Skip to main content
SpringerLink
Log in
Book cover

Mathematics for Life Science and Medicine pp 97–122Cite as

Spatial-Temporal Dynamics in Nonlocal Epidemiological Models

  • Chapter

Part of the book series: Biological and Medical Physics, Biomedical Engineering ((BIOMEDICAL))

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Anderson, R. M. and R. M. May (1991), Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, Oxford.

    Google Scholar 

  2. Aronson, D. G. (1977), The asymptotic speed of propagation of a simple epidemic, in “Nonlinear Diffusion”, eds. by W. E. Fitzgibbon and H. F. Walker, Research Notes in Math. 14, Pitman, London, pp. 1–23.

    Google Scholar 

  3. Aronson, D. G. and H. F. Weinberger (1975), Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in “Partial Differential Equations and Related Topics”, ed. by J. A. Goldstein, Lecture Notes in Math. 446, Springer, Berlin, pp. 5–49.

    Chapter  Google Scholar 

  4. Aronson, D. G. and H. F. Weinberger (1975), Multidimennsional nonlinear diffusion arising in population genetics, Adv. Math. 30: 33–76.

    Article  MathSciNet  Google Scholar 

  5. Atkinson, C. and G. E. H. Reuter (1976), Deterministic epidemic waves, Math. Proc. Camb. Phil. Soc. 80: 315–330.

    MathSciNet  MATH  Google Scholar 

  6. Barbour, A. D. (1977), The uniqueness of Atkinson and Reuter’s epidemic waves, Math. Proc. Camb. Phil. Soc. 82: 127–130.

    MathSciNet  MATH  Google Scholar 

  7. Bartlett, M. S. (1956), Deterministic and stochastic models for recurrent epidemics, Proc. 3rd Berkeley Symp. Math. Stat. Prob. 4: 81–109.

    MathSciNet  Google Scholar 

  8. Bolker, B. M. and B. T. Grenfell (1995), Space, persistence, and dynamics of measles epidemics, Phil. Trans. R. Soc. Lond. B237: 298–219.

    Google Scholar 

  9. Brauer, F. and C. Castillo-Chavez (2000), Mathematical Models in Population Biology and Epidemiology, Springer-Verlag, New York.

    Google Scholar 

  10. Brown, K. J. and J. Carr (1977), Deterministic epidemic waves of critical velocity, Math. Proc. Camb. Phil. Soc. 81: 431–433.

    MathSciNet  MATH  Google Scholar 

  11. Busenberg, S. and K. L. Cooke (1978), Periodic solutions for a vector disease model, SIAM J. Appl. Math. 35: 704–721.

    Article  MathSciNet  MATH  Google Scholar 

  12. Caraco, T., S. Glavanakov, G. Chen, J. E. Flaherty, T. K. Ohsumi and B. K. Szymanski (2002), Stage-structured infection transmission and a spatial epidemic: A model for lyme disease, Am. Nat. 160: 348–359.

    Article  Google Scholar 

  13. Capasso, V. (1978), Global solution for a diffusive nonlinear deterministic epidemic model, SIAM J. Appl. Math. 35: 274–284.

    Article  MathSciNet  MATH  Google Scholar 

  14. Cliff, A., P. Haggett and M. Smallman-Raynor (1993), Measles: An Historical Geography of a Major Human Viral Disease, Blackwell, Oxford.

    Google Scholar 

  15. Cooke, K. L. (1979), Stability analysis for a vector disease model, Rocky Mountain J. Math. 9: 31–42.

    Article  MathSciNet  MATH  Google Scholar 

  16. Cruickshank I., W. S. Gurney and A. R. Veitch (1999), The characteristics of epidemics and invasions with thresholds, Theor. Pop. Biol. 56: 279–92.

    Article  MATH  Google Scholar 

  17. De Monttoni, P., E. Orlandi and A. Tesei (1979), Aysmptotic behavior for a system describing epidemics with migration and spatial spread of infection, Nonlinear Anal. 3: 663–675.

    Article  MathSciNet  Google Scholar 

  18. Diekmann, O. (1978), Thresholds and travelling waves for the geographical spread of infection, J. Math. Biol. 6: 109–130.

    Article  MathSciNet  MATH  Google Scholar 

  19. Diekmann, O. (1979), Run for life: A note on the asymptotic spread of propagation of an epidemic, J. Differential Equations 33: 58–73.

    Article  MathSciNet  MATH  Google Scholar 

  20. Diekmann, O. and J. A. P. Heesterbeek (2000), Mathematical Epidemiology of Infective Diseases: Model Building, Analysis and Interpretation, Wiley, New York.

    Google Scholar 

  21. Diekmann, O., J. A. P. Heesterbeek and H. Metz (1995), The legacy of Kermack and McKendrick, in “Epidemic Models: Their Structure and Relation to Data”, ed. by D. Mollison, Cambridge Univ. Press, Cambridge, pp. 95–115.

