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)
A model of HIV-1 infection with two time delays: mathematical analysis and comparison with patient data. (English) Zbl 1241.92042
Summary: Mathematical models have made considerable contributions to our understanding of HIV dynamics. Introducing time delays to HIV models usually brings challenges to both mathematical analysis of the models and comparison of model predictions with patient data. We incorporate two delays, one is the time needed for infected cells to produce virions after viral entry and the other the time needed for the adaptive immune response to emerge to control viral replication, into an HIV-1 model. We begin the model analysis with proving the positivity and boundedness of the solutions, local stability of the infection-free and infected steady states, and uniform persistence of the system. By developing a few Lyapunov functionals, we obtain conditions ensuring global stability of the steady states. We also fit the model including two delays to viral load data from 10 patients during primary HIV-1 infection and estimate parameter values. Although the delay model provides better fits to patient data (achieving a smaller error between data and modeling prediction) than the one without delays, we could not determine which one is better from the statistical standpoint. This highlights the need of more data sets for model verification and selection when we incorporate time delays into mathematical models to study virus dynamics.
MSC:
92C50Medical applications of mathematical biology
34K20Stability theory of functional-differential equations
62P10Applications of statistics to biology and medical sciences
34K60Qualitative investigation and simulation of models
65C20Models (numerical methods)
References:
[1]Adimy, M.; Crauste, F.; Ruan, S.: Periodic oscillations in leukopoiesis models with two delays, J. theor. Biol. 242, 288 (2006)
[2]Banks, H. T.; Bortz, D. M.: A parameter sensitivity methodology in the context of HIV delay equation models, J. math. Biol. 50, 607 (2005) · Zbl 1083.92025 · doi:10.1007/s00285-004-0299-x
[3]Banks, H. T.; Bortz, D. M.; Holte, S. E.: Incorporation of variability into the modeling of viral delays in HIV infection dynamics, Math. biosci. 183, 63 (2003) · Zbl 1011.92037 · doi:10.1016/S0025-5564(02)00218-3
[4]Beretta, E.; Kuang, Y.: Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. anal. 33, 1144 (2002) · Zbl 1013.92034 · doi:10.1137/S0036141000376086
[5]Bonhoeffer, S.; Coffin, J. M.; Nowak, M. A.: Human immunodeficiency virus drug therapy and virus load, J. virol. 71, 3275 (1997)
[6]Bonhoeffer, S.; May, R. M.; Shaw, G. M.; Nowak, M. A.: Virus dynamics and drug therapy, Proc. natl. Acad. sci. USA 94, 6971 (1997)
[7]Carter, C. C.; Onafuwa-Nuga, A.; Mcnamara, L. A.; Riddell, J.; Bixby, D.; Savona, M. R.; Collins, K. L.: HIV-1 infects multipotent progenitor cells causing cell death and establishing latent cellular reservoirs, Nat. med. 16, 446 (2010)
[8]Chun, T. W.; Stuyver, L.; Mizell, S. B.; Ehler, L. A.; Mican, J. A.; Baseler, M.; Lloyd, A. L.; Nowak, M. A.; Fauci, A. S.: Presence of an inducible HIV-1 latent reservoir during highly active antiretroviral therapy, Proc. natl. Acad. sci. USA 94, 13193 (1997)
[9]Ciupe, M. S.; Bivort, B. L.; Bortz, D. M.; Nelson, P. W.: Estimating kinetic parameters from HIV primary infection data through the eyes of three different mathematical models, Math. biosci. 200, 1 (2006) · Zbl 1086.92022 · doi:10.1016/j.mbs.2005.12.006
