Abstract
Pulse vaccination is an effective and important strategy for the elimination of infectious diseases. A delayed SEIRS epidemic model with pulse vaccination and varying total population size is proposed in this paper. We point out, if R* < 1, the infectious population disappear so the disease dies out, while if R *; > 1, the infectious population persist. Our results indicate that a long period of pulsing or a small pulse vaccination rate is sufficient condition for the permanence of the model.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Agur, Z., Cojocaru, L., Mazor, G., Anderson, R.M., Danon, Y.L., 1993. Pulse mass measles vaccination across age cohorts. Proc. Natl. Acad. Sci. USA 90, 11698–11702.
Al-Showaikh, F.N.M., Twizell, E.H., 2005. One-dimensional measles dynamics. Appl. Math. Comput. 152, 169–194.
Anderson, R.M., May, R.M., 1979. Population biology of infectious diseases I. Nature 180, 361–367.
Bainov, D.D., Simeonov, P.S., 1993. Impulsive Differential Equations: Periodic Solutions and Applications. Longman Scientific and Technical, New York.
Cirino, S., da Silva, J.A.L., 2004. SEIR epidemic model of dengue transmission in coupled populations (in Portuguese). Tend. Mat. Apl. Comput. 5(1), 55–64.
Cooke, K.L., van Den Driessche, P., 1996. Analysis of an SEIRS epidemic model with two delays. J. Math. Biol. 35, 240–260.
Cull, P., 1981. Global stability for population models. Bull. Math. Biol. 43, 47–58.
d'Onofrio, A., 2002. Stability properties of pulse vaccination strategy in SEIR epidemic model. Math. Biosci. 179(1), 57–72.
Esteva, L., Vargas, C., 2003. Coexistence of different serotypes of dengue virus. J. Math. Biol. 46, 31–47.
Hui, J., Chen, L., 2004. Impulsive vaccination of SIR epidemic models with nonlinear incidence rates. Discrete Continuous Dynamical Syst. Ser. B 4, 595–605.
Kuang, Y., 1993. Delay Differential Equation with Application in Population Dynamics. Academic Press, New York, pp. 67–70.
Lakshmikantham, V., Bainov, D.D., Simeonov, P.S., 1989. Theory of Impulsive Differential Equations. World Scientific, Singapore.
Langlais, W., Suppo, C., 2000. A remark on a generic SEIRS model and application to cat retroviruses and fox rabies. Math. Comput. Modell. 31, 117–124.
Murray, A.G., O'Callaghan, M., Jones, B., 2001. Simple models of massive epidemics of herpesvirus in Australian (and New Zealand) pilchards. Environ. Int. 27, 243–248.
Ramsay, M., Gay, N., Miller, E., 1994. The epidemiology of measles in England and Wales: Rationale for 1994 national vaccination campaign. Commun. Dis. Rep. 4(12), 141–146.
Sabin, A.B., 1991. Measles: Killer of millions in developing countries: Strategies of elimination and continuing control. Eur. J. Epidemiol. 7, 1–22.
Schuette, M.C., 2003. A qualitative analysis of a model for the transmission of varicella-zoster virus. Math. Biosci. 182, 113–126.
Shulgin, B., Stone, L., Agur, Z., 1998. Pulse vaccination strategy in the SIR epidemic model. Bull. Math. Biol. 60, 1123–1148.
Stone, L., Shulgin, B., Agur, Z., 2000. Theoretical examination of the pulse vaccination policy in the SIR epidemic model. Math. Comput. Modell. 31, 207–215.
Wang, W., 2002. Global behavior of an SEIRS epidemic model with time delays. Appl. Math. Lett. 15, 423–428.
Xu, J., Zhou, Y., 2004. Hepatitis epidemic models with proportional and pulse vaccination (in Chinese). J. Biomath. 19, 149–155.
Zhou, Y., Liu, H., 2003. Stability of periodic solutions for an SIS model with pulse vaccination. Math. Comput. Modell. 38, 299–308.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Gao, S., Chen, L. & Teng, Z. Impulsive Vaccination of an SEIRS Model with Time Delay and Varying Total Population Size. Bull. Math. Biol. 69, 731–745 (2007). https://doi.org/10.1007/s11538-006-9149-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11538-006-9149-x