[ Log In | Register]
! Skip Navigation Links
Skip Navigation Links
Image of Journal of Applied Nonlinear Dynamics
ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Sufficient Condition of the Bifurcation for a Dynamic System of Three Equations and Application

Journal of Applied Nonlinear Dynamics 12(1) (2023) 99--112 | DOI:10.5890/JAND.2023.03.007

Amine Bernoussi

Laboratory: '{E}quations aux d'{e}riv'{e}es partielles, Alg`{e}bre et G'{e}om'{e}trie spectrales, Faculty of Science, Ibn Tofail University, BP 133, 14000 Kenitra, Morocco

Download Full Text PDF

 

Abstract

In this paper we propose a new sufficient condition of the bifurcation for nonlinear autonomous differential equations with delay in three dimension, which just depends in coefficient on the characteristic equation, who can verify them analytically at the level of the application to a dynamics system. For the application of these conditions we propose a SEIR epidemic model with delay and nonlinear incidence rate. The resulting model has two possible equilibria. By using suitable Lyapunov functionals and LaSalle's invariance principle, the global stability of a disease-free equilibrium is established. Finally we affirm the existence of non constant periodic solutions which bifurcate from the endemic equilibrium when the delay crosses some critical values. In the end, the numerical simulations are proposed to illustrate our results.

