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Discrete Dynamics in Nature and Society
Volume 2015 (2015), Article ID 848623, 10 pages
http://dx.doi.org/10.1155/2015/848623
Research Article

Dynamical Analysis of SIR Epidemic Model with Nonlinear Pulse Vaccination and Lifelong Immunity

1College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China
2State Key Laboratory of Mining Disaster Prevention and Control Cofounded by Shandong Province and the Ministry of Science and Technology, Shandong University of Science and Technology, Qingdao 266590, China

Received 4 November 2014; Accepted 10 February 2015

Academic Editor: Piyapong Niamsup

Copyright © 2015 Wencai Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. The World Health Statistics Report 2013, http://www.who.int/research/en/.
  2. Z. Agur, L. Cojocaru, G. Mazor, R. M. Anderson, and Y. L. Danon, “Pulse mass measles vaccination across age cohorts,” Proceedings of the National Academy of Sciences of the United States of America, vol. 90, no. 24, pp. 11698–11702, 1993. View at Publisher · View at Google Scholar · View at Scopus
  3. C. A. de Quadros, J. K. Andrus, J.-M. Olivé et al., “Eradication of poliomyelitis: progress in the Americas,” The Pediatric Infectious Disease Journal, vol. 10, no. 3, pp. 222–229, 1991. View at Publisher · View at Google Scholar · View at Scopus
  4. A. B. Sabin, “Measles, killer of millions in developing countries: strategy for rapid elimination and continuing control,” European Journal of Epidemiology, vol. 7, no. 1, pp. 1–22, 1991. View at Google Scholar · View at Scopus
  5. M. Ramsay, N. Gay, E. Miller et al., “The epidemiology of measles in England and Wales: rationale for the 1994 national vaccination campaign,” Communicable Disease Report, vol. 4, no. 12, pp. R141–R146, 1994. View at Google Scholar
  6. B. Shulgin, L. Stone, and Z. Agur, “Pulse vaccination strategy in the SIR epidemic model,” Bulletin of Mathematical Biology, vol. 60, no. 6, pp. 1123–1148, 1998. View at Publisher · View at Google Scholar · View at Scopus
  7. L. Stone, B. Shulgin, and Z. Agur, “Theoretical examination of the pulse vaccination policy in the SIR epidemic model,” Mathematical and Computer Modelling, vol. 31, no. 4, pp. 207–215, 2000. View at Google Scholar
  8. A. d'Onofrio, “Stability properties of pulse vaccination strategy in SEIR epidemic model,” Mathematical Biosciences, vol. 179, no. 1, pp. 57–72, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. A. d'Onofrio, “On pulse vaccination strategy in the SIR epidemic model with vertical transmission,” Applied Mathematics Letters, vol. 18, no. 7, pp. 729–732, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. J. Jiao, L. Chen, and S. Cai, “An SEIRS epidemic model with two delays and pulse vaccination,” Journal of Systems Science and Complexity, vol. 21, no. 2, pp. 217–225, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. H. Zhang, L. Chen, and J. J. Nieto, “A delayed epidemic model with stage-structure and pulses for pest management strategy,” Nonlinear Analysis: Real World Applications, vol. 9, no. 4, pp. 1714–1726, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. X. Song, Y. Jiang, and H. Wei, “Analysis of a saturation incidence SVEIRS epidemic model with pulse and two time delays,” Applied Mathematics and Computation, vol. 214, no. 2, pp. 381–390, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. S. Gao, L. Chen, J. J. Nieto, and A. Torres, “Analysis of a delayed epidemic model with pulse vaccination and saturation incidence,” Vaccine, vol. 24, no. 35-36, pp. 6037–6045, 2006. View at Publisher · View at Google Scholar · View at Scopus
  14. X. Liu, Y. Takeuchi, and S. Iwami, “SVIR epidemic models with vaccination strategies,” Journal of Theoretical Biology, vol. 253, no. 1, pp. 1–11, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. Z. Zhao, L. Chen, and X. Song, “Impulsive vaccination of SEIR epidemic model with time delay and nonlinear incidence rate,” Mathematics and Computers in Simulation, vol. 79, no. 3, pp. 500–510, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. W. C. Zhao and X. Z. Meng, “An SIR epidemic disease model with vertical transmission and pulse vaccination,” Mathematica Applicata, vol. 22, no. 3, pp. 676–682, 2009 (Chinese). View at Google Scholar · View at MathSciNet
