Stability results on age-structured SIS epidemic model with coupling impulsive effect
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.
- R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, Oxford, UK, 1991.
- 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.
- T. Bradley, “Malaria and drug resistance,” MBChB special study module project report, 1996, http://www-micro.msb.le.ac.uk/224/Bradley/Bradley.html.
- A. d'Onofrio, “Stability properties of pulse vaccination strategy in SEIR epidemic model,” Mathematical Biosciences, vol. 179, no. 1, pp. 57–72, 2002.
- Z. Feng, D. L. Smith, F. E. McKenzie, and S. A. Levin, “Coupling ecology and evolution: malaria and the -gene across time scales,” Mathematical Biosciences, vol. 189, no. 1, pp. 1–19, 2004.
- 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.
- 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.
- H. W. Hethcote, “The mathematics of infectious diseases,” SIAM Review, vol. 42, no. 4, pp. 599–653, 2000.
- H. Inaba, “Threshold and stability results for an age-structured epidemic model,” Journal of Mathematical Biology, vol. 28, no. 4, pp. 411–434, 1990.
- 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.
- S. A. Levin and J. D. Udovic, “A mathematical model of coevolving populations,” The American Naturalist, vol. 111, no. 980, pp. 657–675, 1977.
- Malaria Foundation International, “Malaria: background information,” 1998, http://www.malaria.org/backgroundinfo.html.
- 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.
- J. H. Pollard, Mathematical Models for the Growth of Human Population, Cambridge University Press, London, 1973.
- 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.
- 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.
- 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.
- WHO, “Malaria, fact sheets,” 1998, http://www.who.int/inf-fs/en/fact094.html.
- 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.
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