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Pulse vaccination delayed SEIRS epidemic model with saturation incidence. (English) Zbl 1182.92064
Summary: A delayed SEIRS epidemic model with pulse vaccination and saturation incidence rate is investigated. Using the discrete dynamical system determined by the stroboscopic map, we obtain the existence of the disease-free periodic solution and its exact expression. Further, using the comparison theorem, we establish sufficient conditions of global attractivity of the disease-free periodic solution and the permanence of the disease. Our results indicate that a long latent period of the disease or a proper pulse vaccination rate will lead to eradication of the disease.

37N25Dynamical systems in biology
34K45Functional-differential equations with impulses
34K60Qualitative investigation and simulation of models
34K13Periodic solutions of functional differential equations
Full Text: DOI
[1] Anderson, R.; May, R.: Infectious disease of humans, dynamical and control. (1992)
[2] Anderson, R.; May, R.: Regulation and stability of host -- parasite population interactions II: Destabilizing process. J. anim. Ecol. 47, 219-267 (1978)
[3] Lu, Z.; Chi, X.; Chen, L.: The effect of constant and pulse vaccination on SIR epidemic model with horizontal and vertical transmission. Math. comput. Model. 36, 1039-1057 (2002) · Zbl 1023.92026
[4] Shulgin, B.; Stone, L.; Agur, Z.: Pulse vaccination strategy in the SIR epidemic model. Bull. math. Biol. 60, 1123-1148 (1998) · Zbl 0941.92026
[5] Kermark, M.; Mckendrick, A.: Contributions to the mathematical theory of epidemics. Part I proc. Roy. soc. A. 115, 700-721 (1927) · Zbl 53.0517.01
[6] Diekmann, O.; Heesterbeek, J. A. P.: Mathematical epidemiology of infectious diseases: model building, analysis and interpretation. (2000) · Zbl 0997.92505
[7] Anderson, R.; May, R.: Population biology of infectious diseases: part I. Nature 280, 361-367 (1979)
[8] Capasso, V.: Mathematical structures of epidemic systems. Lecture notes in biomathematics 97 (1993) · Zbl 0798.92024
[9] Ma, Z.; Zhou, Y.; Wang, W.; Jin, Z.: Mathematical modelling and research of epidemic dynamical systems. (2004)
[10] Mena-Lorca, J.; Hethcote, H. W.: Dynamic models of infectious diseases as regulators of population biology. J. math. Biol. 30, 693-716 (1992) · Zbl 0748.92012
[11] Thieme, H. R.: Epidemic and demographic interaction in the spread of potentially fatal diseases in growing populations. Math. biosci. 111, 99-130 (1992) · Zbl 0782.92018
[12] Wang, W.; Ruan, S.: Bifurcations in an epidemic model with constant removal rate of the infectives. J. math. Anal. appl. 291, 774-793 (2004) · Zbl 1054.34071
[13] Brauer, F.; Den Driessche, P. Van: Models for transmission of disease with immigration of infectives. Math. biosci. 171, 143-154 (2001) · Zbl 0995.92041
[14] Brauer, F.: Epidemic models in populations of varying size. Lecture notes in biomathematics 81 (1989) · Zbl 0684.92016
[15] Gao, L. Q.; Hethcote, H. W.: Disease transmission models with density-dependent demographics. J. math. Biol. 30, 717-731 (1992) · Zbl 0774.92018
[16] Greenhalgh, D.: Some threshold and stability results for epidemic models with a density dependent death rate. Theor. pop. Biol. 42, 130-151 (1992) · Zbl 0759.92009
[17] Bremermann, H.; Thieme, H. R.: A competitive exclusion principle for pathogen virulence. J. math. Biol. 27, 179-190 (1989) · Zbl 0715.92027
[18] Hethcote, H. W.; Den Driessche, P. Van: Some epidemiological models with nonlinear incidence. J. math. Biol. 29, 271-287 (1991) · Zbl 0722.92015
[19] S. Gakkhar, K. Negi, Pulse vaccination SIRS epidemic model with nonmonotonic incidence rate, Chaos Solitons Fractals, doi:10.1016/j.chaos. 2006.05.054. · Zbl 1131.92052
[20] G. Pang, L. Chen, A delayed SIRS epidemic model with pulse vaccination, Chaos Solitons Fractals, doi:10.1016/j.chaos.2006.04.061. · Zbl 1152.34379
[21] Li, G.; Jin, Z.: Global stability of a SEIR epidemic model with in latent, infectious force infected and immune period. Chaos solitons fractals 25, 1177-1184 (2005) · Zbl 1065.92046
[22] Zhang, J.; Ma, Z.: Global dynamics of an SEIR epidemic model with saturating contact rate. Math. biosci. 185, 15-32 (2004) · Zbl 1021.92040
[23] Liu, W. M.; Hethcote, H. W.; Levin, S. A.: Dynamical behavior of epidemiological models with non-linear incidence rate. J. math. Biol. 25, 359 (1987) · Zbl 0621.92014
[24] Greenhalgh, D.: Hopf bifurcation in epidemic models with a latent period and non-permanent immunity. Math. comput. Model. 25, 85 (1997) · Zbl 0877.92023
[25] Li, Michale Y.; Graef, J. R.; Wang, L.; Karsai, J.: Global dynamics of an SEIR model with varying total population size. Math. biosci. 160, 191-213 (1999) · Zbl 0974.92029
[26] Wang, W.; Ruan, S.: Dynamical behavior of an epidemic model with a nonlinear incidence rate. J. diffen. Equat. 188, 135C163 (2003) · Zbl 1028.34046
[27] Cooke, K. L.; Den Driessche, P. Van: Analysis of an SEIRS epidemic model with two delays. J. math. Biol. 35, 240-260 (1996) · Zbl 0865.92019
[28] Orsel, K.; Dekker, A.; Bouma, A.: Vaccination against foot and mouth disease reduces virus transmission in groups of calves. Vaccine 23, 4887-4894 (2005)
[29] Rebecca, J.; Eva, M.; Hakon, S.; Haaheim, L.: The effect of zanamivir treatment on the early immune response to influenza vaccination. Vaccine 19, 4743-4749 (2001)
[30] Zhang, Y.; Peng, B.; Deng, H.: Anti-HBV immune response by electroporation mediated DNA vaccination. Vaccine 24, 897-903 (2006)
[31] Cull, P.: Global stability for population models. Bull. math. Biol. 43, 47-58 (1981) · Zbl 0451.92011
[32] Kuang, Y.: Delay differential equation with application in population dynamics. (1993) · Zbl 0777.34002
[33] Gao, S.; Chen, L.; Nieto, J.; Torres, A.: Analysis of a delayed epidemic model with pulse vaccination and saturation incidence. Vaccine 24, 6037-6045 (2006)