Global dynamics of delayed HCV/HBV co-infection model with antibody immunity

Volume 39, Issue 2, pp 192--232 https://dx.doi.org/10.22436/jmcs.039.02.03

Publication Date: March 29, 2025 Submission Date: October 07, 2024 Revision Date: January 08, 2025 Accteptance Date: January 23, 2025

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 387 Downloads
• 1141 Views

Authors

A. M. Elaiw - Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia. A. D. Al Agha - Department of Mathematical Science, College of Engineering, University of Business and Technology, Jeddah 21361, Saudi Arabia. Gh. Alsaadi - Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia. A. D. Hobiny - Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia.

Abstract

Hepatitis B virus (HBV) and hepatitis C virus (HCV) infect liver cells (hepatocytes) and are responsible for most cases of chronic liver disease. HCV/HBV co-infected patients have a greater risk of severe liver disease, cirrhosis and liver cancer than mono-infected patients. In this paper, we develop a within-host HCV/HBV co-infection model with distributed delays and antibody immunity. The model involves nine delay differential equations (DDEs) that reflect the interactions between uninfected cells, latent HCV-infected cells, active HCV-infected cells, latent HBV-infected cells, active HBV-infected cells, free HCV particles, free HBV particles, HCV-specific antibodies, and HBV-specific antibodies. We test the nonnegativity and boundedness of the solutions. We compute all steady states and affirm the global stability. We confirm the results through numerical simulations. We found that time delays decrease the basic reproduction numbers leading to the stability of uninfected steady state. Also, fewer treatment efficacies will be required when employing a model with time delays. Moreover, increasing the stimulation rate of antibody immunity increases the densities of uninfected cells during co-infection.

Share and Cite

Keywords

• HCV
• HBV
• co-infection
• latent
• distributed
• delay
• antibody
• stability

MSC

•  34K20
•  34K25
•  92D30

References

• [1] A. D. Al Agha, A. M. Elaiw, Global dynamics of SARS-CoV-2/malaria model with antibody immune response, Math. Biosci. Eng., 19 (2022), 8380–8410
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [2] H. Alrabaiah, M. U. Rahman, I. Mahariq, S. Bushnaq, M. Arfan, Fractional order analysis of HBV and HCV co-infection under ABC derivative, Fractals, 30 (2022), 15 pages
  • View Article
  • Google Scholar
  • MATH


• [3] M. Bachraoui, K. Hattaf, N. Yousfi, Qualitative analysis of a fractional model for HBV infection with capsids and adaptive immunity, Int. J. Dyn. Syst. Differ. Equ., 12 (2022), 146–162
  • View Article
  • Google Scholar
  • MATH


• [4] H. Chen,W.Wang, R. Fu, J. Luo, Global analysis of a mathematical model on malaria with competitive strains and immune responses, Appl. Math. Comput., 259 (2015), 132–152
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [5] S. M. Ciupe, R. M. Ribeiro, A. S. Perelson, Antibody responses during hepatitis B viral infection, PLoS Comput. Biol., 10 (2014), 16 pages
  • View Article
  • Google Scholar


• [6] H. Dahari, A. Lo, R. M. Ribeiro, A. S. Perelson, Modeling hepatitis C virus dynamics: liver regeneration and critical drug efficacy, J. Theor. Biol., 247 (2007), 371–381
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [7] J. Danane, K. Allali, Z. Hammouch, Mathematical analysis of a fractional differential model of HBV infection with antibody immune response, Chaos Solitons Fractals, 136 (2020), 9 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [8] A. M. Elaiw, A. D. Al Agha, Global dynamics of a general diffusive HBV infection model with capsids and adaptive immune response, Adv. Differ. Equ., 2019 (2019), 31 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [9] A. M. Elaiw, A. D. Al Agha, Analysis of the in-host dynamics of tuberculosis and SARS-CoV-2 co-infection, Mathematics, 11 (2023), 1–24
  • View Article
  • Google Scholar


• [10] A. M. Elaiw, A. D. Al Agha, S. A. Azoz, E. Ramadan, Global analysis of within-host SARS-CoV-2/HIV co-infection model with latency, Eur. Phys. J. Plus, 137 (2022), 1–22
  • View Article
  • Google Scholar


