Global Stability Analysis of the Original Cellular Model of Hepatitis C Virus Infection Under Therapy
American Journal of Mathematical and Computer Modelling
Volume 4, Issue 3, September 2019, Pages: 58-65
Received: Jun. 12, 2019;
Accepted: Jul. 27, 2019;
Published: Aug. 29, 2019
Views 473 Downloads 117
Alexis Nangue, Department of Mathematics, Higher Teachers' Training College, University of Maroua, Maroua, Cameroon
In this work, we investigate the hepatitis C virus infection under treatment. We first derive a nonlinear ordinary differential equation model for the studied biological phenomenon. The obtained initial value problem is completely analysed. To begin with the analysis of the model, we use the standard theory of ordinary differential equations to prove existence, uniqueness and boundedness of the solution. Morever, the basic reproduction number R0 determining the extinction or the persistence of the HCV infection is computed and used to express the equilibrium points. Also the global asymptotic stability of the HCV-uninfected equilibrium point and the HCV-infected equilibrium point of the model are derived by means of appropriate Lyapunov functions. Finally numerical simulations are carried out to confirm theoretical results obtained at HCV-unfected equilibrium.
Global Stability Analysis of the Original Cellular Model of Hepatitis C Virus Infection Under Therapy, American Journal of Mathematical and Computer Modelling.
Vol. 4, No. 3,
2019, pp. 58-65.
R. M. Anderson and R. M. May, eds. 1991. Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, Oxford, UK.
Chatterjee, A., Guedj, J., & Perelson, A. S. (2012). Mathematical modelling of HCV infection: what can it teach us in the era of direct-acting antiviral agents? Antiviral Therapy, 17 (6PtB), 1171-1182. http://dx.doi.org/10.3851/IMP2428.
Chong, Maureen Siew Fang, Shahrill, Masitah, Crossley, Laurie and Madzvamuse, Anotida, 2015 The stability analyses of the mathematical models of hepatitis C virus infection. Modern Applied Science, 9 (3). pp. 250-271. ISSN 1913-1844.
Dahari, H., Lo, A., Ribeiro, R. M., & Perelson, A. S. (2007). Modeling hepatitis C virus dynamics: Liver regeneration and critical drug efficacy. Journal of Theoretical Biology, 247, 371-381. http://dx.doi.org/10.1016/j.jtbi.2007.03.006.
O. Diekmann, J. A. P. Heesterbeek, and J. A. J. Metz, 1990. On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28, pp. 365-382.
K. Dietz, 1988. Density dependence in parasite transmission dynamics, Parasit. Today, 4, pp. 91-97.
Guedj, J., & Neumann, A. U. (2010). Understanding hepatitis C viral dynamics with direct-acting antiviral agents due to the interplay between intracellular replication and cellular infection dynamics. Journal of Theoretical Biology, 267 (3), 330-340. http://dx.doi.org/10.1016/j.jtbi.2010.08.036.
Khalil, H., 2002. Nonlinear Systems, 3rd edn. Prentice Hall, New York.
Neumann, A. U., Lam, N. P., Dahari, H., Gretch, D. R., Wiley, T. E., Layden, T. J., & Perelson, A. S. (1998). Hepatitis C viral dynamics in vivo and the antiviral efficacy of interferon- therapy. Science, 282, 103-107. http://dx.doi.org/10.1126/science.282.5386.103.
Reluga, T. C., Dahari, H., & Perelson, A. S. (2009). Analysis of hepatitis C virus infection models with hepatocyte homeostasis. SIAM Journal on Applied Mathematics, 69 (4), 999-1023. http://dx.doi.org/10.1137/080714579.
Rong, L., & Perelson, A. S. (2013). Mathematical analysis of multiscale models for hepatitis C virus dynamics under therapy with direct-acting antiviral agents. Mathematical Biosciences, 245 (1), 22-30. http://dx.doi.org/10.1016/j.mbs.2013.04.012.
L. B. Seef, 2002. Natural history of chronic hepatitis C, Hepatology, 36: S35-S46.
Sever Silvestru Dragomir (2013).. Some Gronwall Type Inequalities and Applications, NOVA, Melbourne.
P. van den Driessche and James Watmough, 2002. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical Biosciences 180 29-48.
WHO, Guidelines for the sreening, care and treatement of persons with hepatitis C infection, ISBN 978 92 4 154875 5, NLM classification: WC 536, 2014.