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 stability for an HIV-1 infection model including an eclipse stage of infected cells. (English) Zbl 1223.92024
Summary: We consider the mathematical model for the viral dynamics of HIV-1 introduced by L. Rong et al. [J. Theor. Biol. 247, 804–818 (2007); see also Bull. Math. Biol. 69, No. 6, 2027–2060 (2007; Zbl 05265722)]. One main feature of this model is that an eclipse stage for the infected cells is included and cells in this stage may revert to the uninfected class. The viral dynamics is described by four nonlinear ordinary differential equations. Rong et al. have analyzed the stability of the infected equilibrium locally. We perform a global stability analysis using two techniques, the Lyapunov direct method and the geometric approach to stability, based on a higher-order generalization of Bendixson’s criterion. We obtain sufficient conditions in terms of the system parameters. Numerical simulations are also provided to give a more complete representation of the system dynamics.
MSC:
92C50Medical applications of mathematical biology
34D23Global stability of ODE
34C60Qualitative investigation and simulation of models (ODE)
65C20Models (numerical methods)
References:
[1]Althaus, C. L.; De Vos, A. S.; De Boer, R. J.: Reassessing the human immunodeficiency virus type 1 life cycle through age-structured modeling: life span of infected cells, viral generation time, and basic reproductive number, R0, J. virol. 83, 7659-7667 (2009)
[2]Althaus, C. L.; De Boer, R. J.: Implications of CTL-mediated Killing of HIV-infected cells during the non-productive stage of infection, Plos ONE 6, No. 2, e16468 (2011)
[3]Anderson, R. M.; May, R. M.: Infectious diseases in humans: dynamics and control, (1991)
[4]Arino, J.; Mccluskey, C. C.; Den Driessche, P. Van: Global results for an epidemic model with vaccination that exhibits backward bifurcation, SIAM J. Appl. math. 64, 260-276 (2003) · Zbl 1034.92025 · doi:10.1137/S0036139902413829
[5]Ballyk, M. M.; Mccluskey, C. C.; Wolkowicz, G. K. S.: Global analysis of competition for perfectly substitutable resources with linear response, J. math. Biol. 51, 458-490 (2005) · Zbl 1090.92045 · doi:10.1007/s00285-005-0333-7
[6]Beretta, E.; Solimano, F.; Takeuchi, Y.: Negative criteria for the existence of periodic solutions in a class of delay-differential equations, Nonlinear anal. 50, 941-966 (2002) · Zbl 1087.34542 · doi:10.1016/S0362-546X(01)00794-5
[7]Buonomo, B.; D’onofrio, A.; Lacitignola, D.: Global stability of an SIR epidemic model with information dependent vaccination, Math. biosci. 216, 9-16 (2008) · Zbl 1152.92019 · doi:10.1016/j.mbs.2008.07.011
[8]Buonomo, B.; Lacitignola, D.: General conditions for global stability in a single species population-toxicant model, Nonlinear anal. Real world appl. 5, 749-762 (2004) · Zbl 1074.92036 · doi:10.1016/j.nonrwa.2004.05.002
[9]Buonomo, B.; Lacitignola, D.: On the use of the geometric approach to global stability for three-dimensional ODE systems: a bilinear case, J. math. Anal. appl. 348, 255-266 (2008) · Zbl 1158.34033 · doi:10.1016/j.jmaa.2008.07.021
[10]Buonomo, B.; Lacitignola, D.: On the dynamics of an SEIR epidemic model with a convex incidence rate, Ric. mat. 57, 261-281 (2008) · Zbl 1232.34061 · doi:10.1007/s11587-008-0039-4