[10]Cooke, K. L.; Den Driessche, P. Van: Analysis of an SEIRS epidemic model with two delays, J. math. Bio. 35, 240 (1996) · Zbl 0865.92019 · doi:10.1007/s002850050051
[11]Culshaw, R. V.; Ruan, S.: A delay-differential equation model of HIV infection of CD4(+) T-cells, Math. biosci. 165, 27 (2000) · Zbl 0981.92009 · doi:10.1016/S0025-5564(00)00006-7
[12]Culshaw, R. V.; Ruan, S.; Webb, G.: A mathematical model of cell-to-cell spread of HIV-1 that includes a time delay, J. math. Biol. 46, 425 (2003) · Zbl 1023.92011 · doi:10.1007/s00285-002-0191-5
[13]Daar, E. S.; Moudgil, T.; Meyer, R. D.; Ho, D. D.: Transient high levels of viremia in patients with primary human immunodeficiency virus type 1 infection, N. engl. J. med. 324, 961 (1991)
[14]De Boer, R. J.; Perelson, A. S.: Target cell limited and immune control models of HIV infection: a comparison, J. theor. Biol. 190, 201 (1998)
[15]De Leenheer, P.; Smith, H. L.: Virus dynamics: a global analysis, SIAM J. Appl. math. 63, 1313 (2003) · Zbl 1035.34045 · doi:10.1137/S0036139902406905
[16]Dixit, N. M.; Perelson, A. S.: Complex patterns of viral load decay under antiretroviral therapy: influence of pharmacokinetics and intracellular delay, J. theor. Biol. 226, 95 (2004)
[17]Finzi, D.; Hermankova, M.; Pierson, T.; Carruth, L. M.; Buck, C.; Chaisson, R. E.; Quinn, T. C.; Chadwick, K.; Margolick, J.; Brookmeyer, R.; Gallant, J.; Markowitz, M.; Ho, D. D.; Richman, D. D.; Siliciano, R. F.: Identification of a reservoir for HIV-1 in patients on highly active antiretroviral therapy, Science 278, 1295 (1997)
[18]Gu, K.; Niculescu, S. I.; Chen, J.: On stability crossing curves for general systems with two delays, J. of math. Anal. and appl. 311, 231 (2005) · Zbl 1087.34052 · doi:10.1016/j.jmaa.2005.02.034
[19]Hale, J. K.; Lunel, S. M. Verduyn: Introduction to functional differential equations, (1993)
[20]Haynes, B. F.; Pantaleo, G.; Fauci, A. S.: Toward an understanding of the correlates of protective immunity to HIV infection, Science 271, 324 (1996)
[21]Herz, A. V.; Bonhoeffer, S.; Anderson, R. M.; May, R. M.; Nowak, M. A.: Viral dynamics in vivo: limitations on estimates of intracellular delay and virus decay, Proc. natl. Acad. sci. USA 93, 7247 (1996)
[22]Koup, R. A.; Safrit, J. T.; Cao, Y.; Andrews, C. A.; Mcleod, G.; Borkowsky, W.; Farthing, C.; Ho, D. D.: Temporal association of cellular immune responses with the initial control of viremia in primary human immunodeficiency virus type 1 syndrome, J. virol. 68, 4650 (1994)
[23]Kuang, Y.: Delay differential equations with applications in population dynamics, (1993) · Zbl 0777.34002
[24]Li, J.; Kuang, Y.: Analysis of a model of the glucose-insulin regulatory system with two delays, SIAM J. Appl. math. 67, 757 (2007) · Zbl 1115.92015 · doi:10.1137/050634001
[25]Li, J.; Kuang, Y.; Mason, C.: Modeling the glucose-insulin regulatory system and ultradian insulin secretory oscillations with two time delays, J. theor. Biol. 242, 722 (2006)
[26]Li, M. Y.; Shu, H.: Global dynamics of an in-host viral model with intracellular delay, Bull. math. Biol. 72, 1492 (2010) · Zbl 1198.92034 · doi:10.1007/s11538-010-9503-x
[27]Li, M. Y.; Shu, H.: Impact of intracellular delays and target-cell dynamics on in vivo viral infections, SIAM J. Appl. math. 70, 2434 (2010) · Zbl 1209.92037 · doi:10.1137/090779322
[28]Little, S. J.; Mclean, A. R.; Spina, C. A.; Richman, D. D.; Havlir, D. V.: Viral dynamics of acute HIV-1 infection, J. exp. Med. 190, 841 (1999)
