Nonlinear Dynamics

, Volume 59, Issue 3, pp 503–513 | Cite as

Dynamics of a novel nonlinear SIR model with double epidemic hypothesis and impulsive effects



In this paper, the propagation of a nonlinear delay SIR epidemic using the double epidemic hypothesis is modeled. In the model, a system of impulsive functional differential equations is studied and the sufficient conditions for the global attractivity of the semi-trivial periodic solution are drawn. By use of new computational techniques for impulsive differential equations with delay, we prove that the system is permanent under appropriate conditions. The results show that time delay, pulse vaccination, and nonlinear incidence have significant effects on the dynamics behaviors of the model. The conditions for the control of the infection caused by viruses A and B are given.


Double epidemic hypothesis Permanence Nonlinear incidence Time delay SIR epidemic model 


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  1. 1.
    Enjuanes, L., Sanchez, C., Gebauer, F., Mendez, A., Dopazo, J., Ballesteros, M.L.: Evolution and tropism of transmissible gastroenteritis coronavirus. Adv. Exp. Med. Biol. 342, 35–42 (1993) Google Scholar
  2. 2.
    Beretta, E., Takeuchi, Y.: Global stability of an SIR epidemic model with time delays. J. Math. Biol. 33(3), 250–260 (1995) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Beretta, E., Hara, T., Ma, W.B., Takenchi, Y.: Global asymptotic stability of an SIR epidemic model with distributed time delay. Nonlinear Anal. 47, 4107–4115 (2001) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Takeuchi, Y., Ma, W.B., Beretta, E.: Global asymptotic properties of a delay SIR epidemic model with finite incubation times. Nonlinear Anal. 42, 931–947 (2000) CrossRefMathSciNetGoogle Scholar
  5. 5.
    Ma, W.B., Song, M., Takeuchi, Y.: Global stability of an SIR epidemic model with time delay. Appl. Math. Lett. 17, 1141–1145 (2004) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Song, M., Ma, W.B., Takeuchi, Y.: Permanence of a delayed SIR epidemic model with density dependent birth rate. J. Comput. Appl. Math. 201(2), 389–394 (2007) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Ma, W.B., Takeuchi, Y., Hara, T., Beretta, E.: Permanence of an SIR epidemic model with distributed time delays. Tohoku Math. J. 54, 581–591 (2002) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Zhang, T.L., Teng, Z.D.: Permanence and extinction for a nonautonomous SIRS epidemic model with time delay. Appl. Math. Model. 33(2), 1058–1071 (2009) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Zhang, T.L., Teng, Z.D.: Global behavior and permanence of SIRS epidemic model with time delay. Nonlinear Anal.: Real World Appl. 9(4), 1409–1424 (2008) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Meng, X.Z., Chen, L.S.: The dynamics of a new SIR epidemic model concerning pulse vaccination strategy. Appl. Math. Comput. 197(2), 582–597 (2008) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Shulgin, B., Stone, L., Agur, Z.: Pulse vaccination strategy in the SIR epidemic model. Bull. Math. Biol. 60, 1–26 (1998) CrossRefGoogle Scholar
  12. 12.
    Lu, Z.H., Chi, X.B., Chen, L.S.: The effect of constant and pulse vaccination on SIR epidemic model with horizontal and vertical transmission. Math. Comput. Model. 36, 1039–1057 (2002) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    D’Onofrio, A.: On pulse vaccination strategy in the SIR epidemic model with vertical transmission. Appl. Math. Lett. 18, 729–732 (2005) MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Hui, J., Chen, L.S.: Impulsive vaccination of SIR epidemic models with nonlinear incidence rates. Discrete Continuous Dyn. Syst. Ser. B 4, 595–605 (2004) MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    DeQuadros, C.A., Andrus, J.K., Olive, J.M.: Eradication of the poliomyelitis, progress. Am. Pediatr. Infect. Dis. J. 10(3), 222–229 (1991) CrossRefGoogle Scholar
  16. 16.
    Ramsay, M., Gay, N., Miller, E.: The epidemiology of measles in England and Wales: Rationale for 1994 nation vaccination campaign. Commun. Dis. Rep. 4(12), 141–146 (1994) Google Scholar