  17. X. Meng and L. Chen, “Global dynamical behaviors for an SIR epidemic model with time delay and pulse vaccination,” Taiwanese Journal of Mathematics, vol. 12, no. 5, pp. 1107–1122, 2008. View at Google Scholar · View at MathSciNet · View at Scopus
  18. X. Meng, L. Chen, and B. Wu, “A delay SIR epidemic model with pulse vaccination and incubation times,” Nonlinear Analysis: Real World Applications, vol. 11, no. 1, pp. 88–98, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. X. Meng and L. Chen, “The dynamics of a new SIR epidemic model concerning pulse vaccination strategy,” Applied Mathematics and Computation, vol. 197, no. 2, pp. 582–597, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  20. W. Zhao, T. Zhang, Z. Chang, X. Meng, and Y. Liu, “Dynamical analysis of SIR epidemic models with distributed delay,” Journal of Applied Mathematics, vol. 2013, Article ID 154387, 15 pages, 2013. View at Publisher · View at Google Scholar · View at Scopus
  21. S. Gao, Y. Liu, J. J. Nieto, and H. Andrade, “Seasonality and mixed vaccination strategy in an epidemic model with vertical transmission,” Mathematics and Computers in Simulation, vol. 81, no. 9, pp. 1855–1868, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. R. Shi, X. Jiang, and L. Chen, “The effect of impulsive vaccination on an SIR epidemic model,” Applied Mathematics and Computation, vol. 212, no. 2, pp. 305–311, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  23. J. Li and Y. Yang, “SIR-SVS epidemic models with continuous and impulsive vaccination strategies,” Journal of Theoretical Biology, vol. 280, no. 1, pp. 108–116, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  24. T. Zhang, X. Meng, T. Zhang, and Y. Song, “Global dynamics for a new high-dimensional SIR model with distributed delay,” Applied Mathematics and Computation, vol. 218, no. 24, pp. 11806–11819, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  25. T. Zhang, X. Meng, and T. Zhang, “SVEIRS: a new epidemic disease model with time delays and impulsive effects,” Abstract and Applied Analysis, vol. 2014, Article ID 542154, 15 pages, 2014. View at Publisher · View at Google Scholar
  26. J. Zhang and J. Sun, “A delayed SEIRS epidemic model with impulsive vaccination and nonlinear incidence rate,” International Journal of Biomathematics, vol. 7, no. 3, Article ID 1450032, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  27. Y. Pei, S. Liu, L. Chen, and C. Wang, “Two different vaccination strategies in an SIR epidemic model with saturated infectious force,” International Journal of Biomathematics, vol. 1, no. 2, pp. 147–160, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  28. J. Hou and Z. Teng, “Continuous and impulsive vaccination of SEIR epidemic models with saturation incidence rates,” Mathematics and Computers in Simulation, vol. 79, no. 10, pp. 3038–3054, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  29. G. P. Pang and L. S. Chen, “The SIRS epidemic model with saturated contact rate and pulse vaccination,” Journal of Systems Science and Mathematical Sciences, vol. 27, no. 4, pp. 563–572, 2007 (Chinese). View at Google Scholar · View at MathSciNet
  30. G. Pang and L. Chen, “A delayed SIRS epidemic model with pulse vaccination,” Chaos, Solitons & Fractals, vol. 34, no. 5, pp. 1629–1635, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  31. A. V. Hill, “The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves,” The Journal of Physiology, vol. 40, no. 4, pp. 4–7, 1910. View at Google Scholar
  32. D. Bainov and P. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, CRC Press, Boca Raton, Fla, USA, 1993.
  33. A. Lakmeche and O. Arino, “Bifurcation of non trivial periodic solutions of impulsive differential equations arising chemotherapeutic treatment,” Dynamics of Continuous, Discrete and Impulsive Systems, vol. 7, no. 2, pp. 265–287, 2000. View at Google Scholar · View at MathSciNet