Stability Properties and Hopf Bifurcation of a Delayed HIV Dynamics Model with Saturation Functional Response, Absorption Effect and Cure Rate

Article Sidebar

PDF
DOI: https://doi.org/10.28924/2291-8639-23-2025-92
Published: Apr 14, 2025
Cite as:
N.S. Rathnayaka, J. K. Wijerathna, B.G.S.A. Pradeep, Stability Properties and Hopf Bifurcation of a Delayed HIV Dynamics Model with Saturation Functional Response, Absorption Effect and Cure Rate, Int. J. Anal. Appl., 23 (2025), 92.

Main Article Content

N.S. Rathnayaka, J. K. Wijerathna, B.G.S.A. Pradeep

Abstract

In this paper, stability properties of an HIV infection model with saturation functional response, logistic proliferation term of susceptible CD4+T cells, cure rate of infected CD4+T cell, virus absorption effect, intracellular delay, and maturation delay are investigated. According to our mathematical analysis, the basic reproduction number R0 of the model completely determines its stability features. Using the characteristic equation of the model, we establish that the infection-free equilibrium point and the infected equilibrium point are locally asymptotically stable when R0 ≤ 1 and R0 > 1, respectively. By means of appropriate Lyapunov functionals and LaSalle’s invariance principle for delay models, if R0 ≤ 1, we study the global asymmetric stability of the infection-free equilibrium point of the model. When R0 > 1, we establish the occurrence of Hopf bifurcations and determine conditions for the permanence of the model. Finally, numerical simulations are also presented to confirm the analytical results.

