대한수학회논문집 (Communications of the Korean Mathematical Society)

  • 제22권4호
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  • Pages.597-608
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  • 2007
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  • 1225-1763(pISSN)
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  • 2234-3024(eISSN)
대한수학회 (Korean Mathematical Society)
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A GENERAL UNIQUENESS RESULT OF AN ENDEMIC STATE FOR AN EPIDEMIC MODEL WITH EXTERNAL FORCE OF INFECTION

  • 발행 : 2007.10.31
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초록

We present a general uniqueness result of an endemic state for an S-I-R model with external force of infection. We reduce the problem of finding non-trivial steady state solutions to that of finding zeros of a real function of one variable so that we can easily prove the uniqueness of an endemic state. We introduce an assumption which was usually used to show stability of a non-trivial steady state. It turns out that such an assumption adopted from a stability analysis is crucial for proving the uniqueness as well, and the assumption holds for almost all cases in our model.

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참고문헌

  1. V. Andreassen, Instability in an SIR-model with age-dependent susceptibility, Mathematical Population Dynamics: Analysis of Heterogeneity, vol. 1, Theory of Epidemics, Proc. 3rd International Conference on Mathematical Population Dynamics, Pau, France, 1992, Wuerz Publishing Ltd., Winnipeg, Canada, 1995, 3-14
  2. S. Busenberg, K. Cooke, and M. Iannelli, Endemic thresholds and stability in a class of age-structured epidemics, SIAM J. Appl. Math. 48 (1988), 1379-1395 https://doi.org/10.1137/0148085
  3. S. Busenberg, M. Iannelli, and H. Thieme, Dynamics of an age-structured epidemic model, Dynamical Systems, S.-T. Liao, T.-R. Ding, and Y.-Q. Ye (Eds.), Nankai Series in Pure, Applied Mathematics, and Theoretical Physics, vol. 4, World Scientific, 1993
  4. Y. Cha, Existence and uniqueness of endemic states for an epidemic model with external force of infection, Commun. Korean Math. Soc. 17 (2002), no. 1, 175-187 https://doi.org/10.4134/CKMS.2002.17.1.175
  5. Y. Cha, Local stability of endemic states for an epidemic model with external force of infection, Commun. Korean Math. Soc. 18 (2003), no. 1, 133-149 https://doi.org/10.4134/CKMS.2003.18.1.133
  6. Y. Cha and F. A. Milner, A uniqueness result of an endemic state for an S-I-R epidemic model, submitted to a journal
  7. Y. Cha, M. Iannelli, and F. A. Milner, Existence and uniqueness of endemic states for the age structured S-I-R model, Math. Biosc. 150 (1998), 177-190 https://doi.org/10.1016/S0025-5564(98)10006-8
  8. Y. Cha, Stability change of an epidemic model, Dynamic Systems and Applications 9 (2000), 361-376
  9. K. Dietz and D. Schenzle, Proportionate mixing models for age-dependent infection transmission, J. Math. BioI. 22 (1985), 117-120 https://doi.org/10.1007/BF00276550
  10. El Doma, Analysis of nonlinear integro-differential equations arising in age dependent epidemic models, Nonl. Anal. T. M. A. 11 (1987), 913-937 https://doi.org/10.1016/0362-546X(87)90060-5
  11. M. Iannelli, F. A. Milner, and A. Pugliese, Analytical and Numerical Results for The Age-Structured S-I-S Model with Mixed Inter-Intracohort Transmission, SIAM J. Math. Anal. 23 (1992), 662-688 https://doi.org/10.1137/0523034
  12. H. Inaba, Threshold and stability results for an age-structured epidemic model, J. Math. BioI. 28 (1990), 411-434 https://doi.org/10.1007/BF00178326
  13. A. G. McKendrick, Applications of mathematics to medical problems, Proc. Edinburgh Math. Soc. 44 (1926), 98-130 https://doi.org/10.1017/S0013091500034428
  14. H. Thieme, Stability change for the endemic equilibrium in age-structured models for the spread of S-I-R type infectious diseases, Differential Equations Models in Biology, Epidemiology and Ecology, Lectures Notes in Biomathematics, vol. 92, Springer Verlag, 1991, 139-158
  15. H. von Foerster, The Kinetics of Cellular Proliferation, Grune and Stratton, New York, 1959