[ 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

Stability Analysis of an Seirs Epidemic Model with Relapse, Immune and General Incidence Rates

Journal of Applied Nonlinear Dynamics 11(1) (2022) 217--231 | DOI:10.5890/JAND.2022.03.013

Amine Bernoussi$^1$ , Soufiane Elkhaiar$^2$, Chakib Jerry$^3$

$^1$ 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

$^2$ Department of Mathematics, Faculty of Applied Sciences, PO Box 6146, Ait Melloul, Morocco

$^3$ Moulay Ismail University of Meknes, Team O.M.E.G.A, Faculty of Law, Economics and Socials Sciences, B.P. 3102 Toulal, Meknes, Morocco

Download Full Text PDF

 

Abstract

This paper has the goal to broaden the incidence rate of an SEIRS epidemic model to a wide range of monotonic, concave incidence rates and some non-monotonic or concave cases. These incidence functions could reflect media education or psychological effect or mass action. The model takes into account relapse, recovery and immunity rates but without disease-induced death one. Applying the novel geometric approach we establish the global stability of the SEIRS model. Our analytical results reveal that the basic reproduction number completely determines the global stability of equilibria. Our conclusions are applied to two special incidence functions reflecting media and mass action.

References

  1. [1]  Kermack, W.O. and McKendrick, A.G. (1927), A contribution to the mathematical theory of epidemics, Proc. R Soc. Lond. Ser. A Math. Phys. Eng. Sci., 115(722), 700-721.
  2. [2]  Alexander, M.E. and Moghadas, S.M. (2004), Periodicity in an epidemic model with a generalized nonlinear incidence, Math. Biosci., 189, 75-96.
  3. [3]  Anderson, R.M. and May, R.M. (1979), population biology of infectious disease, I. Nature, 180, 316-367.
  4. [4]  Capasso, V. and Serio, G. (1978), A generalization of the Kermack-McKendrick deterministic epidemic model, Math. Biosci., 42(1), 43-61.
  5. [5]  Cui, J., Sun, Y., and Zhu, H. (2008), The impact of media on the control of infectious diseases, J. Dyn. Differ. Equ., 20(1), 31-53.
  6. [6]  Hethcote, H.W. (2000), The mathematics of infectious diseases, SIAM Rev., 42(4), 599-653.
  7. [7]  Kermack, W.O. and McKendrick, A.G. (1991), Contributions to the mathematical theory of epidemics-II, The problem of endemicity. Bull. Math. Biol., 53(1), 57-87.
  8. [8]  Liu, L., Wang, J., and Liu, X. (2015),Global stability of an SEIR epidemic model with age-dependent latency and relapse, Nonlinear Anal. Real World Appl., 24(1), 18-35.
  9. [9]  Tchuenche, J.M., Dube, N., Bhunu, C.P., Smith, R.J., and Bauch, C.T. (2011), The impact of media coverage on the transmission dynamics of human influenza, BMC Public Health 11(Suppl 1), S5. DOI:10. 1186/1471-2458-11-S1-S5.
  10. [10]  Liu, W.M., Hethcote, H.W., and Levin, S.A. (1987), Dynamical behavior of epidemiological models with nonlinear incidence rates, J. Math. Biol., 25, 359-380.
  11. [11]  Hethcote, H.W. and Yorke, J.A. (1984), Gonorrhea transmission dynamics and control, Lecture Notes in Biomath, vol. 56. Springer, Berlin.
  12. [12]  Hu, Z., Bi, P., Ma, W., and S. Ruan, S. (2011), Bifurcations of an SIRS epidemic model with nonlinear incidence rate, Discret. Contin. Dyn. Syst. Ser. B, 15(1), 93-112.
  13. [13]  Khan, M.A., Badshah, Q., Islam, S., Khan, I., Shafie, S., and Khan, S.A. (2015), Global dynamics of SEIRS epidemic model with non-linear generalized incidences and preventive vaccination, Adv. Differ. Equ., 88.
  14. [14]  Wang, L., Zhang, X., and Liu, Z. (2018),An SEIR Epidemic Model with Relapse and General Nonlinear Incidence Rate with Application to Media Impact, Qual. Theory Dyn. Syst., 17, 309-329.
  15. [15]  Liu, Y. and Cui, J. (2008), The impact of media coverage on the dynamics of infectious disease, Int. J. Biomath., 1(01), 65-74.
  16. [16]  Nyabadza, F., Mukandavire, Z., and Hove-Musekwa, S.D. (2011), Modelling the HIV/AIDS epidemic trends in South Africa: insights from a simple mathematical model, Nonlinear Anal. Real World Appl., 12(4), 2091-2104.
  17. [17]  Kar, T.K. and Batabyal, A. (2010), Modeling and analysis of an epidemic model with non-monotonic incidence rate under treatment, J. Math. Res., 2(1), 103-115.
  18. [18]  Xiao, D. and Ruan, S. (2007),Global analysis of an epidemicmodel with nonmonotoneinc incidence rate, Math. Biosci., 208, 419-429.
  19. [19]  Lahrouz, A., Omari, A.L., and Kiouach, D. (2011), Global analysis of a deterministic and stochastic nonlinear SIRS epidemic model, In: Nonlinear Analysis: Modelling and Control, 16(1), 59-76.
  20. [20]  Korobeinikov, A. (2006),Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission, In: Bulletin of Mathematical biology, 68(3), 615-626.
  21. [21]  Kuang, Y. (1993), Delay Differential Equations with Applications in Population Dynamics, Academic Press, San Diego.
  22. [22]  LaSalle, J.P. (1976), The stability of dynamical systems, Regional Conference Series in Applied Mathematics, SIAM, Philadelphia.
  23. [23]  Smith, R.A. (1986),Some applications of Hausdorff dimension inequalities for ordinary differential equations, Proc. Roy. Soc. Edinburgh Sect. A, 104, 235-259.
  24. [24]  Li, M.Y. and Muldowney, J.S. (1996),A Geometric Approch to Global Stability Problems, SIAM. J. MATH. ANAL., 27(4), pp. 1070-1083.
  25. [25]  Freedman, H.I., Tang, M.X., and Ruan, S.G. (1994), Uniform persistence and flows near a closed positively invariant set, J. Dynam. Differential Equations, 6, pp. 583-600.
  26. [26]  Martin Jr., R.H. (1974),Logarithmic norms and projections applied to linear differential systems, J. Math. Anal. Appl., 45, pp. 432-454.
  27. [27]  Coppel, W.A. (1965), Stability and Asymptotic Behavior of Differential Equations, Heath Boston, (1965).
  28. [28]  Anderson, R.M. and May, R.M. (1978), Regulation and stability of host-parasite population interactions: I Regulatory processes, J. Anim. Ecol., 47(1), 219-267.
  29. [29]  Jiang, Z. and Wei, J. (2008), Stability and bifurcation analysis in a delayed SIR model, Chaos Solitons Fractals, 35, 609-619.
  30. [30]  Zhang, F., Li, Z.Z., and Zhang, F. (2008),lobal stability of an SIR epidemic model with constant infectious period, GAppl. Math. Comput., 199, 285-291.
  31. [31]  Kaddar, A. (2010), Stability analysis in a delayed SIR epidemic model with a saturated incidence rate, Nonlinear Anal. Model. Control, 15(3), 299-306.
  32. [32]  Wei, C. and Chen, L. (2008),A delayed epidemic model with pulse vaccination, Discret. Dyn. Nat. Soc., Article ID 746-951.

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