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Dynamical properties of a kind of SIR model with constant vaccination rate and impulsive state feedback control

Hongjian Guo

College of Mathematics and Statistics, Xinyang Normal University, Xinyang 464000, P. R. China

E-mail Address: xbghj@163.com

Corresponding authors.

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Lansun Chen

Academy of Mathematics and Systems Science, Chinese Academy of Science, Beijing 100080, P. R. China

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, and

Xinyu Song

College of Mathematics and Statistics, Xinyang Normal University, Xinyang 464000, P. R. China

E-mail Address: xysong88@163.com

Corresponding authors.

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https://doi.org/10.1142/S1793524517500930Cited by:14 (Source: Crossref)

Abstract

Considering the fact that the production and provision of some vaccines are ordered and governed by the government according to the history data of disease, a kind of SIR model with constant vaccination rate and impulsive state feedback control is presented. The dynamical properties of semi-continuous three-dimensional SIR system can be obtained by discussing the properties of the corresponding two-dimensional system in the limit set. The existence and uniqueness of order-1 periodic solution are discussed by using the successive function and the compression mapping theorem. A new theorem for the orbital stability of order-1 periodic solution is proved by geometric method. Finally, numerical simulations are given to verify the mathematical results and some conclusions are given. The results show that the disease can be controlled to a lower level by means of impulsive state feedback control strategy, but cannot be eradicated.

Keywords:

