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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


Bifurcation of Periodic Solutions of a Delayed SEIR Epidemic Model with Nonlinear Incidence Rate

Journal of Applied Nonlinear Dynamics 10(3) (2021) 351--367 | DOI:10.5890/JAND.2021.09.001

Amine Bernoussi

'{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

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Abstract

In this paper, we propose the SEIR epidemic model with delay and nonlinear incidence rate. The resulting model has two possible equilibria: if $R_{0} \leq 1,$ then the SEIR epidemic model has a disease-free equilibrium and if $R_{0} > 1,$ then the SEIR epidemic model admits a unique endemic equilibrium. By using suitable Lyapunov functionals and LaSalle's invariance principle, the global stability of a disease-free equilibrium is established. Our main contribution affirms the existence of non constant periodic solutions which bifurcate from the endemic equilibrium when the delay crosses some critical values. Finally, some numerical simulations are presented to illustrate our theoretical results.

References

  1. [1]  Abta, A., Kaddar, A., and Alaoui, H.T. (2011), {Global stability for delay SIR and SEIR epidemic models with saturated incidence rates}, Electronic Journal of Differential Equations, 2012(23), 1-13.
  2. [2]  Kaddar, A., Abta, A., and Alaoui, T.H. (2011), {A comparison of delayed SIR and SEIR epidemic models}, Nonlinear Analysis: Modelling and Control, 16(2), 181-190.
  3. [3]  Bernoussi, A., Kaddar, A., and Asserda, S. (2017), {On the dynamics of an SIRI epidemic model with a generalized incidence function}, the electronic International journal advanced modeling and optimization, 19(1), 87-96.
  4. [4]  Bernoussi, A., Kaddar, A., and Asserda, S. (2016), {Stability Analysis of an SIRI Epidemic Model with Distributed Latent Period}, Journal of Advances in Applied Mathematics, 1(4), 211-221.
  5. [5]  Bernoussi, A., Kaddar, A., and Asserda, S. (2016), {Stability Qualitative analysis of an SEIR epidemic model with a relaps rate}, Asian Journal of Mathematics and computer Research, 7(3), 190-200.
  6. [6]  Bernoussi, A., Kaddar, A., and Asserda, S. (2014), {Stability Global stability of a delayed SIRI epidemic model with nonlinear incidence}, International Journal of Engineering Mathematics, 2014, Article ID 487589.
  7. [7]  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.
  8. [8]  Hattaf, K., Lashari, A.A., Louartassi, Y., and Yousfi, N. (2013), {A delayed SIR epidemic model with a generalized incidence rate}, Electronic Journal of Qualitative Theory of Differential Equations, 2013(3), 1-9.
  9. [9]  Olaniyi, S. and Obabiy, O.S. (2014), {Qualitative Analysis of Malaria Dynamics with nonlinear incidence function}, Appl. Math. Sci., 8(78), 3889-3904.
  10. [10]  Xu, R. and Ma, Z. (2009), {Stability of a delayed SIRS epidemic model with a nonlinear incidence rate}, Chaos, Solitons and Fractals, 41(5), 2319-2325.
  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, 436-444.
  13. [13]  Abta, A., Alaoui, T.H., and Kaddar, A. (2012), {Hopf Bifurcation Analysis In a Delayed SIR Model}, International journal of mathematics and statistics, 11(1).
  14. [14]  Enatsu, Y. and Nakata, Y. (2014), {Stability and bifurcation analysis of epidemic models with saturated incidence rates: an application to a nonmonotone incidence rate}, Mathematical Biosciences and Engineering, 11(4), 785-805.
  15. [15]  Greenhalgh, D. (1997), {Hopf Bifurcation in Epidemic Models with a Latent Period and Nonpermanent Immunity}, Mathematical and Computer Modelling, 25(2), 85-107.
  16. [16]  Hethcote, H.W., Li, Y., and Jing, Z. (1999), {Hopf Bifurcation in Models for Pertussis Epidemiology}, Mathematical and Computer Modelling, 30(11-12), 29-45.
  17. [17]  Jiang, Z. and Wei, J. (2008), {Stability and bifurcation analysis in a delayed SIR model}, Chaos, Solitons and Fractals, 35(3), 609-619.
  18. [18]  Li, M., Wang, X. and Chen, X.(2017), {Analysis of the stability and Hopf bifurcation of a two-delayed SEIR model with general nonlinear incidence rate and saturated recovery rate}, Journal of Multidisciplinary Engineering Science and Technology, 4(10), 8395-8405.
  19. [19]  Sampath Aruna Pradeep, B.G., Ma, M., and Wang, W. (2017), {Stability and Hopf bifurcation analysis of an SEIR model with nonlinear incidence rate and relapse}, Journal of Statistics and Management System, 20(3), 483-497.
  20. [20]  Wang, W. and Ruan, S. (2004), Bifurcation in epidemic model with constant removal rate infectives, Journal of Mathematical Analysis and Applications, 291, 775-793.
  21. [21]  Chitnis, N., Cushing, J.M., and Hyman, J.M. (2006), {Bifurcation analysis of a Mathematical model for Malaria Transmission}, SIAM Journal on Applied Mathematics, 67(1), 24-45.
  22. [22]  Ducrot, A., Sirima, S.B., Some, B., and Zongo, P. (2009), {A mathematical model for malaria involving differential susceptibility, exposedness and infectivity of human host}, Journal of Biological Dynamics, 3(6), 574-598.
  23. [23]  Paula, A., Wysea, P., Bevilacquaa, L., and Rafikovd, M. (2007), {Simulating malaria model for different treatment intensities in a variable environment}, International Journal on Ecological Modelling and Systems Ecology, 206, 322-330.
  24. [24]  Capasso, V. and Serio, G. (1978), {A generalization of Kermack-Mckendrick deterministic epidemic model}, Math. Biosci, 42, 41-61.
  25. [25]  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).
  26. [26]  Kuang, Y. (1993), Delay Differential Equations with Applications in Population Dynamics, Academic Press, San Diego.
  27. [27]  LaSalle, J.P. (1976), The stability of dynamical systems, Regional Conference Series in Applied Mathematics, SIAM, Philadelphia.
  28. [28]  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}, IMA J. Math. App. Med. Biol., 18, 41-52.
  29. [29]  Hale, J.K. and Verduyn Lunel, S.M. (1993), Introduction to Functional Differential Equations, Springer- Verlag, New York.
  30. [30]  Riad, D., Hattaf, K., and Yousfi, N.(2016), {Dynamics of a delayed business cycle model with general investment function}, Chaos, Solitons and Fractals, 85, 110-119.
  31. [31]  Kar, T.K. and Mondal, P.K. (2011), {Global dynamics and bifurcation in delayed SIR epidemic model}, Nonlinear Analysis: Real World Applications, 12(4), 2058-2068.