. Bureau of the Census (1991) Statistical Abstracts of the United States, 111th edition.Google Scholar
  41. 41.
    U.S. Bureau of the Census (1999) Statistical Abstracts of the United States, 119th edition.Google Scholar
  42. 42.
    Wang, W. (2006) Backward bifurcations of an epidemic model with treatment, Math. Biosc. 201: 58–71.MathSciNetCrossRefGoogle Scholar
  43. 43.
    Wang, W. and S. Ruan (2004) Bifurcations in an epidemic model with constant removal rate of the infectives, J. Math. Anal. & Appl. 291: 775–793.MathSciNetCrossRefGoogle Scholar
  44. 44.
    Wearing, H. J., P. Rohani, & M. J. Keeling (2005) Appropriate models for the management of infectious diseases, PLoS Medicine 2 (7), e174.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Fred Brauer
    • 1
  • Carlos Castillo-Chavez
    • 2
  • Zhilan Feng
    • 3
  1. 1.Department of MathematicsUniversity of British ColumbiaVancouverCanada
  2. 2.Mathematical and Computational Modeling Center (MCMSC), Department of Mathematics and StatisticsArizona State UniversityTempeUSA
  3. 3.Department of MathematicsPurdue UniversityWest LafayetteUSA

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