Article Details

References

  1. E. Avila-Vales, N. Chan-Chí, G.E. García-Almeida, C. Vargas-De-León, Stability and Hopf Bifurcation in a Delayed Viral Infection Model with Mitosis Transmission, Appl. Math. Comput. 259 (2015), 293–312. https://doi.org/10.1016/j.amc.2015.02.053.
  2. E. Avila-Vales, A. Canul-Pech, G.E. García-Almeida, Á.G.C. Pérez, Global Stability of a Delay Virus Dynamics Model with Mitotic Transmission and Cure Rate, in: K. Hattaf, H. Dutta (Eds.), Mathematical Modelling and Analysis of Infectious Diseases, Springer, Cham, 2020: pp. 83–126. https://doi.org/10.1007/978-3-030-49896-2_4.
  3. N. Akbari, R. Asheghi, M. Nasirian, Stability and Dynamic of HIV-1 Mathematical Model with Logistic Target Cell Growth, Treatment Rate, Cure Rate and Cell-to-Cell Spread, Taiwan. J. Math. 26 (2022), 411-441. https://doi.org/10.11650/tjm/211102.
  4. J.R. Beddington, Mutual Interference Between Parasites or Predators and Its Effect on Searching Efficiency, J. Animal Ecol. 44 (1975), 331-340. https://doi.org/10.2307/3866.
  5. S. Butler, D. Kirschner, S. Lenhart, Optimal Control of Chemotherapy Affecting the Infectivity of HIV, in: O. Arino, D. Axelrod, M. Kimmel, M. Langlais, (eds.) Advances in Mathematical Population Dynamics: Molecules, Cells, Man, World Scientific, Singapore, 1997, pp. 104–120.
  6. R.V. Culshaw, S. Ruan, A Delay-Differential Equation Model of HIV Infection of CD4+ T-Cells, Math. Biosci. 165 (2000), 27–39. https://doi.org/10.1016/S0025-5564(00)00006-7.
  7. D.L. DeAngelis, R.A. Goldstein, R.V. O’Neill, A Model for Tropic Interaction, Ecology 56 (1975), 881–892. https://doi.org/10.2307/1936298.
  8. P. Essunger, A.S. Perelson, Modeling HIV Infection of CD4+ T-Cell Subpopulations, J. Theor. Biol. 170 (1994), 367–391. https://doi.org/10.1006/jtbi.1994.1199.
  9. K.R. Fister, S. Lenhart, J.S. McNally, Optimizing Chemotherapy in an HIV Model, Electron. J. Differ. Equ. 1998 (1998), 32.
  10. S. Guo, W. Ma, Remarks on a Variant of Lyapunov-LaSalle Theorem, Math. Biosci. Eng. 16 (2019), 1056–1066. https://doi.org/10.3934/mbe.2019050.
  11. S. Guo, W. Ma, X.-Q. Zhao, Global Dynamics of a Time-Delayed Microorganism Flocculation Model with Saturated Functional Responses, J. Dyn. Differ. Equ. 30 (2018), 1247–1271. https://doi.org/10.1007/s10884-017-9605-3.
  12. G. Huang, W. Ma, Y. Takeuchi, Global Properties for Virus Dynamics Model with Beddington–DeAngelis Functional Response, Appl. Math. Lett. 22 (2009), 1690–1693. https://doi.org/10.1016/j.aml.2009.06.004.
  13. K. Hattaf, N. Yousfi, A. Tridane, A Delay Virus Dynamics Model with General Incidence Rate, Differ. Equ. Dyn. Syst. 22 (2014), 181–190. https://doi.org/10.1007/s12591-013-0167-5.
  14. K. Hattaf, N. Yousfi, Global Stability of a Virus Dynamics Model with Cure Rate and Absorption, J. Egypt. Math. Soc. 22 (2014), 386–389. https://doi.org/10.1016/j.joems.2013.12.010.
  15. D.D. Ho, A.U. Neumann, A.S. Perelson, W. Chen, J.M. Leonard, M. Markowitz, Rapid Turnover of Plasma Virions and CD4 Lymphocytes in HIV-1 Infection, Nature 373 (1995), 123–126. https://doi.org/10.1038/373123a0.
  16. J.K. Hale, P. Waltman, Persistence in Infinite-Dimensional Systems, SIAM J. Math. Anal. 20 (1989), 388–395. https://doi.org/10.1137/0520025.
  17. J. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, Rhode Island, 2010. https://doi.org/10.1090/surv/025.
  18. A. Korobeinikov, Global Properties of Basic Virus Dynamics Models, Bull. Math. Biol. 66 (2004), 879–883. https://doi.org/10.1016/j.bulm.2004.02.001.
  19. Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, (1993).
