zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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 a mathematical model for HTLV-I infection of CD4 + T cells with delayed CTL response. (English) Zbl 1239.34086
Summary: Human T-cell leukaemia virus type I (HTLV-I) preferentially infects the CD4 + T cells. The HTLV-I infection causes a strong HTLV-I specific immune response from CD8 + cytotoxic T cells (CTLs). The persistent cytotoxicity of the CTL is believed to contribute to the development of a progressive neurologic disease, HTLV-I associated myelopathy/tropical spastic paraparesis (HAM/TSP). We investigate the global dynamics of a mathematical model for the CTL response to HTLV-I infection in vivo. To account for a series of immunological events leading to the CTL response, we incorporate a time delay in the response term. Our mathematical analysis establishes that the global dynamics are determined by two threshold parameters R 0 and R 1 , basic reproduction numbers for viral infection and for CTL response, respectively. If R 0 1, the infection-free equilibrium P 0 is globally asymptotically stable, and the HTLV-I viruses are cleared. If R 1 1<R 0 , the asymptomatic-carrier equilibrium P 1 is globally asymptotically stable, and the HTLV-I infection becomes chronic but with no persistent CTL response. If R 1 >1, a unique HAM/TSP equilibrium P 2 exists, at which the HTLV-I infection is chronic with a persistent CTL response. We show that the time delay can destabilize the HAM/TSP equilibrium, leading to Hopf bifurcations and stable periodic oscillations. Implications of our results to the pathogenesis of HTLV-I infection and HAM/TSP development are discussed.
34K18Bifurcation theory of functional differential equations
92C60Medical epidemiology
37N25Dynamical systems in biology
34K20Stability theory of functional-differential equations
34K60Qualitative investigation and simulation of models
[1]Robbins, F. W.: A mathematical model of HIV infection: simulating T4, T8, macrophages, antibody, and virus via specific anti-HIV response in the presence of adaptation and tropism, Bull. math. Biol. 72, 1208-1253 (2010) · Zbl 1197.92028 · doi:10.1007/s11538-009-9488-5
[2]Wang, K.; Fan, A.: Global properties of an improved hepatitis B virus model, Nonlinear anal. RWA 11, 3131-3138 (2010) · Zbl 1197.34081 · doi:10.1016/j.nonrwa.2009.11.008
[3]Elaiw, A. M.: Global properties of a class of HIV models, Nonlinear anal. RWA 11, 2253-2263 (2010) · Zbl 1197.34073 · doi:10.1016/j.nonrwa.2009.07.001
[4]Vieira, I. T.; Cheng, R. C. H.: Small world network models of the dynamics of HIV infection, Ann. oper. Res. 178, 173-200 (2010) · Zbl 1197.90294 · doi:10.1007/s10479-009-0571-y
[5]Yu, Y.; Nieto, J. J.: A viral infection model with a nonlinear infection rate, Bound. value probl. 2009 (2009)
[6]Bangham, C. R. M.: The immune response to HTLV-I, Curr. opin. Immunol. 12, 397-402 (2000)
[7]Bangham, C. R. M.: The immune control and cell-to-cell spread of human T-lymphotropic virus type 1, J. gen. Virol. 84, 3177-3189 (2003)
[8]Jacobson, S.: Immunopathogenesis of human T cell lymphotropic virus type I-associated neurologic disease, J. infect. Dis. 186, No. S2, S187-S192 (2002)
[9]Eshima, N.; Tabata, M.: Population dynamics of HTLV-I infection: a discrete-time mathematical epidemic model approach, Math. med. Biol. 20, 29-45 (2003) · Zbl 1042.92029 · doi:10.1093/imammb/20.1.29
[10]Gomez-Acevedo, H.; Li, M. Y.; Jacobson, S.: Multi-stability in a model for CTL response to HTLV-I infection and its consequences in HAM/TSP development and prevention, Bull. math. Biol. 72, 681-696 (2010) · Zbl 1189.92049 · doi:10.1007/s11538-009-9465-z
