zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Global dynamics of an in-host viral model with intracellular delay. (English) Zbl 1198.92034
Summary: The dynamics of a general in-host model with intracellular delay is studied. The model can describe in vivo infections of HIV-I, HCV, and HBV. It can also be considered as a model for HTLV-I infection. We derive the basic reproduction number R 0 for the viral infection, and establish that the global dynamics are completely determined by the values of R 0 . If R 0 1, the infection-free equilibrium is globally asymptotically stable, and the virus are cleared. If R 0 >1, then the infection persists and the chronic-infection equilibrium is locally asymptotically stable. Furthermore, using the method of Lyapunov functionals, we prove that the chronic-infection equilibrium is globally asymptotically stable when R 0 >1. Our results shows that for intercellular delays to generate sustained oscillations in in-host models it is necessary have a logistic mitosis term in target-cell compartments.
MSC:
92C60Medical epidemiology
34D05Asymptotic stability of ODE
34D23Global stability of ODE
92C50Medical applications of mathematical biology
37N25Dynamical systems in biology
References:
[1]Bonhoeffer, S., May, R.M., Shaw, G.M., Nowak, M.A., 1997. Virus dynamics and drug therapy. Proc. Natl. Acad. Sci. USA 94, 6971–6976. · doi:10.1073/pnas.94.13.6971
[2]Breban, R., Blower, S., 2006. Role of parametric resonance in virological failure during HIV treatment interruption therapy. Lancet 367, 1285–1289. · doi:10.1016/S0140-6736(06)68543-7
[3]Culshaw, R.V., Ruan, S.G., 2000. A delay-differential equation model of HIV infection of CD4+ T-cells. Math. Biosci. 165, 27–39. · Zbl 0981.92009 · doi:10.1016/S0025-5564(00)00006-7
[4]Dixit, N.M., Markowitz, M., Ho, D.D., Perelson, A.S., 2004. Estimates of intracellular delay and average drug efficacy from viral load data of HIV-infected individuals under antiretroviral therapy. Antivir. Ther. 9, 237–246.
[5]Hale, J.K., 1977. Theory of Functional Differential Equations. Springer, Berlin.
[6]Hale, J.K., Verduyn Lunel, S., 1993. Introduction to Functional Differential Equations. Springer, New York.
[7]Herz, V., Bonhoeffer, S., Anderson, R., May, R., Nowak, M., 1996. Viral dynamics in vivo: limitations on estimates of intracellular delay and virus decay. Proc. Natl. Acad. Sci. USA 93, 7247–7251. · doi:10.1073/pnas.93.14.7247
[8]Korobeinikov, A., 2004. Global properties of basic virus dynamics models. Bull. Math. Biol. 66, 879–883. · doi:10.1016/j.bulm.2004.02.001
[9]McCluskey, C.C., 2009. Global stability for an SEIR epidemiological model with varying infectivity and infinite delay. Math. Biosci. Eng. 6, 603–610. · Zbl 1190.34108 · doi:10.3934/mbe.2009.6.603
[10]McCluskey, C.C., 2010. Complete global stability for an SIR epidemic model with delay-distributed or discrete. Nonlinear Anal. 11, 55–59. · Zbl 1185.37209 · doi:10.1016/j.nonrwa.2008.10.014
[11]Mittler, J., Sulzer, B., Neumann, A., Perelson, A., 1998. Influence of delayed virus production on viral dynamics in HIV-1 infected patients. Math. Biosci. 152, 143–163. · Zbl 0946.92011 · doi:10.1016/S0025-5564(98)10027-5
[12]Nelson, P.W., Perelson, A.S., 2002. Mathematical analysis of delay differential equation models of HIV-1 infection. Math. Biosci. 179, 73–94. · Zbl 0992.92035 · doi:10.1016/S0025-5564(02)00099-8
[13]Nelson, P.W., Murray, J., Perelson, A., 2000. A model of HIV-1 pathogenesis that includes an intracellular delay. Math. Biosci. 163, 201–215. · Zbl 0942.92017 · doi:10.1016/S0025-5564(99)00055-3
[14]Nowak, M.A., Bangham, C.R.M., 1996. Population dynamics of immune responses to persistent viruses. Science 272, 74–79. · doi:10.1126/science.272.5258.74
[15]Nowak, M.A., May, R.M., 2000. Virus Dynamics. Cambridge University Press, Cambridge.
[16]Nowak, M.A., Bonhoeffer, S., Hill, A.M., Boehme, R., Thomas, H.C., 1996. Viral dynamics in hepatitis B virus infection. Proc. Natl. Acad. Sci. USA 93, 4398–4402. · doi:10.1073/pnas.93.9.4398
[17]Perelson, A.S., Nelson, P.W., 1999. Mathematical analysis of HIV-I dynamics in vivo. SIAM Rev. 41, 3–44. · Zbl 1078.92502 · doi:10.1137/S0036144598335107
[18]Perelson, A.S., Kirschner, D.E., de Boer, R., 1993. Dynamics of HIV infection of CD4 T cells. Math. Biosci. 114, 81–125. · Zbl 0796.92016 · doi:10.1016/0025-5564(93)90043-A
[19]Perelson, A.S., Neumann, A.U., Markowitz, M., Leonard, J.M., Ho, D.D., 1996. HIV-1 dynamics in vivo: virion clearance rate, infected cell life-span, and viral generation time. Science 271, 1582–1586. · doi:10.1126/science.271.5255.1582
[20]Smith, H.L., De Leenheer, P., 2003. Virus dynamics: a global analysis. SIAM J. Appl. Math. 63, 1313–1327. · Zbl 1035.34045 · doi:10.1137/S0036139902406905
[21]Tuckwell, H.C., Wan, F.Y.M., 2004. On the behavior of solutions in viral dynamical models. Biosystems 73, 157–161. · doi:10.1016/j.biosystems.2003.11.004
[22]Wang, L., Li, M.Y., 2006. Mathematical analysis of the global dynamics of a model for HIV infection of CD4+ T cells. Math. Biosci. 200, 44–57. · Zbl 1086.92035 · doi:10.1016/j.mbs.2005.12.026
[23]Wang, L., Li, M.Y., Kirschner, D., 2002. Mathematical analysis of the global dynamics of a model for HTLV-I infection and ATL progression. Math. Biosci. 179, 207–217. · Zbl 1008.92026 · doi:10.1016/S0025-5564(02)00103-7
[24]Wang, Y., Zhou, Y., Wu, J., Heffernan, J., 2009. Oscillatory viral dynamics in a delayed HIV pathogenesis model. Math. Biosci. 219, 104–112. · Zbl 1168.92031 · doi:10.1016/j.mbs.2009.03.003