Periodic Trajectories for HIV Dynamics in a Seasonal Environment With a General Incidence Rate

Main Article Content

Miled El Hajji, Rahmah Mohammed Alnjrani


In this paper, we propose a five-dimensional nonlinear system of differential equations for the human immunodeficiency virus (HIV) including the B-cell functions with a general nonlinear incidence rate. The compartment of infected cells was subdivided into three classes representing the latently infected cells, the short-lived productively infected cells, and the long-lived productively infected cells. The basic reproduction number was established, and the local and global stability of the equilibria of the model were studied. A sensitivity analysis with respect to the model parameters was undertaken. Finally, some numerical simulations are presented to illustrate the theoretical findings.

Article Details


  1. D.C. Douek, M. Roederer, R.A. Koup, Emerging Concepts in the Immunopathogenesis Of AIDS, Annu. Rev. Med. 60 (2009), 471–484.
  2. D. Bernoulli, Essay d’une Nouvelle Analyse de la Mortalité Causée par la Petite Vérole et des Avantages de l’Inoculation pour la Prévenir, Mem. Math. Phys. Acad. R. Sci. Paris, (1766), 1–45.
  3. Y. Nakata, T. Kuniya, Global Dynamics of a Class of SEIRS Epidemic Models in a Periodic Environment, J. Math. Anal. Appl. 363 (2010), 230–237.
  4. N. Bacaër, R. Ouifki, Growth Rate and Basic Reproduction Number for Population Models With a Simple Periodic Factor, Math. Biosci. 210 (2007), 647–658.
  5. N. Bacaër, Approximation of the Basic Reproduction Number R0 for Vector-Borne Diseases With a Periodic Vector Population, Bull. Math. Biol. 69 (2007), 1067–1091.
  6. N. Bacaër, S. Guernaoui, The Epidemic Threshold of Vector-Borne Diseases With Seasonality, J. Math. Biol. 53 (2006), 421–436.
  7. M.E. Hajji, A.H. Albargi, A Mathematical Investigation of an "SVEIR" Epidemic Model for the Measles Transmission, Math. Biosci. Eng. 19 (2022), 2853–2875.
  8. M.E. Hajji, Modelling and Optimal Control for Chikungunya Disease, Theory Biosci. 140 (2020), 27–44.
  9. A.A. Alsolami, M.E. Hajji, Mathematical Analysis of a Bacterial Competition in a Continuous Reactor in the Presence of a Virus, Mathematics. 11 (2023), 883.
  10. A.M. Elaiw, S.F. Alshehaiween, A.D. Hobiny, Global Properties of HIV Dynamics Models Including Impairment of B-Cell Functions, J. Biol. Syst. 28 (2020), 1–25.
  11. S. Alsahafi, S. Woodcock, Exploring HIV Dynamics and an Optimal Control Strategy, Mathematics. 10 (2022), 749.
  12. M.E. Hajji, A. Zaghdani, S. Sayari, Mathematical Analysis and Optimal Control for Chikungunya Virus With Two Routes of Infection With Nonlinear Incidence Rate, Int. J. Biomath. 15 (2021), 2150088.
  13. M. El Hajji, S. Sayari, A. Zaghdani, Mathematical Analysis of an "SIR" Epidemic Model in a Continuous Reactor - Deterministic and Probabilistic Approaches, J. Korean Math. Soc. 58 (2021), 45–67.
  14. D. Xiao, Dynamics and Bifurcations on a Class of Population Model With Seasonal Constant-Yield Harvesting, Discr. Contin. Dyn. Syst. - Series B. 21 (2015), 699–719.
  15. N. Bacaër, M. Gomes, On the Final Size of Epidemics With Seasonality, Bull. Math. Biol. 71 (2009), 1954–1966.
  16. F. Kermack, D. McKendrick, A Contribution to the Mathematical Theory of Epidemics, Proc. R. Soc. Lond. A. 115 (1927), 700–721.
  17. J. Ma, Z. Ma, Epidemic Threshold Conditions for Seasonally Forced SEIR Models, Math. Biosci. Eng. 3 (2006), 161–172.
  18. S. Guerrero-Flores, O. Osuna, C. Vargas-de-Leon, Periodic Solutions for Seasonal SIQRS Models With Nonlinear Infection Terms, Electron. J. Diff. Equ. 2019 (2019), 92.
  19. T. Zhang, Z. Teng, On a Nonautonomous SEIRS Model in Epidemiology, Bull. Math. Biol. 69 (2007), 2537–2559.
  20. W. Wang, X.Q. Zhao, Threshold Dynamics for Compartmental Epidemic Models in Periodic Environments, J. Dyn. Diff. Equ. 20 (2008), 699–717.
  21. M.E. Hajji, D.M. Alshaikh, N.A. Almuallem, Periodic Behaviour of an Epidemic in a Seasonal Environment with Vaccination, Mathematics. 11 (2023), 2350.
  22. O. Diekmann, J. Heesterbeek, On the Definition and the Computation of the Basic Reproduction Ratio R0 in Models for Infectious Diseases in Heterogeneous Populations, J. Math. Biol. 28 (1990), 365–382.
  23. P. van den Driessche, J. Watmough, Reproduction Numbers and Sub-Threshold Endemic Equilibria for Compartmental Models of Disease Transmission, Math. Biosci. 180 (2002), 29–48.
  24. A. Hurwitz, Ueber die Bedingungen, unter Welchen eine Gleichung nur Wurzeln mit Negativen Reellen Theilen Besitzt, Math. Ann. 46 (1895), 273–284.
  25. E.J. Routh, A Treatise on the Stability of a Given State of Motion: Particularly Steady Motion, Macmillan, London, (1877).
  26. J.P. LaSalle, The Stability of Dynamical Systems, SIAM, Philadelphia, (1976).
  27. M.E. Hajji, How Can Inter-Specific Interferences Explain Coexistence or Confirm the Competitive Exclusion Principle in a Chemostat?, Int. J. Biomath. 11 (2018), 1850111.
  28. M.E. Hajji, N. Chorfi, M. Jleli, Mathematical Model for a Membrane Bioreactor Process, Electron. J. Diff. Eqn. 2015 (2015), 315.
  29. M.E. Hajji, N. Chorfi, M. Jleli, Mathematical Modelling and Analysis for a Three-Tiered Microbial Food Web in a Chemostat, Electron. J. Diff. Eqn. 2017 (2017), 255.
  30. F. Zhang, X.Q. Zhao, A Periodic Epidemic Model in a Patchy Environment, J. Math. Anal. Appl. 325 (2007), 496–516.
  31. X. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, (2003).