[11]Gout, O.; Baulac, M.: Medical intelligence: rapid development of myelopathy after HTVL-I infections acquired by transfusion during cardiac transplantation, N. engl. J. med. 322, 383-388 (1990)
[12]Osame, M.; Janssen, R.: Nationwide survey of HTLV-I-associated myelopathy in Japan: association with blood transfusion, Ann. neurol. 28, 50-56 (1990)
[13]Kubota, R.; Osame, M.; Jacobson, S.: Retrovirus: human T-cell lymphotropic virus type I-associated diseases and immune dysfunction, Effects of microbes on the immune system, 349-371 (2000)
[14]Gallo, R. C.: History of the discoveries of the first human retroviruses: HTLV-1 and HTLV-2, Oncogene 24, 5926-5930 (2005)
[15]Coffin, J. M.; Hughes, S. H.; Varmus, H. E.: Retroviruses, (1997)
[16]Wodarz, D.; Nowak, M. A.; Bangham, C. R. M.: The dynamics of HTLV-I and the CTL response, Immunol. today 20, 220-227 (1999)
[17]Nowak, M. A.; May, R. M.: Virus dynamics: mathematical principles of immunology and virology, (2000)
[18]Wodarz, D.; Bangham, C. R. M.: Evolutionary dynamics of HTLV-I, J. mol. Evol. 50, 448-455 (2000)
[19]Perelson, A. S.; Nelson, P. W.: Mathematical analysis of HIV-I dynamics in vivo, SIAM rev. 41, 3-44 (1999) · Zbl 1078.92502 · doi:10.1137/S0036144598335107
[20]De Boer, R. J.; Perelson, A. S.: Towards a general function describing T cell proliferation, J. theoret. Biol. 175, 567-576 (1995)
[21]De Boer, R. J.; Perelson, A. S.: Target cell limited and immune control models of HIV infection: a comparison, J. theoret. Biol. 190, 201-214 (1998)
[22]Nowak, M. A.; Bangham, C. R. M.: Population dynamics of immune responses to persistent viruses, Science 272, 74-79 (1996)
[23]Burić, N.; Mudrinic, M.; Vasović, N.: Time delay in a basic model of the immune response, Chaos solitons fractals 12, 483-489 (2001) · Zbl 1026.92015 · doi:10.1016/S0960-0779(99)00205-2
[24]Wang, K.; Wang, W.: Complex dynamic behavior in a viral model with delayed immune response, Physica D 226, 197-208 (2007) · Zbl 1117.34081 · doi:10.1016/j.physd.2006.12.001
[25]Li, M. Y.; Shu, H.: Impact of intracellular delays and target-cell dynamics on in vivo viral infections, SIAM J. Appl. math. 70, 2434-2448 (2010) · Zbl 1209.92037 · doi:10.1137/090779322
[26]Mccluskey, C. C.: Global stability for an SEIR epidemiological model with varying infectivity and infinite delay, Math. biosci. Eng. 6, 603-610 (2009) · Zbl 1190.34108 · doi:10.3934/mbe.2009.6.603
[27]Lasalle, J.; Lefschetz, S.: Stability by Lyapunov’s direct method, (1961) · Zbl 0098.06102
[28]Hale, J. K.; Lunel, S. V.: Introduction to functional differential equations, (1993)
[29]Hale, J. K.: Theory of functional differential equations, (1977)
[30]Hassard, B. D.; Kazarinoff, N. D.; Wan, Y. H.: Theory and applications of Hopf bifurcation, (1981)
[31]Engelborghs, K.; Luzyanina, T.; Roose, D.: Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL, ACM trans. Math. software 28, 1-21 (2002) · Zbl 1070.65556 · doi:10.1145/513001.513002
[32]K. Engelborghs, T. Luzyanina, G. Samaey, DDE-BIFTOOL v. 2.00: A Matlab package for bifurcation analysis of delay differential equations, Technical Report TW-330, Department of Computer Science, K. U. Leuven, Leuven, Belgium, 2001.
[33]Kato, H.; Koya, Y.: Oral administration of human T-cell leukemia virus type 1 induces immune unresponsiveness with persistent infection in adult rats, J. virol. 72, 7289-7293 (1998)
[34]Lang, J.; Li, M. Y.: Stable and transient periodic oscillations in a mathematical model for CTL response to HTLV-I infection, J. math. Biol. (2011)