  17. 17.
    Sabin, A.B.: Measles, killer of millions in developing countries: Strategies of elimination and continuation control. Eur. J. Epidemiology 7, 1–22 (1991) Google Scholar
  18. 18.
    Zhang, H., Georgescu, P., Chen, L.S.: An impulsive predator–prey system with Beddington–Deangelis functional response and time delay. Int. J. Biomath. 1(1), 1–17 (2008) MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Sun, S.L., Chen, L.S.: Permanence and complexity of the eco-epidemiological model with impulsive perturbation. Int. J. Biomath. 1(2), 121–132 (2008) MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Liu, B., Teng, Z.D., Liu, W.B.: Dynamic behaviors of the periodic Lotka–Volterra competing system with impulsive perturbations. Chaos Solitons Fractals 31(2), 356–370 (2007) MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Li, Z.X., Chen, L.S.: Periodic solution of a turbidostat model with impulsive state feedback control. Nonlinear Dyn. (in press). doi 10.1007/s11071-009-9498-8
  22. 22.
    Wei, C.J., Chen, L.S.: Dynamic analysis of mathematical model of ethanol fermentation with gas stripping. Nonlinear Dyn. 57(1–2), 13–23 (2009) MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Shi, R.Q., Chen, L.S.: The study of a ratio-dependent predator–prey model with stage structure in the prey. Nonlinear Dyn. (in press). doi 10.1007/s11071-009-9491-2
  24. 24.
    Cooke, K.L.: Stability analysis for a vector disease model. Rocky Mt. J. Math. 9, 31–42 (1979) MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Wei, H., Li, X., Martcheva, M.: An epidemic model of a vector-borne disease with direct transmission and time delay. J. Math. Anal. Appl. 342, 895–908 (2008) MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Mukandavire, Z., Garira, W., Chiyaka, C.: Asymptotic properties of an HIV/AIDS model with a time delay. J. Math. Anal. Appl. 330, 916–933 (2007) MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    McCluskey, C.: Global stability for a class of mass action systems allowing for latency in tuberculosis. J. Math. Anal. Appl. 338, 518–535 (2008) MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Beretta, E., Luang, Y.: Modeling and analysis of a marine bacteriophage infection with latency period. Nonlinear Anal.: Real World Appl. 2, 35–74 (2001) MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Ghosh, S., Bhattacharyya, S., Bhattacharya, D.K.: Role of latency period in viral infection: A pest control model. Math. Biosci. 210, 619–646 (2007) MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Meng, X., Chen, L., Cheng, H.: Two profitless delays for the SEIRS epidemic disease model with nonlinear incidence and pulse vaccination. Appl. Math. Comput. 186, 516–529 (2007) MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Wei-min, Levin, S.A., Lwasa, Y.: Influence of nonlinear incidence rates upon the behavior of SIRS Epidemiological models. J. Math. Biol. 25, 359–380 (1987) MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Wei-min, Hethcote, H.W., Levin, S.A.: Dynamical behavior of epidemiological models with nonlinear incidence rates. J. Math. Biol. 23, 187–240 (1986) MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Lakshmikantham, V., Bainov, D., Simeonov, P.: Theory of Impulsive Differential Equations. World Scientific, Singapore (1989) MATHGoogle Scholar
  34. 34.
    Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Academic Press, San Diego (1993) MATHGoogle Scholar

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© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.State Key Laboratory of Vegetation and Environmental ChangeInstitute of Botany, Chinese Academy of SciencesBeijingPeople’s Republic of China
  2. 2.College of Information Science and EngineeringShandong University of Science and TechnologyQingdaoPeople’s Republic of China
  3. 3.College of AgronomyHenan University of Science and TechnologyLuoyangPeople’s Republic of China

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