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Dynamic behavior of a delayed impulsive SEIRS model in epidemiology. (English) Zbl 1195.34122

Summary: A delayed SEIRS epidemic model with pulse vaccination is investigated. Using Krasnoselskii’s fixed-point theorem, the infection-free periodic solution is obtained. Some new threshold values \({\mathcal R}_1, {\mathcal R}_2\) and \({\mathcal R}_3\) are obtained for dynamic behavior of the solutions. We point out, if \({\mathcal R}_1<1\) the infectious population disappear, i.e., the disease dies out, while if \({\mathcal R}_2>1\) or \({\mathcal R}_3>1\), the infectious permanent, the infectious population will ultimately remain above a positive level. An explicit formula is obtained by which the eventual lower bound of infectious individuals can be computed when \({\mathcal R}_2>1\). Our results indicate that a large pulse vaccination rate have some active effects to prevent or curtail the spread of the disease. Furthermore, we only proved the existence of \({\mathcal R}_3\) based upon some abstract theories.

MSC:

34K45 Functional-differential equations with impulses
92D30 Epidemiology
37N25 Dynamical systems in biology
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[1] F. Brauer, Epidemic models in populations of varying size , in Mathematical approaches to problems in resource management and epidemiology , C.C. Carlos, S.A. Levin and C. Shoemaker, eds., Lecture Notes Biomath. 81 , Springer, Berlin, 1989. · Zbl 0684.92016
[2] F. Brauer and P. Van den Driessche, Models for transmission of disease with immigration of infectives , Math. Biosci. 171 (2001), 143-154. · Zbl 0995.92041 · doi:10.1016/S0025-5564(01)00057-8
[3] H. Bremermann and H. Thieme, A competitive exclusion principle for pathogen virulence , J. Math. Biol. 27 (1989), 179-190. · Zbl 0715.92027 · doi:10.1007/BF00276102
[4] T. Burton, A fixed-point theorem of Krasnoselskii , Appl. Math. Lett. 11 (1998), 85-88. · Zbl 1127.47318 · doi:10.1016/S0893-9659(97)00138-9
[5] T. Burton and T. Furumochi, Krasnoselskii’s fixed point theorem and stability , Nonlinear Anal. 49 (2002), 445-454. · Zbl 1015.34046 · doi:10.1016/S0362-546X(01)00111-0
[6] O. Diekmann and J.A.P. Heesterbeek, Mathematical epidemiology of infectious diseases : Model building, analysis and interpretation , John Wiley & Sons, LTD, Chichester, New York, 2000. · Zbl 0997.92505
[7] S. Gakkhar and K. Negi, Pulse vaccination SIRS epidemic model with non monotonic incidence rate , Chaos, Solitons Fractals 35 (2008), 626-638. · Zbl 1131.92052 · doi:10.1016/j.chaos.2006.05.054
[8] S. Gao, L. Chen and Z. Teng, Impulsive vaccination of an SEIRS model with time delay and varying total population size, Bull. Math. Biol. 69 (2007), 731-745. · Zbl 1139.92314 · doi:10.1007/s11538-006-9149-x
[9] ——–, Pulse vaccination of an SEIR epidemic model with time delay, Nonlinear Analysis: Real World Appl. 9 (2008), 599-607. · Zbl 1144.34390 · doi:10.1016/j.nonrwa.2006.12.004
[10] L. Gao and H. Hethcote, Disease transmission models with density-dependent demographics , J. Math. Biol. 30 (1992), 717-731. · Zbl 0774.92018 · doi:10.1007/BF00173265
[11] D. Greenhalgh, Some threshold and stability results for epidemic models with a density dependent death rate , Theoret. Pop. Biol. 42 (1992), 130-151. · Zbl 0759.92009 · doi:10.1016/0040-5809(92)90009-I
[12] H. Hethcote and P. Van den Driessche, Some epidemiological models with nonlinear incidence , J. Math. Biol. 29 (1991), 271-287. · Zbl 0722.92015 · doi:10.1007/BF00160539
[13] M. Kermark and A. Mckendrick, Contributions to the mathematical theory of epidemics , Part I, Proc. Royal Soc. 115 (1927), 700-721. · JFM 53.0517.01
[14] Y. Kuang, Delay differential equation with application in population dynamics , Academic Press, New York, 1993. · Zbl 0777.34002
[15] G. Li and Z. Jin, Global stability of a SEIR epidemic model with infectious force in latent, infected and immune period, Chaos, Solitons Fractals 25 (2005), 1177-1184. · Zbl 1065.92046 · doi:10.1016/j.chaos.2004.11.062
[16] ——–, Global stability of an SEI epidemic model with general contact rate, Chaos, Solitons Fractals 23 (2005), 997-1004. · Zbl 1062.92062 · doi:10.1016/j.chaos.2004.06.012
[17] Z. Ma, Y. Zhou, W. Wang and Z. Jin, Mathematical modelling and research of epidemic dynamaical systems , Science Press, Beijing, 2004 (in Chinese).
[18] J. Mena-Lorca and H. Hethcote, Dynamic models of infectious diseases as regulators of population biology , J. Math. Biol. 30 (1992), 693-716. · Zbl 0748.92012
[19] X. Meng, L. Chen and H. Chen, Two profitless delays for the SEIRS epidemic disease model with nonlinear incidence and pulse vaccination, Appl. Math. Comput. 186 (2007), 516-529. · Zbl 1111.92049 · doi:10.1016/j.amc.2006.07.124
[20] G. Pang and L. Chen, A delayed SIRS epidemic model with pulse vaccination, Chaos, Solitons Fractals 34 (2007), 1629-1635. · Zbl 1152.34379 · doi:10.1016/j.chaos.2006.04.061
[21] B. Shulgin, L. Stone and Z. Agur, Pulse vaccination strategy in the SIR epidemic model , Bull. Math. Biol. 60 (1998), 1123-1148. · Zbl 0941.92026 · doi:10.1016/S0092-8240(98)90005-2
[22] D. Smart, Fixed point theorems , Cambridge University Press, Cambridge, 1980. · Zbl 0427.47036
[23] L. Stone, B. Shulgin and Z. Agur, Theoretical examination of the pulse vaccination policy in the SIR epidemic model , J. Math. Comp. Modelling 31 (2000), 207-215. · Zbl 1043.92527 · doi:10.1016/S0895-7177(00)00040-6
[24] H. Thieme, Epidemic and demographic interaction in the spread of potentially fatal diseases in growing populations , Math. Biosci. 111 (1992), 99-130. · Zbl 0782.92018 · doi:10.1016/0025-5564(92)90081-7
[25] W. Wang and S. Ruan, Bifurcations in an epidemic model with constant removal rate of the infectives , J. Math. Anal. Appl. 291 (2004), 774-793. · Zbl 1054.34071 · doi:10.1016/j.jmaa.2003.11.043
[26] T. Zhang and Z. Teng, Global asymptotic stability of a delayed SEIRS epidemic model with saturation incidence, Chaos, Solitons Fractals · Zbl 1142.34384 · doi:10.1016/j.chaos.2006.10.041
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