References

  1. [1]  Ruan, S. and Wei, J. (2001), On the zeros of a third degree exponential polynomial with applications to a delayed model for the control of testosterone secretion, Mathematical Medicine and Biology, 18, 41-52.
  2. [2]  Kaddar, A. and Talibi Alaoui, H. (2008), Fluctuations in a Mixed IS-LM Business cycle Model, Electronic Journal of Differential Equations, 2008(134), 1-9.
  3. [3]  Raji-allah, A. and Talibi Alaoui, H. (2015), Global dynamics of a delayed SEIR epidemic model with relapse effect, International Journal of Academic Studies, 1(1), 78-91.
  4. [4]  Sampath Aruna, P. (2015), Local stability properties of a delayed SIR model with relaps effect, International Journal of Scientific and Research Publication, 5(10), 1-8.
  5. [5]  Smith, H.L. (2010), An Introduction to Delay Differential Equations With Applications to the Life Sciences, 57, New York: Springer.
  6. [6]  Hassard, B.D., Kazarinoff, N.D., and Wan, Y.H. (1981), Theory and Applications of Hopf Bifurcation, Cambridge: Cambridge University Press.
  7. [7]  Kuang, Y. (1993), Delay Differential Equations with Applications in Population Dynamics, Academic Press, San Diego.
  8. [8]  Ruan, S. and Wei, J. (2003), On the zeroes of transcendental functions with applications to the stability of delay differential equations, Journal Dynamics of Continuous, Discrete and Impulsive Systems, SER. A: Mathematical Analysis, 10, 863-874.
  9. [9]  Hale, J.K. and Verduyn Lunel, S.M. (1993), Introduction to Functional Differential Equations, Springer- Verlag, New York.
  10. [10]  Hbid, M. (2006), Introduction to Hopf Bifurcation Theory for Delay Differential Equations, Springer, Dordrecht, 161-191.
  11. [11]  Xu, R. (2013), Global dynamic of a delayed epidemic model with latency and relapse, Nonlinear Analysis: Modelling and control, 18(2), 250-263.
  12. [12]  Xu, R. (2014), Global dynamics of an SEIRI epidemiological model with time delay, Applied Mathematics and Computation, 232(1), 436-444.
  13. [13]  Anderson, R.M. and May, R.M. (1978), Regulation and stability of host-parasite population interactions: I. Regulatory processes, The Journal of Animal Ecology, 47(1), 219-267.
  14. [14]  Chen, L.S. and Chen, J. (1993), Nonlinear Biological Dynamics System, Scientific Press, China.
  15. [15]  Gao, S. Chen, L. Nieto, J.J., and Torres, A. (2006), Analysis of a delayed epidemic model with pulse vaccination and saturation incidence, Vaccine, 24(35-36), 6037-6045.
  16. [16]  Wei, C. and Chen, L. (2008), A delayed epidemic model with pulse vaccination, Discrete Dynamics in Nature and Society, 1-2.
  17. [17]  Zhang, J.Z., Jin, Z., Liu, Q.X., and Zhang, Z.-Y. (2008), Analysis of a delayed SIR model with nonlinear incidence rate, Discrete Dynamics in Nature and Society, 1-6.
  18. [18]  Gabriela, M., Gomes, M., White, L.J., and Medley, G.F. (2005), The reinfection threshold, Journal of Theoretical Biology, 236, 111-113.
  19. [19]  Wang, W. and Ruan, S. (2004), Bifurcation in epidemic model with constant removal rate infectives, Journal of Mathematical Analysis and Applications, 291, 775-793.
  20. [20]  Zhang, F., Li, Z.Z., and Zhang, F. (2008), Global stability of an SIR epidemic model with constant infectious period, Applied Mathematics and Computation, 199(1), 285-291.
  21. [21]  Xu, C. and Liao, M. (2011), Stability and bifurcation analysis in a SEIR epidemic model with non linear incidence rates, IAENG International Journal of Applied Mathematics, 41(3), 191-198.
  22. [22]  Bernoussi, A. (2019), Global stability analysis of an SEIR epidemic model with relapse and general incidence rates, Electronique journal of Mathematical, Analysis and Applications, 7(2), 168-180.
  23. [23]  Enatsu, Y., Nakata, Y., and Muroya, Y. (2011), Global stability of SIR epidemic models with a wide class of nonlinear incidence and distributed delays, Discrete and Continuous Dynamical Systems Series B, 15(1), 61-74.
  24. [24]  LaSalle, J.P. (1976), The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics, SIAM, Philadelphia. %
  25. [25]  Ruan, S. and Wei, J. (2001), On the zeros of a third degree exponential polynomial with applications to a delayed model for the control of testosterone secretion, % Mathematical Medicine and Biology, 18, 41-52. % %
  26. [26]  Kaddar, A. and Talibi Alaoui, H. (2008), Fluctuations in a Mixed IS-LM Business cycle Model, Electronic Journal of %Differential Equations, 2008(134), 1-9. % %
  27. [27]  Raji-allah, A. and Talibi Alaoui, H. (2015), Global dynamics of a delayed SEIR epidemic model with relapse effect, International Journal of Academic Studies, % 1(1), 78-91. % %
  28. [28]  Sampath Aruna, P. (2015), Local stability properties of a delayed SIR model with relaps effect, International Journal of Scientific and Research Publication, % 5(10), 1-8. % %
  29. [29]  Smith, H.L. (2010), An Introduction to Delay Differential Equations With Applications to the Life Sciences, 57, New York: Springer. % %
  30. [30]  Hassard, B.D., Kazarinoff, N.D., and Wan, Y.H. (1981), %Theory and Applications of Hopf Bifurcation, Cambridge: Cambridge University Press. % %
  31. [31]  Kuang, Y. (1993), Delay Differential Equations with Applications in Population Dynamics, Academic Press, San Diego. % %
  32. [32]  LaSalle, J.P. (1976), The Stability of Dynamical Systems, Regional Conference Series in Applied %Mathematics, SIAM, Philadelphia. % %
  33. [33]  Ruan, S. and Wei, J. (2003), On the zeroes of transcendental functions with applications to the stability of delay differential equations, Journal Dynamics of Continuous, Discrete and Impulsive Systems, SER. A: Mathematical Analysis, % 10, 863-874. % %
  34. [34]  Hale, J.K. and Verduyn Lunel, S.M. (1993), %Introduction to Functional Differential Equations, Springer- Verlag, New York. % %
  35. [35]  Hbid, M. (2006), Introduction to Hopf Bifurcation Theory for Delay Differential Equations, Springer, Dordrecht, 161-191. % %
  36. [36]  Xu, R. (2013), Global dynamic of a delayed epidemic model with latency and relapse, %Nonlinear Analysis: Modelling and control, 18(2), 250-263. % %
  37. [37]  Xu, R. (2014), Global dynamics of an SEIRI epidemiological model with time delay, Applied %Mathematics and Computation, 232(1), 436-444. % %
  38. [38]  Anderson, R.M. and May, R.M. (1978), Regulation and stability of host-parasite population interactions: I. %Regulatory processes, The Journal of Animal Ecology, 47(1), 219-267. % %
  39. [39]  Chen, L.S. and Chen, J. (1993), Nonlinear Biological Dynamics System, Scientific Press, China. % %
  40. [40]  Gao, S. Chen, L. Nieto, J.J., and Torres, A. (2006), Analysis of a delayed epidemic model with pulse vaccination and saturation incidence, Vaccine, % 24(35-36), 6037-6045. % %
  41. [41]  Wei, C. and Chen, L. (2008), A delayed epidemic model with pulse vaccination, %Discrete Dynamics in Nature and Society, 1-2. % %
  42. [42]  Zhang, J.Z., Jin, Z., Liu, Q.X., and Zhang, Z.-Y. (2008), Analysis of a delayed SIR model with nonlinear %incidence rate, Discrete Dynamics in Nature and Society, 1-6. % %
  43. [43]  Gabriela, M., Gomes, M., White, L.J., and Medley, G.F. (2005), The reinfection threshold, Journal of Theoretical Biology, 236, 111-113. % %
  44. [44]  Wang, W. and Ruan, S. (2004), Bifurcation in epidemic model with constant removal rate infectives, Journal %of Mathematical Analysis and Applications, 291, 775-793. % %
  45. [45]  Zhang, F., Li, Z.Z., and Zhang, F. (2008), Global stability of an SIR epidemic model with constant %infectious period, Applied Mathematics and Computation, 199(1), 285-291. % %
  46. [46]  Xu, C. and Liao, M. (2011), Stability and bifurcation analysis in a SEIR epidemic model with non linear %incidence rates, IAENG International Journal of Applied Mathematics, 41(3), 191-198. % %
  47. [47]  Bernoussi, A. (2019), Global stability analysis of an SEIR epidemic model with relapse and general incidence rates, Electronique journal of Mathematical, Analysis and Applications, 7(2), 168-180. % % % % %
  48. [48]  Enatsu, Y., Nakata, Y., and Muroya, Y. (2011), Global stability of SIR epidemic models with a wide class of nonlinear incidence and distributed delays, Discrete and Continuous Dynamical Systems Series B, 15(1), 61-74. % %

Copyright © 2011-2023 L & H Scientific Publishing. All rights reserved. Wildcard SSL Certificates