Journal of the Korean Society for Industrial and Applied Mathematics

Korean Society for Industrial and Applied Mathematics (한국산업응용수학회)
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STABILITY OF AN SIRS EPIDEMIC MODEL WITH A VARIABLE INCIDENCE RATE AND TIME DELAY

  • Seo, Young Il (NATIONAL FISHERIES RESEARCH AND DEVELOPMENT INSTITUTE) ;
  • Cho, Gi Phil (DEPARTMENT OF MATHEMATICS, PUSAN NATIONAL UNIVERSITY) ;
  • Chae, Kyoung Sook (DEPARTMENT OF MATHEMATICS, PUSAN NATIONAL UNIVERSITY) ;
  • Jung, Il Hyo (DEPARTMENT OF MATHEMATICS, PUSAN NATIONAL UNIVERSITY)
  • Received : 2012.08.25
  • Accepted : 2013.03.06
  • Published : 2013.03.25
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Abstract

The purpose of this paper is to prove existence of solutions of an SIRS epidemic model with time delay of continuous type and the variable incidence rate and to investigate some asymptotic behaviors of the SIRS epidemic model. An example illustrating the stability of the model is given. The results extend the corresponding results in the literature.

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References

  1. N. T. J. Bailey, The Mathematical Theory of Infectious Diseases, 2nd ed., Hafner, New York, 1975.
  2. E. Beretta, T. Hara, W. Ma, and Y. Takeuchi, Convergence results in SIR epidemic models with varying population sizes, J. Appl. Nonlinear Anal. 28 (1997), 1909-1921. https://doi.org/10.1016/S0362-546X(96)00035-1
  3. F. Brauer and C. Castillo-Charvez, Mathematical Models in Population Biology and Epidemiology, Springer- Verlag, New York, 2001.
  4. W. H. Herbert, The mathematics of infectious diseases, SIAM Rev. 42 (2000), 599-653. https://doi.org/10.1137/S0036144500371907
  5. W. O. Kermack and A. G. Mckendrick, Contribution to the mathematical theory of epidemics, Proc. Roy. Soc. London Ser. A 115, (1927), 700-721. https://doi.org/10.1098/rspa.1927.0118
  6. W. Ma, M. Song, and Y. Takeuchi, Global stability of an SIR epidemic model with time delay, Appl. Math. Lett. 17 (2004), 1141-1145. https://doi.org/10.1016/j.aml.2003.11.005
  7. G. Schneckenreither, N. Popper, G. Zauner, and F. Breitenecker, Modelling SIR-type epidemics by ODEs, PDEs, difference equations and cellular automata - A comparative study, Simulation Modelling Practice and Theory, 16 (2008), 1014-1023. https://doi.org/10.1016/j.simpat.2008.05.015
  8. Z. Teng and Z. Zhang, Global behavior and permanence of an SIRS epidemic model with time delays, J. Nonlinear Anal. 9 (2008), 1409-1424. https://doi.org/10.1016/j.nonrwa.2007.03.010
  9. S. Ruan and W. Wang, Dynamical behavior of an epidemic model with a nonlinear incidence rate, J. Differential Equations 188 (2003), 135-163. https://doi.org/10.1016/S0022-0396(02)00089-X
  10. G. Zaman, K. H. Kang, and I. H. Jung, Stability analysis and optimal vaccination of an SIR epidemic model, BioSystem 93 (2008), 240-249. https://doi.org/10.1016/j.biosystems.2008.05.004
  11. Y. Jin, W. Wang, and S. Xiao, An SIRS model with a nonlinear incidence rate, J. Chaos, Solitons and Fractals 34 (2007), 1482-1497. https://doi.org/10.1016/j.chaos.2006.04.022
  12. P. van den Driessche and J. Watmough, A simple SIS epidemic model with a backward bifurcation, J. Math. Biol. 40 (2000), 525-540. https://doi.org/10.1007/s002850000032