Discrete Dynamics in Nature and Society

Discrete Dynamics in Nature and Society / 2006 / Article

Open Access

Volume 2006 |Article ID 83489 | 11 pages | https://doi.org/10.1155/DDNS/2006/83489

Stability results on age-structured SIS epidemic model with coupling impulsive effect

Received29 Jun 2005
Accepted10 Oct 2005
Published14 Feb 2006

Abstract

We develop an age-structured epidemic model for malaria with impulsive effect, and consider the effect of blood transfusion and infected-vector transmission. Transmission rates depend on age. We derive the condition in which eradication solution is locally asymptotically stable. The condition shows that large enough pulse reducing proportion and relatively small interpulse time lead to the eradication of the diseases.

References

  1. R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, Oxford, UK, 1991. View at: Google Scholar
  2. W. Arendt, A. Grabosch, G. Greiner, U. Groh, H. P. Lotz, U. Moustakas, R. Nagel, F. Neubrander, and U. Schlotterbeck, One-Parameter Semigroups of Positive Operators, vol. 1184 of Lecture Notes in Mathematics, Springer, Berlin, 1986. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  3. T. Bradley, “Malaria and drug resistance,” MBChB special study module project report, 1996, http://www-micro.msb.le.ac.uk/224/Bradley/Bradley.html. View at: Google Scholar
  4. A. d'Onofrio, “Stability properties of pulse vaccination strategy in SEIR epidemic model,” Mathematical Biosciences, vol. 179, no. 1, pp. 57–72, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  5. Z. Feng, D. L. Smith, F. E. McKenzie, and S. A. Levin, “Coupling ecology and evolution: malaria and the S-gene across time scales,” Mathematical Biosciences, vol. 189, no. 1, pp. 1–19, 2004. View at: Publisher Site | S-gene%20across%20time%20scales&author=Z. Feng&author=D. L. Smith&author=F. E. McKenzie&author=&author=S. A. Levin&publication_year=2004" target="_blank">Google Scholar | PubMed | MathSciNet
  6. D. Greenhalgh, “Analytical threshold and stability results on age-structured epidemic models with vaccination,” Theoretical Population Biology, vol. 33, no. 3, pp. 266–290, 1988. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  7. D. Greenhalgh, “Threshold and stability results for an epidemic model with an age-structured meeting rate,” IMA: Journal of Mathematics Applied in Medicine and Biology, vol. 5, no. 2, pp. 81–100, 1988. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  8. H. W. Hethcote, “The mathematics of infectious diseases,” SIAM Review, vol. 42, no. 4, pp. 599–653, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  9. H. Inaba, “Threshold and stability results for an age-structured epidemic model,” Journal of Mathematical Biology, vol. 28, no. 4, pp. 411–434, 1990. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  10. H. Inaba and H. Sekine, “A mathematical model for Chagas disease with infection-age-dependent infectivity,” Mathematical Biosciences, vol. 190, no. 1, pp. 39–69, 2004. View at: Publisher Site | Google Scholar | PubMed | Zentralblatt MATH | MathSciNet
  11. S. A. Levin and J. D. Udovic, “A mathematical model of coevolving populations,” The American Naturalist, vol. 111, no. 980, pp. 657–675, 1977. View at: Publisher Site | Google Scholar
  12. Malaria Foundation International, “Malaria: background information,” 1998, http://www.malaria.org/backgroundinfo.html. View at: Google Scholar
  13. M. Martcheva and H. R. Thieme, “Progression age enhanced backward bifurcation in an epidemic model with super-infection,” Journal of Mathematical Biology, vol. 46, no. 5, pp. 385–424, 2003. View at: Publisher Site | Google Scholar | PubMed | Zentralblatt MATH | MathSciNet
  14. J. H. Pollard, Mathematical Models for the Growth of Human Population, Cambridge University Press, London, 1973. View at: Google Scholar | Zentralblatt MATH
  15. H. R. Thieme, “Quasi-compact semigroups via bounded perturbation,” in Advances in Mathematical Population Dynamics—Molecules, Cells and Man (Houston, TX, 1995), O. Arino, D. E. Axelrod, and M. Kimmel, Eds., vol. 6 of Ser. Math. Biol. Med., pp. 691–711, World Scientific, New Jersey, 1997. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  16. G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, vol. 89 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, 1985. View at: Google Scholar | MathSciNet
  17. W. H. Wernsdorfer, “The importance of malaria in the world,” in Malaria. Volume 1: Epidemiology, Chemotherapy, Morphology, and Metabolism, J. P. Kreier, Ed., pp. 1–93, Academic Press, New York, 1980. View at: Google Scholar
  18. WHO, “Malaria, fact sheets,” 1998, http://www.who.int/inf-fs/en/fact094.html. View at: Google Scholar
  19. Y. Zhou and H. Liu, “Stability of periodic solutions for an SIS model with pulse vaccination,” Mathematical and Computer Modelling, vol. 38, no. 3-4, pp. 299–308, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet

Copyright © 2006 Helong Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

0 Views | 0 Downloads | 0 Citations
 PDF  Download Citation  Citation
 Order printed copiesOrder