• [11] A. M. Elaiw, A. D. Al Agha, G. Alsaadi, A. D. Hobiny, Global analysis of HCV/HBV codynamics model with antibody immunity, Eur. Phys. J. Plus, 139 (2024),
  • View Article
  • Google Scholar


• [12] A. M. Elaiw, N. H. AlShamrani, Analysis of a within-host HIV/HTLV-I co-infection model with immunity, Virus Res., 295 (2021),
  • View Article
  • Google Scholar


• [13] S. Foko, Dynamical analysis of a general delayed HBV infection model with capsids and adaptive immune response in presence of exposed infected hepatocytes, J. Math. Biol., 88 (2024), 55 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [14] Y. Geng, J. Xu, J. Hou, Discretization and dynamic consistency of a delayed and diffusive viral infection model, Appl. Math. Comput., 316 (2018), 282–295
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [15] S. A. Gourley, Y. Kuang, J. D. Nagy, Dynamics of a delay differential equation model of hepatitis B virus infection, J. Biol. Dyn., 2 (2008), 140–153
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [16] J. K. Hale, S. M. Verduyn Lunel, Introduction to functional-differential equations, Springer-Verlag, New York (1993)
  • View Article
  • MathSciNet
  • Google Scholar


• [17] S. Harroudi, A. Meskaf, K. Allali, Modelling the adaptive immune response in HBV infection model with HBV DNAcontaining capsids, Differ. Equ. Dyn. Syst., 31 (2023), 371–393
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [18] K. Hattaf, A new class of generalized fractal and fractal-fractional derivatives with non-singular kernels, Fractals Fract., 7 (2023), 16 pages
  • View Article
  • Google Scholar


• [19] K. Hattaf, A new mixed fractional derivative with applications in computational biology, Computation, 12 (2024), 17 pages
  • View Article
  • Google Scholar


• [20] , Hepatitis, World Health Organization (WHO), (2022),
  • View Article


• [21] G. Huang, Y. Takeuchi, W. Ma, Lyapunov functionals for delay differential equations model of viral infections, SIAM J. Appl. Math., 70 (2010), 2693–2708
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [22] R. T. Ibarra, Treatment of HBV-HCV co-infection, Ann. Hepatol., 5 (2006), S49–S52
  • View Article
  • Google Scholar


• [23] M. N. Jan, G. Zaman, N. Ali, I. Ahmad, Z. Shah, Optimal control application to the epidemiology of HBV and HCV co-infection, Int. J. Biomath., 15 (2022), 36 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [24] H. K. Khalil, Nonlinear Systems, Prentice Hall, Upper Saddle River (2002)
  • View Article


• [25] K. Kitagawa, T. Kuniya, S. Nakaoka, Y. Asai, K. Watashi, S. Iwami, Mathematical analysis of a transformed ODE from a PDE multiscale model of hepatitis C virus infection, Bull. Math. Biol., 81 (2019), 1427–1441
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [26] D. Konstantinou, M. Deutsch, The spectrum of HBV/HCV co-infection: epidemiology, clinical characteristics, viral interactions and management, Ann. Gastroenterol., 28 (2015), 221–228
  • View Article
  • Google Scholar


• [27] A. Korobeinikov, Global properties of basic virus dynamics models, Bull. Math. Biol., 66 (2004), 879–883
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [28] S. R. Lewin, R. M. Ribeiro, T. Walters, G. K. Lau, S. Bowden, S. Locarnini, A. S. Perelson, Analysis of hepatitis B viral load decline under potent therapy: complex decay profiles observed, Hepatology, 34 (2001), 1012–1020
  • View Article
  • Google Scholar


• [29] Z. Liu, J. Hou, Hepatitis B virus (HBV) and hepatitis C virus (HCV) dual infection, Int. J. Med. Sci., 3 (2006), 57–62
  • View Article
  • Google Scholar