    Google Scholar 

  22. Dushoff, J. and S. A. Levin (1995), The effects of population heterogeneityon disease invasion, Math. Biosci. 128: 25–40.

    Article  MATH  Google Scholar 

  23. Fenichel, N. (1979), Geometric singular perturbation theory for ordinary differential equations. J. Differential Equations 31: 53–98.

    Article  MathSciNet  MATH  Google Scholar 

  24. Fisher, R. A. (1937), The wave of advance of advantageous genes, Ann. Eugenics 7: 353–369.

    Google Scholar 

  25. Grenfell, B. T., O. N. Bjornstad and J. Kappey (2001), Travelling waves and spatial hierarchies in measles epidemics, Nature 414: 716–723.

    Article  Google Scholar 

  26. Hethcote, H. W. (2000), The mathematics of infectious disease, SIAM Rev. 42: 599–653.

    Article  MathSciNet  MATH  Google Scholar 

  27. Hosono, Y. and B. Ilyas (1995), Traveling waves for a simple diffusive epidemic model, Math. Models Methods Appl. Sci. 5: 935–966.

    Article  MathSciNet  MATH  Google Scholar 

  28. Källen, A., P. Arcuri and J. D. Murray (1985), A simple model for the spatial spread and control of rabies, J. Theor. Biol. 116: 377–393.

    Article  Google Scholar 

  29. Kendall, D. G. (1957), Discussion of ‘Measles periodicity and community size’ by M. S. Bartlett, J. Roy. Stat. Soc. A120: 64–76.

    Google Scholar 

  30. Kendall, D. G. (1965), Mathematical models of the spread of injection, in “Mathematics and ComputerScience in Biology and Medicine”, Medical Research Council, London, pp. 213–225

    Google Scholar 

  31. Kermack, W. O. and A. G. McKendrik (1927), A contribution to the mathematical theory of epidemics, Proc. Roy. Soc. A115: 700–721.

    Google Scholar 

  32. Kolmogorov, A., I. Petrovskii, and N. S. Piskunov (1937), étude de l’équation de le diffusion avec croissance de la quantité de mateère et son application à un problème biologique, Moscow Univ. Bull. 1: 1–25.

    Google Scholar 

  33. Lajmanovich, A. and J. A. Yorke (1976), A deterministic model for gonorrhea in a non-homogeneous population, Math. Biosci. 28: 221–236.

    Article  MathSciNet  MATH  Google Scholar 

  34. Lloyd, A. L. and R. M. May (1996), Spatial heterogeneity in epidemic models, J. Theor. Biol. 179: 1–11.

    Article  Google Scholar 

  35. Lounibos, L. P. (2002), Invasions by insect vectors of human disease, Ann. Rev. Entomol. 47: 233–266.

    Article  Google Scholar 

  36. Marcati, P. and M. A. Pozio (1980), Global asymptotic stability for a vector disease model with spatial spread, J. Math. Biol. 9: 179–187.

    Article  MathSciNet  MATH  Google Scholar 

  37. Medlock, J. and M. Kot (2003), Spreding disease: integro-differential equations old and new, Math. Biosci. 184: 201–222.

    Article  MathSciNet  MATH  Google Scholar 

  38. Mollison, D. (1972), Possible velocities for a simple epidemic, Adv. Appl. Prob. 4: 233–257.

    Article  MathSciNet  MATH  Google Scholar 

  39. Mollison, D.(1991), Dependence of epidemic and population velocities on basic parameters, Math. Biosci. 107: 255–87.

    Google Scholar 

  40. Murray, J. D. (1989), Mathematical Biology, Springer-Verlag, Berlin.

    MATH  Google Scholar 

  41. Murray, J. D. and W. L. Seward (1992), On the spatial spread of rabies among foxes with immunity, J. Theor. Biol. 156: 327–348.

    Article  Google Scholar 

  42. Murray, J. D., E. A. Stanley and D. L. Brown (1986), On the spatial spread of rabies among foxes, Proc. R. Soc. Lond. B229(1255): 111–150.

    Article  Google Scholar 

  43. O’Callagham, M. and A. G. Murray (2002), A tractable deterministic model with realistic latent periodic for an epidemic in a linear habitat, J. Math. Biol. 44: 227–251.

    Article  MathSciNet  Google Scholar 

  44. Pozio, M. A. (1980), Behavior of solutions of some abstract functional differential equations and application to predator-prey dynamics, Nonlinear Anal. 4: 917–938.

    Article  MathSciNet  MATH  Google Scholar 

  45. Pozio, M. A. (1983), Some conditions for global asymptotic stability of equilibria of integrodifferential equations, J. Math. Anal. Appl. 95: 501–527.

    Article  MathSciNet  MATH  Google Scholar 

  46. Rass, L. and J. Radcliffe (2003), Spatial Deterministic Epidemics, Math. Surveys Monogr. 102, Amer. Math. Soc., Providence.

