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Global properties of an improved hepatitis B virus model. (English) Zbl 1197.34081
Summary: We introduce an improved HBV model with standard incidence function and cytokine-mediated ‘cure’ based on empirical evidences. By carrying out a global analysis of the modified model and studying the stability of the equilibria, we show that the infection-free equilibrium is globally asymptotically stable if the basic reproduction number of virus is less than one and, conversely, the infection equilibrium is globally asymptotically stable if the basic reproduction number of virus is greater than one.

MSC:
34C60 Qualitative investigation and simulation of ordinary differential equation models
34D20 Stability of solutions to ordinary differential equations
92C60 Medical epidemiology
92D30 Epidemiology
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[1] Lai, C.L.; Ratziu, V.; Yuen, M.F.; Poynard, T., Viral hepatitis B, Lancet, 362, 2089-2094, (2003)
[2] Ferrari, C.; Penna, A.; Bertoletti, A.; Valli, A.; Antoni, A.D.; Giuberti, T.; Cavalli, A.; Petit, M.A.; Fiaccadori, F., Cellular immune response to hepatitis B virusencoded antigens in acute and chronic hepatitis B virus infection, The journal of immunology, 145, 3442-3449, (1990)
[3] Beasley, R.P.; Lin, C.C.; Wang, K.Y.; Hsieh, F.J.; Hwang, L.Y.; Stevens, C.E.; Sun, T.S.; Szmuness, W., Hepatocellular carcinoma and hepatitis B virus, Lancet, 2, 1129-1133, (1981)
[4] Nowak, M.A.; May, R.M., Virus dynamics, (2000), Oxford University Press Oxford · Zbl 1101.92028
[5] Weissberg, J.I.; Andres, L.L.; Smith, C.I.; Weick, S.; Nichols, J.E.; Garcia, G.; Robinson, W.S.; Merigan, T.C.; Gregory, P.B., Survival in chronic hepatitis B: an analysis of 379 patients, Annals of internal medicine, 101, 613-616, (1984)
[6] de Leenheer, P.; Smith, H.L., Virus dynamics: A global analysis, SIAM journal on applied mathematics, 63, 1313-1327, (2003) · Zbl 1035.34045
[7] Elaiw, A.M., Global properties of a class of HIV models, Nonlinear analysis. real world applications, (2009) · Zbl 1197.34073
[8] Perelson, A.S., Modelling viral and immune system dynamics, Nature reviews immunology, 2, 28-36, (2002)
[9] Perelson, A.S.; Herrmann, E.; Micol, F.; Zeuzem, S., New kinetic models for the hepatitis C virus, Hepatology, 42, 749-754, (2005)
[10] Srivastava, P.Kr.; Chandra, P., Modeling the dynamics of HIV and CD4^{+} T cells during primary infection, Nonlinear analysis. real world applications, (2008) · Zbl 1181.37122
[11] Nowak, M.A.; Bonhoeffer, S.; Hill, A.M.; Boehme, R.; Thomas, H.C., Viral dynamics in hepatitis B virus infection, Proceedings of the national Academy of sciences of the united states of America, 93, 4398-4402, (1996)
[12] Korobeinikov, A., Global properties of basic virus dynamics models, Bulletin of mathematical biology, 66, 879-883, (2004) · Zbl 1334.92409
[13] Eikenberry, S.; Hews, S.; Nagy, J.D.; Kuang, Y., The dynamics of a delay model of HBV infection with logistic hepatocyte growth, Mathematical biosciences and engineering, 6, 283-299, (2009) · Zbl 1167.92013
[14] Gourley, S.A.; Kuang, Y.; Nagy, J.D., Dynamics of a delay differential equation model of hepatitis B virus infection, Journal of biological dynamics, 2, 140-153, (2008) · Zbl 1140.92014
[15] Hews, S.; Eikenberry, S.; Nagy, J.D.; Kuang, Y., Rich dynamics of a hepatitis B viral infection model with logistic hepatocyte growth, Journal of mathematical biology, (2009) · Zbl 1167.92013
[16] Min, L.; Su, Y.; Kuang, Y., Mathematical analysis of a basic virus infection model with application to HBV infection, Rocky mountain journal of mathematics, 38, 1573-1585, (2008) · Zbl 1194.34107
[17] Lau, G.K.; Tsiang, M.; Hou, J.; Yuen, S.; Carman, W.F.; Zhang, L.; Gibbs, C.S.; Lam, S., Combination therapy with lamivudine and famciclovir for chronic hepatitis B-infected Chinese patients: A viral dynamics study, Hepatology, 32, 394-399, (2000)