[29]Liu, S.; Wang, S.; Wang, L.: Global dynamics of delay epidemic models with nonlinear incidence rate and relapse, Nonlinear anal. Real world app. 12, 119 (2011) · Zbl 1208.34125 · doi:10.1016/j.nonrwa.2010.06.001
[30]Liu, S.; Beretta, E.: Stage-structured predator-prey model with the beddington-deangelis functional response, SIAM J. Appl. math. 66, 1101 (2006) · Zbl 1110.34059 · doi:10.1137/050630003
[31]Liu, W.: Nonlinear oscillation in models of immune response to persistent viruses, Theor. popul. Biol. 52, 224 (1997) · Zbl 0890.92015 · doi:10.1006/tpbi.1997.1334
[32]Mccluskey, C. C.: Global stability for an SEIR epidemiological model with varying infectivity and infinite delay, Math. biosci. And eng. 6, 603 (2009) · Zbl 1190.34108 · doi:10.3934/mbe.2009.6.603
[33]Mellors, J. W.; Jr., C. R. Rinaldo; Gupta, P.; White, R. M.; Todd, J. A.; Kingsley, L. A.: Prognosis in HIV-1 infection predicted by the quantity of virus in plasma, Science 272, 1167 (1996)
[34]Mittler, J. E.; Markowitz, M.; Ho, D. D.; Perelson, A. S.: Improved estimates for HIV-1 clearance rate and intracellular delay, Aids 13, 1415 (1999)
[35]Mittler, J. E.; Sulzer, B.; Neumann, A. U.; Perelson, A. S.: Influence of delayed viral production on viral dynamics in HIV-1 infected patients, Math. biosci. 152, 143 (1998) · Zbl 0946.92011 · doi:10.1016/S0025-5564(98)10027-5
[36]Nelson, P. W.; Murray, J. D.; Perelson, A. S.: A model of HIV-1 pathogenesis that includes an intracellular delay, Math. biosci. 163, 201 (2000) · Zbl 0942.92017 · doi:10.1016/S0025-5564(99)00055-3
[37]Nowak, M. A.; Bangham, C. R.: Population dynamics of immune responses to persistent viruses, Science 272, 74 (1996)
[38]Nowak, M. A.; May, R. M.: Virus dynamics: mathematical principles of immunology and virology, (2000)
[39]Nelson, P. W.; Perelson, A. S.: Mathematical analysis of delay differential equation models of HIV-1 infection, Math. biosci. 179, 73 (2002) · Zbl 0992.92035 · doi:10.1016/S0025-5564(02)00099-8
[40]Perelson, A. S.: Modeling viral and immune system dynamics, Nat. rev. Immunol. 2, 28 (2002)
[41]Perelson, A. S.; Essunger, P.; Cao, Y.; Vesanen, M.; Hurley, A.; Saksela, K.; Markowitz, M.; Ho, D. D.: Decay characteristics of HIV-1-infected compartments during combination therapy, Nature 387, 188 (1997)
[42]Perelson, A. S.; Nelson, P. W.: Mathematical analysis of HIV-1 dynamics in vivo, SIAM rev. 41, 3 (1999) · Zbl 1078.92502 · doi:10.1137/S0036144598335107
[43]Perelson, A. S.; Neumann, A. U.; Markowitz, M.; Leonard, J. M.; Ho, D. D.: HIV-1 dynamics in vivo: virion clearance rate, infected cell life-span, and viral generation time, Science 271, 1582 (1996)
[44]Phillips, A. N.: Reduction of HIV concentration during acute infection: independence from a specific immune response, Science 271, 497 (1996)
[45]Jr., M. Piatak; Saag, M. S.; Yang, L. C.; Clark, S. J.; Kappes, J. C.; Luk, K. C.; Hahn, B. H.; Shaw, G. M.; Lifson, J. D.: High levels of HIV-1 in plasma during all stages of infection determined by competitive PCR, Science 259, 1749 (1993)
[46]Ramratnam, B.; Bonhoeffer, S.; Binley, J.; Hurley, A.; Zhang, L.; Mittler, J. E.; Markowitz, M.; Moore, J. P.; Perelson, A. S.; Ho, D. D.: Rapid production and clearance of HIV-1 and hepatitis C virus assessed by large volume plasma apheresis, Lancet 354, 1782 (1999)
[47]Regoes, R. R.; Wodarz, D.; Nowak, M. A.: Virus dynamics: the effect of target cell limitation and immune responses on virus evolution, J. theor. Biol. 191, 451 (1998)