  20. L.M. Cai, B.Z. Guo, X.Z. Li, Global Stability for a Delayed HIV-1 Infection Model with Nonlinear Incidence of Infection, Appl. Math. Comput. 219 (2012), 617–623. https://doi.org/10.1016/j.amc.2012.06.051.
  21. D. Li, W. Ma, Asymptotic Properties of a HIV-1 Infection Model with Time Delay, J. Math. Anal. Appl. 335 (2007), 683–691. https://doi.org/10.1016/j.jmaa.2007.02.006.
  22. C. Connell McCluskey, Y. Yang, Global Stability of a Diffusive Virus Dynamics Model with General Incidence Function and Time Delay, Nonlinear Anal.: Real World Appl. 25 (2015), 64–78. https://doi.org/10.1016/j.nonrwa.2015.03.002.
  23. M.A. Nowak, C.R.M. Bangham, Population Dynamics of Immune Responses to Persistent Viruses, Science 272 (1996), 74–79. https://doi.org/10.1126/science.272.5258.74.
  24. P.W. Nelson, A.S. Perelson, Mathematical Analysis of Delay Differential Equation Models of HIV-1 Infection, Math. Biosci. 179 (2002), 73–94. https://doi.org/10.1016/S0025-5564(02)00099-8.
  25. B.G.S.A. Pradeep, W. Ma, Stability Properties of a Delayed HIV Dynamics Model with Beddington-DeAngelis Functional Response and Absorption Effect, Dyn. Contin. Discrete Impuls. Syst. A: Math. Anal. 21 (2014), 421–434.
  26. A.S. Perelson, A.U. Neumann, M. Markowitz, J.M. Leonard, D.D. Ho, HIV-1 Dynamics in Vivo: Virion Clearance Rate, Infected Cell Life-Span, and Viral Generation Time, Science 271 (1996), 1582–1586. https://doi.org/10.1126/science.271.5255.1582.
  27. B.S.A. Pradeep, W. Ma, S. Guo, Stability Properties of a Delayed HIV Model with Nonlinear Functional Response and Absorption Effect, J. Nat. Sci. Found. Sri Lanka 43 (2015), 235. https://doi.org/10.4038/jnsfsr.v43i3.7953.
  28. A.S. Perelson, D.E. Kirschner, R. De Boer, Dynamics of HIV Infection of CD4+ T Cells, Math. Biosci. 114 (1993), 81–125. https://doi.org/10.1016/0025-5564(93)90043-A.
  29. N.S. Rathnayaka, J.K. Wijerathna, B.G.S.A. Pradeep, Global Stability of a Delayed HIV-1 Dynamics Model With Saturation Response With Cure Rate, Absorption Effect and Two Time Delays, Commun. Math. Biol. Neurosci. 2023 (2023), 16. https://doi.org/10.28919/cmbn/7877.
  30. R.R. Regoes, D. Ebert, S. Bonhoeffer, Dose–Dependent Infection Rates of Parasites Produce the Allee Effect in Epidemiology, Proc. R. Soc. London. Ser. B: Biol. Sci. 269 (2002), 271–279. https://doi.org/10.1098/rspb.2001.1816.
  31. N.S. Rathnayaka, J.K. Wijerathna, B.G.S.A. Pradeep, P.D.N. Silva, Stability of a Delayed HIV-1 Dynamics Model with Beddington-DeAngelis Functional Response and Absorption Effect With Two Delays, Commun. Math. Biol. Neurosci. 2022 (2022), 105. https://doi.org/10.28919/cmbn/7702.
  32. L. Rong, M.A. Gilchrist, Z. Feng, A.S. Perelson, Modeling Within-Host HIV-1 Dynamics and the Evolution of Drug Resistance: Trade-Offs between Viral Enzyme Function and Drug Susceptibility, J. Theor. Biol. 247 (2007), 804–818. https://doi.org/10.1016/j.jtbi.2007.04.014.
  33. A. Rezounenko, Continuous Solutions to a Viral Infection Model with General Incidence Rate, Discrete StateDependent Delay, CTL and Antibody Immune Responses, Electron. J. Qual. Theory Differ. Equ. 2016 (2016), 79. https://doi.org/10.14232/ejqtde.2016.1.79.
  34. L. Rong, M.A. Gilchrist, Z. Feng, A.S. Perelson, Modeling Within-Host HIV-1 Dynamics and the Evolution of Drug Resistance: Trade-Offs between Viral Enzyme Function and Drug Susceptibility, J. Theor. Biol. 247 (2007), 804–818. https://doi.org/10.1016/j.jtbi.2007.04.014.