[18] Ciupe, S.M.; Ribeiro, R.M.; Nelson, P.W.; Perelson, A.S., Modeling the mechanisms of acute hepatitis B virus infection, Journal of theoretical biology, 247, 23-35, (2007)
[19] Dahari, H.; Shudo, E.; Ribeiro, R.M.; Perelson, A.S., Modeling complex decay profiles of hepatitis B virus during antiviral therapy, Hepatology, 49, 32-38, (2009)
[20] Guidotti, L.G.; Rochford, R.; Chung, J.; Shapiro, M.; Purcell, R.; Chisari, F.V., Viral clearance without destruction of infected cells during acute HBV infection, Science, 284, 825-829, (1999)
[21] Li, M.Y.; Muldowney, J.S., A geometric approach to the global-stability problems, SIAM journal on mathematical analysis, 27, 1070-1083, (1996) · Zbl 0873.34041
[22] Thieme, H.R., Persistence under relaxed point-dissipativity (with application to an endemic model), SIAM journal on mathematical analysis, 24, 407-435, (1993) · Zbl 0774.34030
[23] Arino, J.; Mccluskey, C.C.; van den Driessche, P., Global results for an epidemic model with vaccination that exhibits backward bifurcation, SIAM journal on applied mathematics, 64, 260-276, (2003) · Zbl 1034.92025
[24] Fan, M.; Li, M.Y.; Wang, K., Global stability of an SEIS epidemic model with recruitment and a varying total population size, Mathematical biosciences, 170, 199-208, (2001) · Zbl 1005.92030
[25] Li, M.Y.; Smith, H.L.; Wang, L., Global dynamics of an SEIR epidemic model with vertical transmission, SIAM journal on applied mathematics, 62, 58-69, (2001) · Zbl 0991.92029
[26] Wang, K.; Wang, W.; Liu, X., Global stability in a viral infection model with lytic and nonlytic immune responses, Computers and mathematics with applications, 51, 1593-1610, (2006) · Zbl 1141.34034
[27] Martin, R.H., Logarithmic norms and projections applied to linear differential systems, Journal of mathematical analysis and applications, 45, 432-454, (1974) · Zbl 0293.34018
[28] Coppel, W.A., Stability and asymptotic behavior of differential equations, (1965), Health Boston, MA · Zbl 0154.09301
[29] Ciupe, S.M.; Ribeiro, R.M.; Nelson, P.W.; Dusheiko, G.; Perelson, A.S., The role of cells refractory to productive infection in acute hepatitis B viral dynamics, Proceedings of the national Academy of sciences of the united states of America, 104, 5050-5055, (2007)
[30] Lewin, S.R.; Ribeiro, R.M.; Walters, T.; Lau, G.K.; Bowden, S.; Locarnini, S.; Perelson, A.S., Analysis of hepatitis B viral load decline under potent therapy: complex decay profiles observed, Hepatology, 34, 1012-1020, (2001)
[31] Li, D.; Ma, W., Asymptotic properties of a HIV-1 infection model with time delay, Journal of mathematical analysis and applications, 335, 683-691, (2007) · Zbl 1130.34052
[32] Shi, X.; Zhou, X.; Song, X., Dynamical behavior of a delay virus dynamics model with CTL immune response, Nonlinear analysis. real world applications, (2009)
[33] Wang, K.; Wang, W.; Liu, X., Viral infection model with periodic lytic immune response, Chaos, solitons and fractals, 28, 90-99, (2006) · Zbl 1079.92048
[34] Wang, K.; Wang, W.; Song, S., Dynamics of an HBV model with diffusion and delay, Journal of theoretical biology, 253, 36-44, (2008)
[35] Wodarz, D., Hepatitis C virus dynamics and pathology: the role of CTL and antibody responses, Journal of general virology, 84, 1743-1750, (2003)
[36] Yu, Y.; Nieto, J.J.; Torres, A.; Wang, K., A viral infection model with a nonlinear infection rate, Boundary value problems, (2009) · Zbl 1187.34062
[37] Lai, C.L.; Yuen, M.F., The natural history and treatment of chronic hepatitis B: A critical evaluation of standard treatment criteria and end points, Annals of internal medicine, 147, 58-61, (2007)
[38] Liu, G.T., Bicyclol: A novel drug for treating chronic viral hepatitis B and C, Medicinal chemistry, 5, 29-43, (2009)
[39] Wang, L.; Chen, L.; Nieto, J.J., The dynamics of an epidemic model for pest control with impulsive effect, Nonlinear analysis. real world applications, (2009)
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