AMSC: 34C25, 34D20, 92B05

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References

1. D. D. Bainov and P. S. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, Pitman Monographs and Surveys in Pure and Applied Mathematics (Longman Scientific and Technical, New York, 1993). Google Scholar
2. L. S. Chen, Pest control and geometric theory of semi-continuous dynamical system, J. Beihua Univ. Nat. Sci. Ed. 1 (2011) 1–9. Google Scholar
3. M. De la Sen and S. Alonso-Quesada, Vaccination strategies based on feedback control techniques for a general SEIR-epidemic model, Appl. Math. Comput. 218(7) (2011) 3888–3904. Web of Science, Google Scholar
4. A. D'Onofrio, On pulse vaccination strategy in the SIR epidemic model with vertical transmission, Appl. Math. Lett. 18(7) (2005) 729–732. Crossref, Web of Science, Google Scholar
5. S. J. Gao et al., Analysis of an SIR epidemic model with pulse vaccination and distributed time delay, J. Biomed. Biotechnol. 2007 (2007) 64870. Crossref, Google Scholar
6. H. J. Guo and L. S. Chen, Periodic solution of a turbidostat system with impulsive state feedback control, J. Math. Chem. 46(4) (2009) 1074–1086. Crossref, Web of Science, Google Scholar
7. H. J. Guo and L. S. Chen, Periodic solution of a chemostat model with monod growth rate and impulsive state feedback control, J. Theor. Biol. 260(4) (2009) 502–509. Crossref, Web of Science, Google Scholar
8. H. J. Guo and L. S. Chen, Qualitative analysis of a variable yield turbidostat model with impulsive state feedback control, J. Appl. Math. Comput. 33(1–2) (2010) 193–208. Crossref, Google Scholar
9. H. J. Guo, L. S. Chen and X. Y. Song, Geometric properties of solution of a cylindrical dynamic system with impulsive state feedback control, Nonlinear Anal. Hybrid Syst. 15 (2015) 98–111. Crossref, Web of Science, Google Scholar
10. H. J. Guo, X. Y. Song and L. S. Chen, Qualitative analysis of a Korean pine forest model with impulsive thinning measure, Appl. Math. Comput. 234 (2014) 203–213. Crossref, Web of Science, Google Scholar
11. G. R. Jiang, Q. S. Lu and L. N. Qian, Complex dynamics of a Holling-type II prey–predator system with state feedback control, Chaos, Solitons Fractals 31(2) (2007) 448–461. Crossref, Web of Science, Google Scholar
12. G. R. Jiang and Q. G. Yang, Bifurcation analysis in an SIR epidemic model with birth pulse and pulse vaccination, Appl. Math. Comput. 215(3) (2009) 1035–1046. Web of Science, Google Scholar
13. V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations (World Scientific, Singapore, 1989). Link, Google Scholar
14. Z. X. Li, T. Y. Wang and L. S. Chen, Periodic solution of a chemostat model with Beddington–DeAngelis uptake function and impulsive state feedback control, J. Theor. Biol. 261(1) (2009) 23–32. Crossref, Web of Science, Google Scholar
15. Z. H. Lu, X. B. Chi and L. S. Chen, The effect of constant and pulse vaccination on SIR epidemic model with horizontal and vertical transmission, Math. Comput. Model. 36(9) (2002) 1039–1057. Crossref, Web of Science, Google Scholar
16. X. Z. Meng, L. S. Chen and B. Wu, A delay SIR epidemic model with pulse vaccination and incubation times, Nonlinear Anal. Real World Appl. 11(1) (2010) 88–98. Crossref, Web of Science, Google Scholar
17. L. F. Nie, Z. D. Teng and A. Torres, Dynamic analysis of an SIR epidemic model with state-dependent pulse vaccination, Nonlinear Anal. Real World Appl. 13(4) (2012) 1621–1629. Crossref, Web of Science, Google Scholar
18. P. Ogren and M. Clyde, Vaccination strategies for epidemics in highly mobile populations, Appl. Math. Comput. 127(2)(2002) 261–276. Web of Science, Google Scholar
19. G. P. Pang and L. S. Chen, Periodic solution of the system with impulsive state feedback control, Nonlinear Dynam. 78(1) (2014) 743–753. Crossref, Web of Science, Google Scholar
20. B. Shulgin, L. Stone and Z. Agur, Pulse vaccination strategy in the SIR epidemic model, Bull. Math. Biol. 60(6) (1998) 1123–1148. Crossref, Web of Science, Google Scholar
21. P. S. Simeonov and D. D. Bainov, Orbital stability of the periodic solutions of autonomous systems with impulse effect, Publ. RIMS Kyoto Univ. 25 (1989) 321–346. Crossref, Web of Science, Google Scholar
22. M. J. Sun et al., A novel method for analyzing the stability of periodic solution of impulsive state feedback model, Appl. Math. Comput. 273 (2016) 425–434. Crossref, Web of Science, Google Scholar
23. S. Y. Tang and R. A. Cheke, State-dependent impulsive models of integrated pest management (IPM) strategies and their dynamic consequences, J. Math. Biol. 50(3) (2005) 257–292. Crossref, Web of Science, Google Scholar
24. S. Y. Tang, W. H. Pang and A. Robert, Global dynamics of a state-dependent feedback control system, Adv. Differ. Equations 2015 (2015) 322. Crossref, Web of Science, Google Scholar
25. Y. Tian et al., Studies on the dynamics of a continuous bioprocess with impulsive state feedback control, Chem. Engrg. J. 157(2–3) (2010) 558–567. Crossref, Web of Science, Google Scholar
26. Y. Tian, K. B. Sun and L. S. Chen, Geometric approach to the stability analysis of the periodic solution in a semi-continuous dynamic system, Int. J. Biomath. 7(2) (2014) 1450018. Link, Web of Science, Google Scholar
27. C. J. Wei and L. S. Chen, Periodic solution of prey–predator model with Beddington–DeAngelis functional response and impulsive state feedback control, J. Appl. Math. 2012 (2012) 1–17. Google Scholar
28. Y. N. Xiao and S. Y. Tang, Dynamics of infection with nonlinear incidence in a simple vaccination model, Nonlinear Anal. Real World Appl. 11(5) (2010) 4154–4163. Crossref, Web of Science, Google Scholar
29. J. Y. Zhang and B. Y. Feng, Geometrical Theory and Bifurcation Problems of Ordinary Differential Equations (Peking University Press, 2000), pp. 262–272. Google Scholar