[48]Rong, L.; Feng, Z.; Perelson, A. S.: Emergence of HIV-1 drug resistance during antiretroviral treatment, Bull. math. Biol. 69, 2027 (2007)
[49]Rong, L.; Perelson, A. S.: Asymmetric division of activated latently infected cells May explain the decay kinetics of the HIV-1 latent reservoir and intermittent viral blips, Math. biosci. 217, 77 (2009) · Zbl 1158.92028 · doi:10.1016/j.mbs.2008.10.006
[50]Rong, L.; Perelson, A. S.: Modeling latently infected cell activation: viral and latent reservoir persistence, and viral blips in HIV-infected patients on potent therapy, Plos comput. Biol. 5, e1000533 (2009)
[51]Rong, L.; Perelson, A. S.: Modeling HIV persistence, the latent reservoir, and viral blips, J. theor. Biol. 260, 308 (2009)
[52]Ruan, S.; Wei, J.: On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, dynamics of continuous, Discrete and impulsive systems 10, 863 (2003) · Zbl 1068.34072
[53]Ruan, S.; Wei, J.: Periodic solutions of planar systems with two delays, Proc. R. Soc. Edinburgh ser. A 129, 1017 (1999) · Zbl 0946.34062 · doi:10.1017/S0308210500031061
[54]Röst, G.; Wu, J.: SEIR epidemiological model with varying infectivity and infinite delay, Math. biosci. And eng. 5, 389 (2008) · Zbl 1165.34421 · doi:10.3934/mbe.2008.5.389 · doi:http://aimsciences.org/journals/redirecting.jsp?paperID=3256
[55]Sedaghat, A. R.; Dinoso, J. B.; Shen, L.; Wilke, C. O.; Siliciano, R. F.: Decay dynamics of HIV-1 depend on the inhibited stages of the viral life cycle, Proc. natl. Acad. sci. USA 105, 4832 (2008)
[56]Sedaghat, A. R.; Siliciano, J. D.; Brennan, T. P.; Wilke, C. O.; Siliciano, R. F.: Limits on replenishment of the resting CD4+T cell reservoir for HIV in patients on HAART, Plos pathog. 3, e122 (2007)
[57]Smith, H.; Zhao, X.: Robust persistence for semidynamical systems, Nonlinear anal. 46, 6169 (2001) · Zbl 1042.37504 · doi:10.1016/S0362-546X(01)00678-2
[58]Smith, H. L.: An introduction to delay differential equations with applications to the life sciences, texts in applied mathematics, (2011)
[59]Stafford, M. A.; Corey, L.; Cao, Y.; Daar, E. S.; Ho, D. D.; Perelson, A. S.: Modeling plasma virus concentration during primary HIV infection, J. theor. Biol. 203, 285 (2000)
[60]Wang, L.; Li, M. Y.: Mathematical analysis of the global dynamics of a model for HIV infection of CD4+T cells, Math. biosci. 200, 44 (2006) · Zbl 1086.92035 · doi:10.1016/j.mbs.2005.12.026
[61]Wang, Y.; Zhou, Y.; Wu, J.; Heffernan, J.: Oscillatory viral dynamics in a delayed HIV pathogenesis model, Math. biosci. 219, 104 (2009) · Zbl 1168.92031 · doi:10.1016/j.mbs.2009.03.003
[62]White, M.; Zhao, X.: Threshold dynamics in a time-delayed epidemic model with dispersal, Math. biosci. 218, 121 (2009) · Zbl 1160.92037 · doi:10.1016/j.mbs.2009.01.004
[63]Wodarz, D.: Killer cell dynamics: mathematical and computational approaches to immunology, (2007)
[64]Wong, J. K.; Hezareh, M.; Günthard, H. F.; Havlir, D. V.; Ignacio, C. C.; Spina, C. A.; Richman, D. D.: Recovery of replication-competent HIV despite prolonged suppression of plasma viremia, Science 278, 1291 (1997)
[65]Zhao, X.: Dynamical systems in population biology, (2003)
[66]Zhu, H.; Zou, X.: Dynamics of a HIV-1 infection model with cell-mediated immune response and intracellular delay, Disc. cont. Dyan. syst. B. 12, 511 (2009) · Zbl 1169.92033 · doi:10.3934/dcdsb.2009.12.511
[67]Zurakowski, R.; Teel, A. R.: A model predictive control based scheduling method for HIV therapy, J. theor. Biol. 238, 368 (2006)