Dynamical analysis of an HCV model with cell-to-cell transmission and cure rate in the presence of adaptive immunity

2022;
: pp. 579–593
https://doi.org/10.23939/mmc2022.03.579
Received: February 10, 2022
Accepted: May 15, 2022
1
Laboratory of Mathematics, Computer Science and Applications, FST Mohammedia, University Hassan II of Casablanca
2
ENCG of Casablanca, University Hassan II
3
Laboratory of Mathematics, Computer Science and Applications, FST Mohammedia, University Hassan II of Casablanca

In this paper, we will study mathematically and numerically the dynamics of the hepatitis C virus disease with the consideration of two fundamental modes of transmission of the infection, namely virus-to-cell and cell-to-cell.  In our model, we will take into account the role of cure rate of the infected cells and the effect of the adaptive immunity.  The model consists of five nonlinear differential equations, describing the interaction between the uninfected cells, the infected cells, the hepatitis C virions and the adaptive immunity.  This immunity will be represented by the humoral and cellular immune responses.  This work begins with proving the non-negativity and the boundedness of solutions and determining the basic reproduction number.  Secondly, five equilibria are established, the local stability analysis for all the equilibria is demonstrated theoretically and numerically.  Finally, we have concluded that the numerical results are coherent with our theoretical postulations.

  1. https://www.who.int/news-room/fact-sheets/detail/hepatitis-c.
  2. Neumann A. U., Lam N. P., Dahari H., Gretch D. R., Wiley T. E., Layden T. J., Perelson A. S.  Hepatitis C Viral Dynamics in Vivo and the Antiviral Efficacy of Interferon-$\alpha$ Therapy.  Science. 282 (5386), 103–107 (1998).
  3. Hattaf K., Yousfi N.  A Delay Differential Equation Model of HIV with Therapy and Cure Rate.  International Journal of Nonlinear Science. 12, 503–512 (2011).
  4. Hattaf K., Yousfi N., Tridane A.  Mathematical analysis of a virus dynamics model with general incidence rate and cure rate.  Nonlinear Analysis: Real World Applications. 13 (4), 1866–1872 (2012).
  5. Liu X., Wang H., Hu Z., Ma W.  Global stability of an HIV pathogenesis model with cure rate.  Nonlinear Analysis: Real World Applications. 12, 2947–2961 (2011).
  6. Srivastava P. K., Banerjee M., Chandra P.  Modeling the drug therapy for HIV infection. Journal of Biological Systems.  Journal of Biological Systems. 17 (2), 213–223 (2009).
  7. Tian Y., Liu X.  Global dynamics of a virus dynamical model with general incidence rate and cure rate.  Nonlinear Analysis: Real World Application. 16, 17–26 (2014).
  8. Zhou X., Song X., Shi X.  A differential equation model of HIV infection of CD4$^+$ $\mathcal{T}$-cells with cure rate.  Journal of Mathematical Analysis and Applications. 342 (2), 1342–1355 (2008).
  9. Dahari H., Major M., Zhang X., Mihalik K., Rice M. C., Perelson S. A., Feinstone M. S., Neumann U. A.  Mathematical modeling of primary hepatitis C infection: Noncytolytic clearance and early blockage of virion production. Gastroenterology. 128 (4), 1056–1066 (2005).
  10. Reluga T. C., Dahari H., Perelson A. S.  Analysis of hepatitis C virus infection models with hepatocyte homeostasis.  SIAM Journal on Applied Mathematics. 69 (4), 999–1023 (2009).
  11. Lai X., Zou X.  Modeling cell-to-cell spread of HIV-1 with logistic target cell growth.  Journal of Mathematical Analysis and Applications. 426 (1), 563–584 (2015).
  12. Mojaver A., Kheiri H.  Dynamical analysis of a class of hepatitis C virus infection models with application of optimal control.  International Journal of Biomathematics. 9 (3), 3997–4008 (2016).
  13. Pan S., Chakrabarty S. P.  Threshold dynamics of HCV model with cell-to-cell transmission and a non-cytolytic cure in the presence of humoral immunity.  Communications in Nonlinear Science and Numerical Simulation. 61, 180–197 (2018).
  14. Avendano R., Esteva L., Flores J. A., Fuentes Allen J. L., Gómez G., López-Estrada Je.  A mathematical model for the dynamics of hepatitis C.  Journal of Theoretical Medicine. 4, 109–118 (2002).
  15. Meskaf A., Tabit Y., Allali K.  Global analysis of a HCV model with CTL antibody responses and therapy.  Applied Mathematical Sciences. 9 (81), 3997–4008 (2015).
  16. Nabi K. N., Podder C. N.  Sensitivity analysis of chronic hepatitis C virus infection with immune response and cell proliferation.  International Journal of Biomathematics. 13 (3), 301–319 (2020).
  17. Wodarz D.  Hepatitis C virus dynamics and pathology: The role of CTL and antibody responses.  Journal of General Virology. 84 (7), 1743–1750 (2003).
  18. Wodarz D.  Mathematical models of immune effector responses to viral infections: Virus control versus the development of pathology.  Journal of Computational and Applied Mathematics. 184 (1), 301–319 (2005).
  19. Yousfi N., Hattaf K., Rachik M.  Analysis of a HCV model with CTL and antibody responses.  Applied Mathematical Sciences. 3, 2835–2847 (2009).
  20. Banerjee S., Keval R., Gakkhar S.  Modeling the dynamics of hepatitis C virus with combined antiviral drug therapy: Interferon and Ribavirin.  Mathematical Biosciences. 245 (2), 235–248 (2013).
  21. Chen S.-S., Cheng C.-Y., Takeuchi Y.  Stability analysis in delayed within-host viral dynamics with both viral and cellular infections.  Journal of Mathematical Analysis and Applications. 442 (2), 642–672 (2016).
  22. Dahari H., Lo A., Ribeiro R. M., Perelson A. S.  Modeling hepatitis C virus dynamics: Liver regeneration and critical drug efficacy.  Journal of Theoretical Biology. 247 (2), 371–381 (2007).
  23. Dubey B., Dubey P., Dubey S. U.  Modeling the intracellular pathogen-immune interaction with cure rate.  Communications in Nonlinear Science and Numerical Simulation. 38, 72–90 (2016).
  24. Zwillinger D., Jeffrey A.  Table of Integrals, Series, and Products. Elsevier (2007).
  25. Hattaf K., Yousfi N.  A generalized virus dynamics model with cell-to-cell transmission and cure rate.  Advances in Difference Equations. 2016, 174 (2016).
  26. Li J., Men K., Yang Y., Li D.  Dynamical analysis on a chronic hepatitis C virus infection model with immune response.  Journal of Theoretical Biology. 365, 337–346 (2015).
  27. Perasso A.  An introduction to the basic reproduction number in mathematical epidemiology.  ESAIM: Proceedings and Surveys. 62, 123–138 (2018).
  28. Reyes-Silveyra J., Mikler A. R.  Modeling immune response and its effect on infectious disease outbreak dynamics.  Theoretical Biology and Medical Modelling. 13, 10 (2016).
  29. Van der Driessche P., Watmough J.  Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission.  Mathematical Biosciences. 180 (1–2), 29–48 (2002).
  30. Vargas-De-León C.  Stability analysis of a model for HBV infection with cure of infected cells and intracellular delay.  Applied Mathematics and Computation. 219 (1), 389–398 (2012).
Mathematical Modeling and Computing, Vol. 9, No. 3, pp. 579–593 (2022)