Influence of backward bifurcation in a model of hepatitis B and C viruses. (English) Zbl 1188.92017

Summary: In the 1990s, liver transplantation for hepatitis B and C virus (HBV and HCV) related-liver diseases was a very controversial issue since recurrent infection of the graft was inevitable. Significant progress has been made in the prophylaxis and treatment of recurrent hepatitis B/C (or HBV/HCV infection) after liver transplantation. We propose a mathematical model of ordinary differential equations describing the dynamics of the HBV/HCV and its interactions with both liver and blood cells. A single model is used to describe infection of either virus since the dynamics in-host (infected of the liver) are similar. Analyzing the model, we observe that the system shows either a transcritical or a backward bifurcation. Explicit conditions on the model parameters are given for the backward bifurcation to be present. Consequently, we investigate possible factors that are responsible for HBV/HCV infection and assess control strategies to reduce HBV/HCV reinfection and improve graft survival after liver transplantation.


92C50 Medical applications (general)
34C60 Qualitative investigation and simulation of ordinary differential equation models
34D20 Stability of solutions to ordinary differential equations
92C60 Medical epidemiology
37N25 Dynamical systems in biology
Full Text: DOI


[1] Brauer, F.; Castillo-Chávez, C., Mathematical Models in Population Biology and Epidemiology. Mathematical Models in Population Biology and Epidemiology, Texts in Applied Mathematics (2001), Springer: Springer New York, NY · Zbl 0967.92015
[2] Berenguer, M.; Prieto, M.; San Juan, F.; Rayon, J. M.; Martinez, F.; Carrasco, D., Contribution of donor age to the recent decrease in patient survival among HCV-infected liver transplant recipients, Hepatology, 36, 202 (2002)
[3] Carr, J., Applications of centre manifolds, (Applied Mathematical Sciences (1981), Springer: Springer NY, USA), 35 · Zbl 0051.45301
[4] Castillo-Chávez, C.; Song, B., Dynamical models of tuberculosis and their applications, Math. Biosci. Eng., 1, 2, 361 (2004) · Zbl 1060.92041
[5] Dahari, H.; Feliu, A.; Garcia-Retortillo, M.; Forns, X.; Neumann, A. U., Second hepatitis C replication compartment indicated by viral dynamics during liver transplantation, J. Hepatol., 42, 491 (2005)
[6] Davies, S. E.; Portmann, B. C.; Grady, O’J. G.; Aldis, P. M.; Chaggar, K.; Alexander, G. J., including an unique pattern of fibrosing cholestatic hepatitis, Hepatology, 13, 150 (1991)
[7] Diekmann, O.; Heesterbeek, J. A.P.; Metz, J. A.J., On the definition and computation of the basic reproduction ratio \(R_0\) in models for infections diseases in heterogeneous populations, J. Math. Biol., 28, 365 (1990) · Zbl 0726.92018
[8] van den Driessche, P.; Watmough, J., Reproduction number and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180, 29 (2002) · Zbl 1015.92036
[9] Feng, Z.; Castillo-Chavez, C.; Capurro, A., A model for tuberculosis with exogenous reinfection, Theor. Populat. Biol., 57, 235 (2000) · Zbl 0972.92016
[10] Fried, M. W.; Shiffman, M. L.; Reddy, K. R.; Smith, C.; Marinos, G.; Goncales, F. L.; Haussinger, D.; Diago, M.; Carosi, G.; Dhumeaux, D.; Craxi, A.; Lin, A.; Hoffman, J.; Yu, J., Peginterferon alfa-2a plus ribavirin for chronic hepatitis C virus infection, N. Engl. J. Med., 347, 975 (2002)
[11] O’Grady, J. G.; Smith, H. M.; Davies, S. E.; Daniels, H. M.; Donaldson, P. T.; Tan, K. C., J. Hepatol., 14, 104 (1992)