  35. P.K. Srivastava, P. Chandra, Modeling the Dynamics of HIV and T Cells during Primary Infection, Nonlinear Anal.: Real World Appl. 11 (2010), 612–618. https://doi.org/10.1016/j.nonrwa.2008.10.037.
  36. Q. Sun, L. Min, Dynamics Analysis and Simulation of a Modified HIV Infection Model with a Saturated Infection Rate, Comput. Math. Methods Med. 2014 (2014), 145162. https://doi.org/10.1155/2014/145162.
  37. N. Sachsenberg, A.S. Perelson, S. Yerly, G.A. Schockmel, D. Leduc, B. Hirschel, L. Perrin, Turnover of CD4+ and CD8+ T Lymphocytes in HIV-1 Infection as Measured by Ki-67 Antigen, J. Exp. Med. 187 (1998), 1295–1303. https://doi.org/10.1084/jem.187.8.1295.
  38. H.L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Society, 1995.
  39. L.F. Shampine, S. Thompson, Solving DDEs in Matlab, Appl. Numer. Math. 37 (2001), 441–458. https://doi.org/10.1016/S0168-9274(00)00055-6.
  40. C. Vargas-De-León, N.C. Chí, E.Á. Vales, Global Analysis of Virus Dynamics Model with Logistic Mitosis, Cure Rate and Delay in Virus Production, Math. Methods Appl. Sci. 38 (2015), 646–664. https://doi.org/10.1002/mma.3096.
  41. 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. https://doi.org/10.1016/S0025-5564(02)00108-6.
  42. R.A. Weiss, How Does HIV Cause AIDS?, Science 260 (1993), 1273–1279. https://doi.org/10.1126/science.8493571.
  43. K. Wang, A. Fan, A. Torres, Global Properties of an Improved Hepatitis B Virus Model, Nonlinear Anal.: Real World Appl. 11 (2010), 3131–3138. https://doi.org/10.1016/j.nonrwa.2009.11.008.
  44. J. Wang, C. Qin, Y. Chen, X. Wang, Hopf Bifurcation in a CTL-Inclusive HIV-1 Infection Model with Two Time Delays, Math. Biosci. Eng. 16 (2019), 2587–2612. https://doi.org/10.3934/mbe.2019130.
  45. H.M. Wei, X.Z. Li, M. Martcheva, An Epidemic Model of a Vector-Borne Disease with Direct Transmission and Time Delay, J. Math. Anal. Appl. 342 (2008), 895–908. https://doi.org/10.1016/j.jmaa.2007.12.058.
  46. R. Xu, Global Stability of an HIV-1 Infection Model with Saturation Infection and Intracellular Delay, J. Math. Anal. Appl. 375 (2011), 75–81. https://doi.org/10.1016/j.jmaa.2010.08.055.
  47. R. Xu, Global Dynamics of a Delayed Hiv-1 Infection Model With Absorption and Saturation Infection, Int. J. Biomath. 05 (2012), 1260012. https://doi.org/10.1142/S1793524512600121.
  48. H. Xiang, L.X. Feng, H.F. Huo, Stability of the Virus Dynamics Model with Beddington–DeAngelis Functional Response and Delays, Appl. Math. Model. 37 (2013), 5414–5423. https://doi.org/10.1016/j.apm.2012.10.033.
  49. Y. Yu, J.J. Nieto, A. Torres, K. Wang, A Viral Infection Model with a Nonlinear Infection Rate, Bound. Value Probl. 2009 (2009), 958016. https://doi.org/10.1155/2009/958016.
  50. J.A. Zack, S.J. Arrigo, S.R. Weitsman, et al. HIV-1 Entry into Quiescent Primary Lymphocytes: Molecular Analysis Reveals a Labile, Latent Viral Structure, Cell 61 (1990), 213–222. https://doi.org/10.1016/0092-8674(90)90802-L.
  51. J.A. Zack, A.M. Haislip, P. Krogstad, I.S. Chen, Incompletely Reverse-Transcribed Human Immunodeficiency Virus Type 1 Genomes in Quiescent Cells Can Function as Intermediates in the Retroviral Life Cycle, J. Virol. 66 (1992), 1717–1725. https://doi.org/10.1128/jvi.66.3.1717-1725.1992.
  52. X. Zhou, X. Song, X. Shi, A Differential Equation Model of HIV Infection of CD4+ T-Cells with Cure Rate, J. Math. Anal. Appl. 342 (2008), 1342–1355. https://doi.org/10.1016/j.jmaa.2008.01.008.
  53. T. Zhang, X. Meng, T. Zhang, Global Dynamics of a Virus Dynamical Model with Cell-to-Cell Transmission and Cure Rate, Comput. Math. Methods Med. 2015 (2015), 758362. https://doi.org/10.1155/2015/758362.