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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


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:

Global Dynamics of an SIRSI Epidemic Model with Discrete Delay and General Incidence Rate

Journal of Applied Nonlinear Dynamics 10(3) (2021) 547--562 | DOI:10.5890/JAND.2021.09.013

Amine Bernoussi$^1$ , Khalid Hattaf$^2$

$^1$ 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

$^2$ Centre R$mbox{'e}$gional des M$mbox{'e}$tiers de l'Education et de la Formation (CRMEF), 20340 Derb Ghalef, Casablanca, Morocco

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In this paper, we propose the global dynamics of an SIRSI epidemic model with discrete latent period and general nonlinear incidence function. By analyzing the corresponding characteristic equations, the local stability of the endemic equilibrium is established. By using suitable Lyapunov functionals and LaSalle's invariance principle, the global stability of the disease-free equilibrium and the endemic equilibrium are established for the SIRSI epidemic model with discrete latent period.


  1. [1]  Beretta, E. and Takeuchi, Y. (1995), Global stability of an SIR epidemic model with time delays, Journal of Mathematical Biology, 33, 250-260.
  2. [2]  Capasso, V. and Serio, G. (1978), A generalization of Kermack-Mckendrick deterministic epidemic model, Math. Biosci, 42, 41-61.
  3. [3]  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.
  4. [4]  Hale, J. and Verduyn Lunel, S.M. (1993), Introduction to Functional Differential Equations, Springer-Verlag.
  5. [5]  Hethcote, H.M. (2000), The Mathematics of Infectious Disease, SIAM review, 42, 599-653.
  6. [6]  Korobeinikov, A. and Maini, P.K. (2005), Nonlinear incidence and stability of infectious disease models, Math. Med. Biol, 22, 113-128.
  7. [7]  Kuang, Y. (1993), Delay Differential Equations with Applications in Population Dynamics, Academic Press, San Diego.
  8. [8]  LaSalle, J. and Lefschetz, S. (1961), Stability by Liapunovs Direct Method, with Applications, Mathematics in Science and Engineering, 4, Academic Press, New York-London.
  9. [9]  Li, M. and Liu, X. (2014), An SIR Epidemic Model with Time Delay and General Nonlinear Incidence Rate, Hindawi Publishing Corporation, Abstract and Applied Analysis, 2014, Article ID 131257.
  10. [10]  Takeuchi, Y., Ma, W., and Beretta, E. (2000), Global asymptotic properties of a delay SIR epidemic model with finite incubation times, Nonlinear Anal, 42, 931-947.
  11. [11]  Xu, R. and Ma, Z. (2009), Global stability of a SIR epidemic model with nonlinear incidence rate and time delay, Nonl. Anal. RWA., 10, 3175-3189.
  12. [12]  Hu, Z., Bi, P., Ma, W., and 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]  Korobeinikov, A. (2006), Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission, Bull. Math. Biol, 68, 615-626.
  15. [15]  Nakata, Y., Enatsu, Y., and Muroya, Y. (2011), On the global stability of an SIRS epidemic model with distributed delays, Published in Discrete and Continuous Dynamical Systems Supplement, 1119-1128.
  16. [16]  Wang, W. and Ruan, S. (2004), Bifurcation in epidemic model with constant removal rate infectives, Journal of Mathematical Analysis and Applications, 291, 775-793.
  17. [17]  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.
  18. [18]  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.
  19. [19]  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.
  20. [20]  Bernoussi, A., Kaddar, A., and Asserda, S. (2014), Global Stability of a Delayed SIRI Epidemic Model with Nonlinear Incidence, International Journal of Engineering Mathematics, 2014, ID 487589, 1-6.
  21. [21]  Liu, S. Wang, S., and Wang, L. (2011), Global dynamics of delay epidemic models with nonlinear incidence rate and relapse, Nonlinear Analysis: Real World Applications, 12, 119-127.
  22. [22]  van den Driessche, P., Wang, L., and Zou, X. (2007), Modeling diseases with latency and relapse, Math. Biosci. Eng, 4, 205-219.
  23. [23]  VanLandingham, K.E., Marsteller, H.B., Ross, G.W., and Hayden, F.G. (1988), Relapse of herpes simplex encephalitis after conventional acyclovir therapy, JAMA, 259, 1051-1053.
  24. [24]  Wang, J., Pang, J., and Liua, X. (2014), Modelling diseases with relapse and nonlinear incidence of infection: a multi-group epidemic model, Journal of Biological Dynamics, 8(1), 99-116.
  25. [25]  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.
  26. [26]  Gabriela, M., Gomes, M., White, L.J., and Medley, G.F. (2005), The reinfection threshold, Journal of Theoretical Biology, 236, 111-113.
  27. [27]  Jiang, Z. and Wei, J. (2008), Stability and bifurcation analysis in a delayed SIR model, Chaos, Solitons and Fractals, 35(3), 609-619.
  28. [28]  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.
  29. [29]  Zhou, Y. and Liu, H. (2003), Stability of periodic solutions for an SIS model with pulse vaccination, Mathematical and Computer Modelling, 38, 299-308.
  30. [30]  Chen, L.S. and Chen, J. (1993), Nonlinear biologicl dynamics system, Scientific Press, China.
  31. [31]  Kaddar, A. (2010), Stability analysis in a delayed SIR epidemic model with a saturated incidence rate, Nonlinear Analysis: Modelling and Control, 15(3), 299-306.
  32. [32]  Wei, C. and Chen, L. (2008), A delayed epidemic model with pulse vaccination, Discrete Dynamics in Nature and Society, 2008, Article ID 746-951.
  33. [33]  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, 2008, Article ID 66153.
  34. [34]  de Jong, M.C.M., Diekmann, O., and Heesterbeek, H. (1995), How does transmission of infection depend on population size? In: Epidemic models: their structure and relation to data, Mollison D. (Ed.), Cambridge University Press, Cambridge, 84-94.
  35. [35]  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.