• [30] K. Manna, Global properties of a HBV infection model with HBV DNA-containing capsids and CTL immune response, Int. J. Appl. Comput. Math., 3 (2017), 2323–2338
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [31] K. Manna, S. P. Chakrabarty, Chronic hepatitis B infection and HBV DNA-containing capsids: modeling and analysis, Commun. Nonlinear Sci. Numer. Simul., 22 (2015), 383–395
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [32] M. G. Mavilia, G. Y. Wu, HBV-HCV co-infection: viral interactions, management, and viral reactivation, J. Clin. Transl. Hepatol., 6 (2018), 296–305
  • View Article
  • Google Scholar


• [33] Q. Maqsood, A. Sumrin, M. Iqbal, S. Younas, N. Hussain, M. Mahnoor, A. Wajid, Hepatitis C virus/hepatitis B virus co-infection: current prospectives, Antiviral Ther., 28 (2023), 1–18
  • View Article
  • Google Scholar


• [34] K. G. Mekonen, S. F. Balcha, L. L. Obsu, A. Hassen, Mathematical modeling and analysis of TB and COVID-19 co-infection, J. Appl. Math., 2022 (2022), 20 pages
  • View Article
  • Google Scholar
  • MATH


• [35] A. Meskaf, K. Allali, Y. Tabit, Optimal control of a delayed hepatitis B viral infection model with cytotoxic T-lymphocyte and antibody responses, Int. J. Dyn. Control, 5 (2017), 893–902
  • View Article
  • MathSciNet
  • Google Scholar


• [36] A. U. Neumann, N. P. Lam, H. Dahari, D. R. Gretch, T. E. Wiley, T. J. Layden, A. S. Perelson, Hepatitis C viral dynamics in vivo and the antiviral efficacy of interferon-alpha therapy, Science, 282 (1998), 103–107
  • View Article
  • Google Scholar


• [37] M. A. Nowak, C. R. M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74–79
  • View Article
  • Google Scholar


• [38] M. A. Nowak, S. Bonhoeffer, A. M. Hill, R. Boehme, H. C. Thomas, H. McDade, Viral dynamics in hepatitis B infection, Proc. Natl. Acad. Sci. U.S.A., 93 (1996), 4398–4402
  • View Article
  • Google Scholar


• [39] M. A. Nowak, R. M. May, Virus Dynamics: Mathematical Principles of Immunology and Virology, Oxford University Press, Oxford (2000)
  • View Article
  • Google Scholar
  • MATH


• [40] S. Pan, S. P. Chakrabarty, Threshold dynamics of HCV model with cell-to-cell transmission and a non-cytolytic cure in the presence of humoral immunity, Commun. Nonlinear Sci. Numer. Simul., 61 (2018), 180–197
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [41] S. Pan, S. P. Chakrabarty, Analysis of a reaction-diffusion HCV model with general cell-to-cell incidence function incorporating B cell activation and cure rate, Math. Comput. Simul., 193 (2022), 431–450
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [42] L. Pinky, H. M. Dobrovolny, SARS-CoV-2 co-infections: could influenza and the common cold be beneficial?, J. Med. Virol., (2020), 2623–2630
  • View Article
  • Google Scholar


• [43] R. Qesmi, S. ElSaadany, J. M. Heffernan, J. Wu, A hepatitis B and C virus model with age since infection that exhibits backward bifurcation, SIAM J. Appl. Math., 71 (2011), 1509–1530
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [44] G. Raimondo, G. Cacciamo, C. Saitta, Hepatitis B virus and hepatitis C virus co-infection: additive players in chronic liver disease?, Ann. Hepatol., 4 (2005), 100–106
  • View Article
  • Google Scholar


• [45] J. Reyes-Silveyra, A. R. Mikler, Modeling immune response and its effect on infectious disease outbreak dynamics, Theor. Biol. Med. Model., 13 (2016), 1–21
  • View Article
  • Google Scholar


• [46] F. A. Rihan, H. J. Alsakaji, Analysis of a stochastic HBV infection model with delayed immune response, Math. Biosci. Eng., 18 (2021), 5194–5220
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [47] F. A. Rihan, A. A. Arafa, R. Rakkiyappan, C. Rajivganthi, Y. Xu, Fractional-order delay differential equations for the dynamics of hepatitis C virus infection with IFN-α treatment, Alex. Eng. J., 60 (2021), 4761–4774
  • View Article
  • Google Scholar