[12] Grellier, L.; Multimer, D.; Ahmed, M., Lamivudine prophylaxis against reinfection in liver transplantation for hepatitis B cirrhosis, Lancet, 1, 1212 (1996)
[13] Hadeler, K.; van den Driessche, P., Backward bifurcation in epidemic control, Math. Biosci., 146, 15 (1997) · Zbl 0904.92031
[14] Hale, J. K., Ordinary Differential Equations (1969), Wiley: Wiley New York · Zbl 0186.40901
[15] Heffernan, J. M.; Smith, R. J.; Wahl, L. M., Perspectives on the basic reproductive ratio, J.R. Soc. Interf., 2, 281 (2005)
[16] Heffernan, J. M.; Wahl, L. M., Monte Carlo estimates of natural variation in HIV infection, J. Theor. Biol., 236, 137 (2005) · Zbl 1442.92167
[17] Hollinger, F. B.; Liang, T. J., Hepatitis B virus, (Knipe, D. M.; Howley, P. M., Fields Virology, vol. 2 (2001), Lippincott Williams and Wilkins: Lippincott Williams and Wilkins Philadelphia, PA), 2971
[18] Holmes, K. K.; Sparling, P. F.; Stamm, W. E.; Piot, P.; Wasserheit, J. N.; Corey, L.; Cohen, M. S.; Watts, D. H., Sexually Transmitted Diseases (2008), McGraw Hill: McGraw Hill Toronto, Canada
[19] Ichikawa, T., Clearance of hepatitis C virus after living-donor liver transplantation in spite of residual viremia on end date of interferon therapy before transplantation, World J. Gastroenterol., 13, 30, 4149 (2007)
[20] Powers Kimberly, A.; Ruy Ribeiro, M., Kinetics of hepatitis C virus reinfection after liver transplantation, Liver Transplant., 12, 207 (2006)
[21] Lau, G.; Tsiang, M.; Hou, J.; Yuen, S.; Carman, W.; Zhang, L.; Gibbs, C.; Lam, S., Combination therapy with lamivudine and famciclovir for chronic hepatitis B infected Chinese patients: a viral dynamics study, Hepatology, 32, 394 (2000)
[22] Lewin, S.; Ribeiro, R.; Walters, T.; Lau, G.; Bowden, S.; Locarnini, S.; Perelson, A., Analysis of hepatitis B viral load decline under potent therapy: complex decay profiles observed, Hepatology, 34, 1012 (2001)
[23] Manns, M. P.; McHutchison, J. G.; Gordon, S. C.; Rustgi, V. K.; Shiffman, M.; Reindollar, R.; Goodman, Z. D.; Koury, K.; Ling, M.; Albrecht, J. K., Peginterferon alfa-2b plus ribavirin compared with interferon alfa-2b plus ribavirin for initial treatment of chronic hepatitis C: a randomised trial, Lancet, 358, 958 (2001)
[24] Mazzaferro, V., Risk of HBV reinfection after liver transplantation in HBsAg-positive cirrhosis. Primary hepatocellular carcinoma is not a predictor for HBV recurrence. The European Cooperative Study Group on Liver Cancer and Transplantation, Liver, 16, 2, 117 (1996)
[25] Nowak, M. A.; Bonhoeffer, S.; Hill, A. M.; Boehme, R.; Thomas, H. C.; McDade, H., Viral dynamics in Hepatitis B Virus infection, Proc. Natl. Acad. Sci. USA, 93, 4398 (1996)
[26] Nowak, M. A.; May, R. M., Virus Dynamics (2000), Oxford University · Zbl 1101.92028
[27] Ribeiro, R.; Lo, A.; Perelson, A., Dynamics of hepatitis B virus infection, Microb. Infect., 4, 829 (2002)
[28] Todo, S.; Demetris, A. J.; Van Thiel, D.; Teperman, L.; Fung, J. J.; Starzl, T. E., Orthotopic liver transplantation for patients with hepatitis B virus-related liver disease, Hepatology, 13, 619 (1991)
[29] Tsiang, M.; Rooney, J.; Toole, J.; Gibbs, C., Biphasic clearance kinetics of hepatitis B virus from patients during adefovir dipivoxil therapy, Hepatology, 29, 1863 (1999)
[30] World Health Organization. Heptatitis B fact sheet No. 204, WHO website, 2009.; World Health Organization. Heptatitis B fact sheet No. 204, WHO website, 2009.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.