×

Pulse vaccination of an SEIR epidemic model with time delay. (English) Zbl 1144.34390

Summary: A delayed epidemic model with pulse vaccination is formulated in this paper. It is proved that the disease-free periodic solution is globally attractive if the vaccination rate is larger than \(\theta ^{*}\), and the disease is uniformly persistent if the vaccination rate is less than \(\theta _{*}\). The permanence of the model is investigated analytically. Our results indicate that large vaccination rate or short pulse of vaccination or long latent period is sufficient condition for the extinction of the disease.

MSC:

34K60 Qualitative investigation and simulation of models involving functional-differential equations
34K20 Stability theory of functional-differential equations
34K25 Asymptotic theory of functional-differential equations
92D30 Epidemiology
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Agur, Z.; Cojocaru, L.; Mazor, G.; Anderson, R.M.; Danon, Y.L., Pulse mass measles vaccination across age cohorts, Proc. natl. acad. sci. USA, 90, 11698-11702, (1993)
[2] Bainov, D.D.; Simeonov, P.S., Impulsive differential equations: periodic solutions and applications, (1995), World Scientific Singapore · Zbl 0828.34002
[3] Cirino, S.; da Silva, J.A.L., SEIR epidemic model of dengue transmission in coupled populations (in portuguese), TEMA tend. mat. apl. comput., 5, 1, 55-64, (2004) · Zbl 1208.92063
[4] Cooke, K.L.; van Den Driessche, P., Analysis of an SEIRS epidemic model with two delays, J. math. biol., 35, 240-260, (1996) · Zbl 0865.92019
[5] Cull, P., Global stability for population models, Bull. math. biol., 43, 47-58, (1981) · Zbl 0451.92011
[6] d’Onofrio, A., Stability properties of pulse vaccination strategy in SEIR epidemic model, Math. biosci., 179, 57-72, (2002) · Zbl 0991.92025
[7] D’Onofrio, A., Pulse vaccination strategy in the SIR epidemic model: global asymptotic stable eradication in presence of vaccine failures, Math. comput. modelling, 36, 473-489, (2002) · Zbl 1025.92011
[8] d’Onofrio, A., Mixed pulse vaccination strategy in epidemic model with realistic distributed infectious and latent times, Appl. math. comput., 151, 181-187, (2004) · Zbl 1043.92033
[9] d’Onofrio, A., On pulse vaccination strategy in the SIR epidemic model with vertical transmission, Appl. math. lett., 18, 729-732, (2005) · Zbl 1064.92041
[10] Esteva, L.; Vargas, C., Coexistence of different serotypes of dengue virus, J. math. biol., 46, 31-47, (2003) · Zbl 1015.92023
[11] Hersh, B.S.; Tambini, G.; Norgueira, A.C.; Carrasco, P.; de Quadros, C.A., Review of regional measles surveillance in the americas, 1996-1999, Lancet, 355, 1943-1948, (2000)
[12] Hui, J.; Chen, L., Impulsive vaccination of SIR epidemic models with nonlinear incidence rates, Discrete continuous dyn. syst. ser. B, 4, 595-605, (2004) · Zbl 1100.92040
[13] Kuang, Y., Delay differential equation with application in population dynamics, (1993), Academic Press New York, pp. 67-70
[14] Lakshmikantham, V.; Bainov, D.D.; Simeonov, P.S., Theory of impulsive differential equations, (1989), World Scientific Singapore · Zbl 0719.34002
[15] Langlais, M.; Suppo, C., A remark on a generic SEIRS model and application to cat retroviruses and fox rabies, Math. comput. modelling, 31, 117-124, (2000)
[16] Lu, Z.; Chi, X.; Chen, L., The effect of constant and pulse vaccination on SIR epidemic model with horizontal and vertical transmission, Math. comput. modelling, 36, 1039-1057, (2002) · Zbl 1023.92026
[17] Murray, A.G.; O’Callaghan, M.; Jones, B., Simple models of massive epidemics of herpesvirus in Australian (and New Zealand) pilchards, Environ. int., 27, 243-248, (2001)
[18] Nokes, D.J.; Swinton, J., The control of a childhood viral infection by pulse vaccination, IMA J. math. appl. med. biol., 12, 29-53, (1995) · Zbl 0832.92024
[19] Nokes, D.J.; Swinton, J., Vaccination in pulses: a strategy for global eradication of measles and polio?, Trends. microbiol., 5, 14-19, (1997)
[20] M. Ramsay, N. Gay, E. Miller, The epidemiology of measles in England and Wales: rationale for 1994 national vaccination campaign, Commun. Dis. Rep. 4 (1994) 141-146.
[21] Sabin, A.B., Measles: killer of millions in developing countries: strategies of elimination and continuing control, Eur. J. epidemial, 7, 1-22, (1991)
[22] Schuette, M.C., A qualitative analysis of a model for the transmission of varicella-zoster virus, Math. biosci., 182, 113-126, (2003) · Zbl 1012.92025
[23] Shulgin, B.; Stone, L.; Agur, Z., Pulse vaccination strategy in the SIR epidemic model, Bull. math. biol., 60, 1123-1148, (1998) · Zbl 0941.92026
[24] Shulgin, B.; Stone, L.; Agur, Z., Theoretical examinations of pulse vaccination policy in the SIR epidemic model, Math. comput. modelling, 30, 207-215, (2000) · Zbl 1043.92527
[25] Stone, L.; Shulgin, B.; Agur, Z., Theoretical examination of the pulse vaccination policy in the SIR epidemic model, Math. comput. modelling, 31, 207-215, (2000) · Zbl 1043.92527
[26] Xiao, Y.; Chen, L., Modelling and analysis of a predator-prey model with disease in the prey, Math. biosci., 171, 59-82, (2001) · Zbl 0978.92031
[27] Yan, J.; Zhao, A.; Nieto, J.J., Existence and global attractivity of positive periodic solution of periodic single-species impulsive lotka – volterra systems, Math. comput. modelling, 40, 509-518, (2004) · Zbl 1112.34052
[28] Zhang, B.; Liu, Y., Global attractivity for certain impulsive delay differential equations, Nonlinear anal., 52, 725-736, (2003) · Zbl 1027.34086
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.