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Stability of periodic solutions for an SIS model with pulse vaccination. (English) Zbl 1045.92042

Summary: Pulse vaccination is an important strategy for the elimination of infectious diseases. A mathematical SIS model with pulse vaccination is formulated in this paper. The dynamical behavior of the model is studied, and the basic reproductive number \(R_0\) is defined. It is proved that the disease-free periodic solution is stable if \(R_0 < 1\), and is unstable if \(R_0 > 1\). The global stability of the disease-free periodic solution is studied and a sufficient condition is obtained. The existence and stability of endemic periodic solutions are investigated analytically and numerically.

MSC:

92D30 Epidemiology
34C25 Periodic solutions to ordinary differential equations
34D23 Global stability of solutions to ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
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References:

[1] Braur, F., Basical ideal of mathematical epidemiology, (Castillo-Chavez, C.; Blower, S.; etal., Mathematical Approaches for Emerging and Reemerging Infectious Diseases, IMA Volume 125 (2002), Springer-Verlag: Springer-Verlag New York), 31-65
[2] Bailey, N. T.J., The Mathematical Theory of Infectious Diseases (1975), Hafner: Hafner New York · Zbl 0115.37202
[3] Capasso, V., (Mathematical Structures of Epidemic Systems, Volume 97, Lectures Notes in Biomathematics (1993), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0798.92024
[4] Hethcote, H. W., Mathematics of infectious diseases, SIAM Review, 42, 4, 599-653 (2000) · Zbl 0993.92033
[5] Hethcote, H. W., Three basic epidemic models, (Levin, S. A.; Hallam, T. G.; Gross, L. J., Applied Mathematical Ecology (1989), Springer-Verlag: Springer-Verlag Berlin), 119-144
[6] Kribs-Zaleta, C. M.; Valesco-Hernández, J. X., A simple vaccination model with multiple endemic states, Mathematical Biosciences, 164, 2, 183-201 (2000) · Zbl 0954.92023
[7] Nokes, D. J.; Swinton, J., The control of childhood viral infections by pulse vaccination, IMA Journal of Mathematics Applied in Medicine & Biology, 12, 29-53 (1995) · Zbl 0832.92024
[8] Stone, L.; Shulgin, B.; Agur, Z., Theoretical examination of the pulse vaccination policy in the SIR epidemic modelc, Mathl. Comput. Modelling, 31, 4/5, 207-215 (2000) · Zbl 1043.92527
[9] d’Onofrio, A., Pulse vaccination strategy in the SIR epidemic model: Global asymptotic stable eradication in presence of vaccine failures, Mathl. Comput. Modelling, 36, 4/5, 473-489 (2002) · Zbl 1025.92011
[10] d’Onofrio, A., Stability properties of pulse vaccination strategy in SEIR epidemic model, Mathematical Biosciences, 179, 1, 57-72 (2002) · Zbl 0991.92025
[11] Lakshmikantham, V.; Bailov, D. D.; Simeonov, P. S., Theory of Impulsive Differential Equations (1989), World Scientific: World Scientific Singapore · Zbl 0719.34002
[12] Bailov, D. D.; Simeonov, P. S., The Stability Theory of Impulsive Differential Equations, Asymptotic Properties of the Solutions (1995), World Scientific: World Scientific Singapore · Zbl 0828.34002
[13] LaSalle, J. P., The Stability of Dynamical Systems (1976), SIAM: SIAM Philadelphia, PA · Zbl 0364.93002
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