    MATH  Google Scholar 

  47. Ross, R. and H. P. Hudson (1917), Ann appliicationn of the theory of probability to the study of a priori pathometry-Part III, Proc. R. Soc. Lond. A93: 225–240.

    Google Scholar 

  48. Ruan, S. and J. Wu (1994), Reaction-diffusion equations with infinite delay, Can. Appl. Math. Quart. 2: 485–550.

    MathSciNet  MATH  Google Scholar 

  49. Ruan, S. and D. Xiao (2004), Stability of steady states and existence of traveling waves in a vector disease model, Proc. Royal Soc. Edinburgh Sect. A134: 991–1011.

    MathSciNet  Google Scholar 

  50. Schumacher, K. (1980a), Travelling-front solution for integro-differential equation. I, J. Reine Angew. Math. 316: 54–70.

    MathSciNet  MATH  Google Scholar 

  51. Schumacher, K. (1980b), Travelling-front solution for integro-differential equation. II, in “Biological Growth and Spread”, Lecture Notes in Biomathematics 38, eds. by W. Jäger, H. Rost and P. Tautu, Springer, Berlin, pp. 296–309.

    Google Scholar 

  52. Thieme, H. R. (1977a), A model for the spatial spread of an epidemic, J. Math. Biol. 4: 337–351.

    Article  MathSciNet  MATH  Google Scholar 

  53. Thieme H. R. (1977b), The asymptotic behavior of solutions of nonlinear integral equations, Math. Zeitchrift 157: 141–154.

    Article  MathSciNet  MATH  Google Scholar 

  54. Thieme, H. R. (1979), Asymptotic estimates of the solutions of nonlinear integral equation and asymptotic speeds for the spread of populations, J. Reine Angew. Math. 306: 94–121.

    MathSciNet  MATH  Google Scholar 

  55. Thieme, H. R. (1980), Some mathematical considerations of how to stop the spatial spread of a rabies epidemics, in “Biological Growth and Spread,” Lecture Notes in Biomathematics 38, eds. by W. Jäger, H. Rost and P. Tautu, Springer, Berlin, pp. 310–319.

    Google Scholar 

  56. Thieme, H. R. (1985), Renewal theorems for some mathematical models in epidemiology, J. Integral Equations 8: 185–216.

    MathSciNet  MATH  Google Scholar 

  57. Thieme, H. R. (1988), Asymptotic proportionality (weak ergodicity) and conditional asymptotic equality of solutions to time-heterogeneous sublinear difference and differential equations, J. Differential Equations 73: 237–268.

    Article  MathSciNet  MATH  Google Scholar 

  58. Thieme, H. R. (2003), Mathematics in Population Biology, Princeton University Press, Princeton.

    MATH  Google Scholar 

  59. Thieme, H. R. and X.-Q. Zhao (2003), Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, J. Differential Equations 195: 430–470.

    Article  MathSciNet  MATH  Google Scholar 

  60. Van den Bosch, F., J. A. J. Metz and O. Diekmann (1990), The velocity of spatial population expansion, J. Math. Biol. 28: 529–565.

    Article  MathSciNet  MATH  Google Scholar 

  61. Volz, R. (1982), Global asymptotic stability of a periodic solution to an epidemic model, J. Math. Biol. 15: 319–338.

    Article  MathSciNet  MATH  Google Scholar 

  62. Wang, Z.-C., W.-T. Li and S. Ruan (2006), Traveling wave fronts in reaction-diffusion systems with spatio-temporal delays, J. Differential Equations, 222: 185–232.

    Article  MathSciNet  MATH  Google Scholar 

  63. Webb, G. F. (1981), A reaction-diffusion model for a deterministic diffusive epidemic, J. Math. Anal. Appl. 84 (1981), 150–161.

    Article  MathSciNet  MATH  Google Scholar 

  64. Wu, J. (1996), Theory and Applications of Partial Functional Differential Equations, Springer, New York.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Miami, USA

    Shigui Ruan

Authors
  1. Shigui Ruan
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Faculty of Engineering, Department of Systems Engineering, Shizuoka University, Hamamatsu 3-5-1, 432-8561, Shizuoka, Japan

    Yasuhiro Takeuchi  & Kazunori Sato  & 

  2. Department of Biology, Kyushu University, 812-8581, Fukuoka, Japan

    Yoh Iwasa

Rights and permissions

Reprints and permissions

Copyright information

© 2007 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Ruan, S. (2007). Spatial-Temporal Dynamics in Nonlocal Epidemiological Models. In: Takeuchi, Y., Iwasa, Y., Sato, K. (eds) Mathematics for Life Science and Medicine. Biological and Medical Physics, Biomedical Engineering. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-34426-1_5

Download citation

Publish with us

Policies and ethics

Search

Navigation