Mathematical analysis of a virus dynamics model with general incidence rate and cure rate. (English) Zbl 1254.92065

Summary: The rate of infection in many virus dynamics models is assumed to be bilinear in the virus and uninfected target cells. In this paper, the dynamical behavior of a virus dynamics model with general incidence rate and cure rate is studied. Global dynamics of the model is established. We prove that the virus is cleared and the disease dies out if the basic reproduction number \(R_{0}\leq 1\) while the virus persists in the host and the infection becomes endemic if \(R_{0}>1\).


92C60 Medical epidemiology
Full Text: DOI


[1] 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, 9, 4398-4402, (1996)
[2] Nowak, M.A.; May, R.M., Viral dynamics, (2000), Oxford University Press Oxford
[3] Perelson, A.; Nelson, P., Mathematical models of HIV dynamics in vivo, SIAM rev., 41, 3-44, (1999) · Zbl 1078.92502
[4] Perelson, A.; Neumann, A.; Markowitz, M.; Leonard, J.; Ho, D., HIV-1 dynamics in vivo: virion clearance rate, infected cell life-span, and viral generation time, Science, 271, 1582-1586, (1996)
[5] Min, L.; Su, Y.; Kuang, Y., Mathematical analysis of a basic model of virus infection with application to HBV infection, Rocky mountain J. math., 38, 5, 1573-1585, (2008) · Zbl 1194.34107
[6] Eikenberry, S.; Hews, S.; Nagy, J.D.; Kuang, Y., The dynamics of a delay model of HBV infection with logistic hepatocyte growth, Math. biosci. eng., 6, 283-299, (2009) · Zbl 1167.92013
[7] Gourley, S.A.; Kuang, Y.; Nagy, J.D., Dynamics of a delay differential model of hepatitis B virus, J. biol. dyn., 2, 140-153, (2008) · Zbl 1140.92014
[8] Hattaf, K.; Yousfi, N., Hepatitis B virus infection model with logistic hepatocyte growth and cure rate, Appl. math. sci., 5, 47, 2327-2335, (2011) · Zbl 1242.92040
[9] Hews, S.; Eikenberry, S.; Nagy, J.D.; Kuang, Y., Rich dynamics of a hepatitis B viral infection model with logistic hepatocyte growth, J. math. biol., 60, 573-590, (2010) · Zbl 1311.92117
[10] Yousfi, N.; Hattaf, K.; Tridane, A., Modeling the adaptative immune response in HBV infection, J. math. biol., 63, 933-957, (2011) · Zbl 1234.92040
[11] Tian, X.; Xu, R., Asymptotic properties of a hepatitis B virus infection model with time delay, Discrete dyn. nat. soc., (2010), 21 pages · Zbl 1204.34111
[12] Wang, K.; Fan, A.; Torres, A., Global properties of an improved hepatitis B virus model, Nonlinear anal. RWA, 11, 3131-3138, (2010) · Zbl 1197.34081
[13] Zhou, X.; Cui, J., Global stability of the viral dynamics with crowley – martin functional response, Bull. Korean math. soc., 48, 3, 555-574, (2011) · Zbl 1364.34076
[14] Crowley, P.H.; Martin, E.K., Functional responses and interference within and between year classes of a dragonfly population, J. north. am. benthol. soc., 8, 211-221, (1989)
[15] Li, D.; Ma, W., Asymptotic properties of an HIV-1 infection model with time delay, J. math. anal. appl., 335, 683-691, (2007) · Zbl 1130.34052
[16] Song, X.; Neumann, A., Global stability and periodic solution of the viral dynamics, J. math. anal. appl., 329, 281-297, (2007) · Zbl 1105.92011
[17] Yu, Y.; Nieto, J.J.; Torres, A.; Wang, K., A viral infection model with a nonlinear infection rate, Bound. value probl., (2009), Article ID 958016, 19 pages · Zbl 1187.34062
[18] Shi, X.; Zhou, X.; Song, X., Dynamical behavior of a delay virus dynamics model with CTL immune response, Nonlinear anal. RWA, 11, 1795-1809, (2010) · Zbl 1204.34110
[19] Wang, L.; Chen, L.; Nieto, J.J., The dynamics of an epidemic model for pest control with impulsive effect, Nonlinear anal. RWA, 11, 1374-1386, (2010) · Zbl 1188.93038
[20] Wu, C.; Weng, P., Stability analysis of a stage structured SIS model with general incidence rate, Nonlinear anal. RWA, 11, 1826-1834, (2010) · Zbl 1198.34186
[21] Yuan, Z.; Wang, L., Global stability of epidemiological models with group mixing and nonlinear incidence rates, Nonlinear anal. RWA, 11, 995-1004, (2010) · Zbl 1254.34075
[22] Dahari, H.; Shudo, E.; Ribeiro, R.M.; Perelson, A.S., Modeling complex decay profiles of hepatitis B virus during antiviral therapy, Hepatology, 49, 1, 32-38, (2009)
[23] 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)
[24] Hattaf, K.; Yousfi, N., Dynamics of HIV infection model with therapy and cure rate, Int. J. tomogr. stat., 74-80, (2011)
[25] Liu, X.; Wang, H.; Hu, Z.; Ma, W., Global stability of an HIV pathogenesis model with cure rate, Nonlinear anal. RWA, 12, 2947-2961, (2011) · Zbl 1231.34094
[26] Srivastava, P.K.; Chandra, P., Modeling the dynamics of HIV and \(\operatorname{CD} 4^+\) T cells during primary infection, Nonlinear anal. RWA, 11, 612-618, (2010) · Zbl 1181.37122
[27] Zhou, X.; Song, X.; Shi, X., A differential equation model of HIV infection of \(\operatorname{CD} 4^+\) T-cells with cure rate, J. math. anal. appl., 342, 2, 1342-1355, (2008) · Zbl 1188.34062
[28] Gradshteyn, I.S.; Ryzhik, I.M., ()
[29] Tridane, A.; Kuang, Y., Modeling the interaction of cytotoxic T lymphocytes and influenza virus infected epithelial cells, Math. biosci. eng., 7, 1, 175-189, (2010) · Zbl 1192.92026
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.