• [48] L. Rong, A. S. Perelson, Mathematical analysis of multiscale models for hepatitis C virus dynamics under therapy with direct-acting antiviral agents, Math. Biosci., 245 (2013), 22–30
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [49] M. Sadki, S. Harroudi, K. Allali, Local and global stability of an HCV viral dynamics model with two routes of infection and adaptive immunity, Comput. Methods Biomech. Biomed. Eng., 27 (2024), 1510–1537
  • View Article
  • Google Scholar


• [50] W.-Y. Shen, Y.-M. Chu, M. ur. Rahman, I. Mahariq, A. Zeb, Mathematical analysis of HBV and HCV co-infection model under nonsingular fractional order derivative, Results Phys., 28 (2021), 1–7
  • View Article
  • Google Scholar


• [51] Y.-F. Shih, C.-J. Liu, Hepatitis C virus and hepatitis B virus co-infection, Viruses, 12 (2020), 1–11
  • View Article
  • Google Scholar


• [52] H. L. Smith, P. Waltman, The theory of the chemostat: dynamics of microbial competition, Cambridge University Press, (1995)
  • View Article
  • Google Scholar
  • MATH


• [53] M. Tsiang, J. F. Rooney, J. J. Toole, C. S. Gibbs, Biphasic clearance kinetics of hepatitis B virus from patients during adefovir dipivoxil therapy, Hepatology, 29 (1999), 1863–1869
  • View Article
  • Google Scholar


• [54] W.Wang,W. Ma, Block effect on HCV infection by HMGB1 released from virus-infected cells: an insight from mathematical modeling, Commun. Nonlinear Sci. Numer. Simul., 59 (2018), 488–514
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [55] X. Wang, X. Meng, L. Rong, Global dynamics of a multiscale model for hepatitis C virus infection, Appl. Math. Lett., 149 (2024), 5 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [56] G. J. M. Webster, S. Reignat, M. K. Maini, S. A. Whalley, G. S. Ogg, A. King, D. Brown, P. L. Amlot, R. Williams, D. Vergani, G. M. Dusheiko, A. Bertoletti, Incubation phase of acute hepatitis B in man: dynamic of cellular immune mechanism, Hepatology, 32 (2000), 1117–1124
  • View Article
  • Google Scholar


• [57] D. Wodarz, Hepatitis C virus dynamics and pathology: the role of CTL and antibody responses, J. Gen. Virol., 84 (2003), 1743–1750
  • View Article
  • Google Scholar


• [58] D. Wodarz, Mathematical models of immune effector responses to viral infections: virus control versus the development of pathology, J. Comput. Appl. Math., 184 (2005), 301–319
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [59] D. Wodarz, R. M. May, M. A. Nowak, The role of antigen-independent persistence of memory cytotoxic T lymphocytes, Int. Immunol., 12 (2000), 467–477
  • View Article
  • Google Scholar


• [60] R. Xu, Z. Ma, An HBV model with diffusion and time delay, J. Theor. Biol., 257 (2009), 499–509
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [61] X. Yang, Y. Su, H. Li, X. Zhuo, Optimal control of a cell-to-cell fractional-order model with periodic immune response for HCV, Symmetry, 13 (2021), 14 pages
  • View Article
  • Google Scholar


• [62] N. Yousfi, K. Hattaf, A. Tridane, Modeling the adaptive immune response in HBV infection, J. Math. Biol., 63 (2011), 933–957
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [63] S. Zeinali, M. Shahrokhi, A. Pishro, Observer-based controller for treatment of hepatitis C infection using fractional order model, Math. Methods Appl. Sci., 45 (2022), 10689–10709
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [64] Z. Zhang, G. Liang, K. Chang, Stationary distribution of a reaction-diffusion hepatitis B virus infection model driven by the Ornstein-Uhlenbeck process, PLoS ONE, 18 (2023), 21 pages
  • View Article
  • Google Scholar


• [65] H. Zhong, K. Wang, Dynamics on hepatitis B virus infection in vivo with interval delay, SIAM J. Appl. Dyn. Syst., 23 (2024), 205–235
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH