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Stability properties of pulse vaccination strategy in SEIR epidemic model. (English) Zbl 0991.92025
Summary: The problem of the applicability of the pulse vaccination strategy (PVS) for the stable eradication of some relevant general class of infectious diseases is analyzed in terms of the study of local asymptotic stability (LAS) and global asymptotic stability (GAS) of the periodic eradication solution for the SEIR epidemic model in which is included the PVS. Demographic variations due or not to disease-related fatalities are also considered. Due to the non-triviality of the Floquet’s matrix associate to the studied model, the LAS is studied numerically and in this way it is found a simple approximate (but analytical) sufficient criterion which is an extension of the LAS constraint for the stability of the trivial equilibrium in SEIR models without vaccination. The numerical simulations also seem to suggest that the PVS is slightly more efficient than the continuous vaccination strategy. Analytically, the GAS of the eradication solutions is studied and it is demonstrated that the above criteria for the LAS guarantee also the GAS.

92D30 Epidemiology
34D05 Asymptotic properties of solutions to ordinary differential equations
92C60 Medical epidemiology
34D23 Global stability of solutions to ordinary differential equations
Full Text: DOI
[1] Agur, Z.; Cojocaru, L.; Mazor, G.; Anderson, R.M.; Danon, Y.L., Pulse mass measles vaccination across age cohorts, Proc. nat. acad. sci. USA, 90, 11698, (1993)
[2] Shulgin, B.; Stone, L.; Agur, Z., Pulse vaccination strategy in the SIR epidemic model, Bull. math. biol., 60, 1123, (1998) · Zbl 0941.92026
[3] Shulgin, B.; Stone, L.; Agur, Z., Theoretical examination of pulse vaccination policy in the SIR epidemic model, Math. comp. model., 31, 4/5, 207, (2000) · Zbl 1043.92527
[4] Nokes, D.J.; Swinton, J., The control of childhood viral infections by pulse vaccination, IMA J. math. appl. med. biol., 12, 29, (1995) · Zbl 0832.92024
[5] Earn, J.D.; Rohani, P.; Grenfell, B.T., Persistence, chaos and synchrony in ecology and epidemiology, Proc. roy. soc. lond. B, 265, 7, (1998)
[6] Hetcote, H.W., Qualitative analyses of communicable disease models, Math. biosci., 28, 335, (1976) · Zbl 0326.92017
[7] Agur, Z., Randomness synchrony and population resistence, J. theor. biol., 112, 677, (1985)
[8] de Quadros, C.A.; Andrus, J.K.; Olivé, J.M., Eradication of poliomyelitis progress, The am. pediat. inf. dis. J., 10, 3, 222, (1991)
[9] Sabin, A.B., Measles killer of millions in developing countries strategies of elimination and continuing control, Eur. J. epid., 7, 1, (1991)
[10] Ramsay, M.; Gay, N.; Miller, E., The epidemiology of measles in england and wales rationale for 1994 national vaccination campaign, Eur. J. epid., 7, 1, (1991)
[11] Hyman, J.M.; Li, J., An intuitive formulation for the reproductive number for the spread of diseases in heterogeneous populations, Math. biosci., 167, 65, (2000) · Zbl 0962.92033
[12] Thieme, H.R., Convergence results and a poincare – bendixons thricothomy for asymptotically autonomous differential equations, J. math. bio., 30, 755, (1992)
[13] Li, M.Y.; Muldowney, J.S., Global stability for SEIR model in epidemiology, Math. biosci., 164, 125, (1995)
[14] Arnold, V.I., Ordinary differential equations, (1979), MIR Editions Moscow
[15] Hale, J.K., Ordinary differential equations, (1969), Wiley-Interscience New York · Zbl 0186.40901
[16] Prodi, G., Analisi matematica, (1972), Boringhieri Torino
[17] Oppenheim, A.V.; Willsky, A.S., Signals and systems, (1983), Prentice Hall NJ, USA · Zbl 0539.93001
[18] Landau, L.D., Theoretical physics course mechanics, (1966), Mir Editions Moscow
[19] Cesari, L., Stability of differential equations, (1963), Springer Heidelberg · Zbl 0111.08701
[20] Farkas, W., Periodic motions, lecture notes on applied mathematics, (1991), Springer Heidelberg
[21] Kalivianakis, M.; Mous, S.L.J.; Grasman, J., Reconstruction of the seasonally varying contact rate for measles, Math. biosci., 124, 225, (1994) · Zbl 0818.92021
[22] Pugliese, A., Population models for diseases with no recovery, J. math. bio., 28, 65, (1990) · Zbl 0727.92023
[23] Coppel, W.A., Stability and asymptotic behaviour of differential equations, (1965), Springer Heath, Boston, MA · Zbl 0154.09301
[24] A. d’Onofrio, The pulse vaccination strategy and the SIR epidemic models: global asymptotic stability and vaccine failures, submitted for publication, 2001
[25] A. d’Onofrio, The pulse vaccination strategy and AIDS models, in preparation, 2002
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