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Extinction and permanence for a pulse vaccination delayed SEIRS epidemic model. (English) Zbl 1197.34090
Summary: A delayed SEIRS epidemic model with pulse vaccination and bilinear incidence rate is investigated. Using Krasnoselskii’s fixed-point theorem, we obtain the existence of disease-free periodic solution (DFPS for short) of the delayed impulsive epidemic system. Further, using the comparison method, we prove that under the condition $R^{*} < 1$, the DFPS is globally attractive, and that $R_{*} > 1$ implies that the disease is permanent. Theoretical results show that the disease will be extinct if the vaccination rate is larger than $\theta ^{*}$ and the disease is uniformly persistent if the vaccination rate is less than $\theta _{*}$. Our results indicate that a long latent period of the disease or a large pulse vaccination rate will lead to eradication of the disease. Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.

34D05Asymptotic stability of ODE
34K45Functional-differential equations with impulses
Full Text: DOI
[1] Anderson, R. M.; May, R. M.: Infectious disease of humans, dynamical and control, (1992)
[2] Anderson, R. M.; May, R. M.: 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 · doi:10.1016/S0895-7177(02)00257-1
[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 · doi:10.1016/S0092-8240(98)90005-2
[5] Kermark, M. D.; Mckendrick, A. G.: Contributions to the mathematical theory of epidemics. Part I, Proc roy soc A 115, 700-721 (1927)
[6] Diekmann, O.; Heesterbeek, J. A. P.: Mathematical epidemiology of infectious diseases: model building, analysis and interpretation, (2000) · Zbl 0997.92505
[7] Anderson, R. M.; May, R. M.: Population biology of infectious diseases: part I, Nature 280, 361-367 (1979)
[8] Capasso, V.: Mathematical structures of epidemic systems, Berlin: 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 · doi:10.1016/0025-5564(92)90081-7
[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 · doi:10.1016/j.jmaa.2003.11.043
[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 · doi:10.1016/S0025-5564(01)00057-8
[14] Brauer, F.: Epidemic models in populations of varying size, Berlin: 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 · doi:10.1007/BF00173265
[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 · doi:10.1016/0040-5809(92)90009-I
[17] Bremermann, H. J.; Thieme, H. R.: A competitive exclusion principle for pathogen virulence, J math biol 27, 179-190 (1989) · Zbl 0715.92027 · doi:10.1007/BF00276102
[18] Hethcote, H. W.; Den Driessche, P. Van: Some epidemiological models with nonlinear incidence, J math biol 29, 271-287 (1991) · Zbl 0722.92015 · doi:10.1007/BF00160539
[19] Gakkhar, S.; Negi, K.: Pulse vaccination SIRS epidemic model with non monotonic incidence rate, Chaos, solitons & fractals 35, 626-638 (2008) · Zbl 1131.92052 · doi:10.1016/j.chaos.2006.05.054
[20] Pang, G.; Chen, L.: A delayed SIRS epidemic model with pulse vaccination, Chaos, solitons & fractals 34, 1629-1635 (2007) · Zbl 1152.34379 · doi:10.1016/j.chaos.2006.04.061
[21] Li, G.; Jin, Z.: Global stability of a SEIR epidemic model with infectious force in latent, infected and immune period, Chaos, solitons & fractals 25, 1177-1184 (2005) · Zbl 1065.92046 · doi:10.1016/j.chaos.2004.11.062
[22] Li, G.; Jin, Z.: Global stability of an SEI epidemic model with general contact rate, Chaos, solitons & fractals 23, 997-1004 (2005) · Zbl 1062.92062
[23] Zhang, T.; Teng, Z.: Global asymptotic stability of a delayed SEIRS epidemic model with saturation incidence, Chaos, solitons & fractals 37, 1456-1468 (2008) · Zbl 1142.34384 · doi:10.1016/j.chaos.2006.10.041
[24] 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 · doi:10.1007/s002850050051
[25] Orsel, K.; Dekker, A.; Bouma, A.: Vaccination against foot and mouth disease reduces virus transmission in groups of calves, Vaccine 23, 4887-4894 (2005)
[26] Rebecca, J. C.; Eva, M.; Hakon, S.; Haaheim, L. R.: The effect of zanamivir treatment on the early immune response to influenza vaccination, Vaccine 19, 4743-4749 (2001)
[27] Zhang, Y. G.; Peng, B.; Deng, H.: Anti-HBV immune response by electroporation mediated DNA vaccination, Vaccine 24, 897-903 (2006)
[28] Gao, S. J.; Chen, L. S.; Nieto, J.; Torres, A.: Analysis of a delayed epidemic model with pulse vaccination and saturation incidence, Vaccine 24, 6037-6045 (2006)
[29] Meng, X. Z.; Chen, L. S.; Chen, H. D.: Two profitless delays for the SEIRS epidemic disease model with nonlinear incidence and pulse vaccination, Appl math comput (2006) · Zbl 1111.92049
[30] Gao, S. J.; Chen, L. S.; Teng, Z. D.: Impulsive vaccination of an SEIRS model with time delay and varying total population size, Bull math biol (2006) · Zbl 1139.92314
[31] Smart, D. R.: Fixed point theorems, (1980) · Zbl 0427.47036
[32] Burton, T. A.: A fixed-point theorem of Krasnoselskii, Appl math lett 11, C85-C88 (1998) · Zbl 1127.47318 · doi:10.1016/S0893-9659(97)00138-9
[33] Burton, T. A.; Furumochi, T.: Krasnoselskii’s fixed point theorem and stability, Nonlinear anal 49, 445-454 (2002) · Zbl 1015.34046 · doi:10.1016/S0362-546X(01)00111-0
[34] Kuang, Y.: Delay differential equation with application in population dynamics, (1993) · Zbl 0777.34002
[35] Cull, P.: Global stability for population models, Bull math biol 43, 47-58 (1981) · Zbl 0451.92011
[36] Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. S.: Theory of impulsive differential equations, (1989) · Zbl 0718.34011