[11]Buonomo, B.; Lacitignola, D.: Analysis of a tuberculosis model with a case study in uganda, J. biol. Dyn. 4, 571-593 (2010)
[12]Buonomo, B.; Lacitignola, D.: Global stability for a four dimensional epidemic model, Note mat. 30, 81-93 (2010)
[13]Dieckmann, O.; Heesterbeek, J. A. P.: Mathematical epidemiology of infectious diseases, Wiley series in mathematical and computational biology (2000)
[14]Freedman, H. I.; Ruan, S.; Tang, M.: Uniform persistence and flows near a closed positively invariant set, J. differential equations 6, 583-600 (1994) · Zbl 0811.34033 · doi:10.1007/BF02218848
[15]Funk, G. A.; Fischer, M.; Joos, B.; Opravil, M.; Guenthard, H. F.; Ledergerber, B.; Bonhoeffer, S.: Quantification of in vivo replicative capacity of HIV-1 in different compartments of infected cells, J. acquir. Immune defic. Syndr. 26, 397-404 (2001)
[16]Gumel, A. B.; Mccluskey, C. C.; Watmough, J.: Modelling the potential impact of a SARS vaccine, Math. biosci. Eng. 3, 485-512 (2006)
[17]Hutson, V.; Schmitt, K.: Permanence and the dynamics of biological systems, Math. biosci. 111, 1-71 (1992) · Zbl 0783.92002 · doi:10.1016/0025-5564(92)90078-B
[18]Korobeinikov, A.: Global properties of basic virus dynamics models, Bull. math. Biol. 66, 879-883 (2004)
[19]Korobeinikov, A.: Lyapunov functions and global properties for SEIR and SEIS epidemic models, IMA J. Math. appl. Med. biol. 21, 75-83 (2004) · Zbl 1055.92051 · doi:10.1093/imammb/21.2.75
[20]Korobeinikov, A.; Wake, G. C.: Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models, Appl. math. Lett. 15, 955-960 (2002) · Zbl 1022.34044 · doi:10.1016/S0893-9659(02)00069-1
[21]Li, M. Y.; Graef, J. R.; Wang, L.; Karsai, J.: Global stability of a SEIR model with varying total population size, Math. biosci. 160, 191-213 (1999) · Zbl 0974.92029 · doi:10.1016/S0025-5564(99)00030-9
[22]Li, M. Y.; Muldowney, J. S.: On Bendixson’s criterion, J. differential equations 106, 27-39 (1993) · Zbl 0786.34033 · doi:10.1006/jdeq.1993.1097
[23]Li, M. Y.; Muldowney, J. S.: On R.A. Smith’s autonomous convergence theorem, Rocky mountain J. Math. 25, 365-379 (1995) · Zbl 0841.34052 · doi:10.1216/rmjm/1181072289
[24]Li, M. Y.; Muldowney, J. S.: Global stability for the SEIR model in epidemiology, Math. biosci. 125, 155-164 (1995) · Zbl 0821.92022 · doi:10.1016/0025-5564(95)92756-5
[25]Li, M. Y.; Muldowney, J. S.: A geometric approach to global-stability problems, SIAM J. Math. anal. 27, 1070-1083 (1996) · Zbl 0873.34041 · doi:10.1137/S0036141094266449
[26]Li, M. Y.; Smith, H. L.; Wang, L.: Global dynamics of an SEIR epidemic model with vertical transmission, SIAM J. Math. anal. 62, 58-69 (2001) · Zbl 0991.92029 · doi:10.1137/S0036139999359860
[27]Lyapunov, A. M.: The general problem of the stability of motion, (1992) · Zbl 0786.70001
[28]Margheri, A.; Rebelo, C.: Some examples of persistence in epidemiological models, J. math. Biol. 46, 564-570 (2003) · Zbl 1023.92027 · doi:10.1007/s00285-002-0193-3
[29]Jr., R. H. Martin: Logarithmic norms and projections applied to linear differential systems, J. math. Anal. appl. 45, 432-454 (1974) · Zbl 0293.34018 · doi:10.1016/0022-247X(74)90084-5
[30]Mccluskey, C. C.: A strategy for constructing Lyapunov functions for non-autonomous linear differential equations, Linear algebra appl. 409, 100-110 (2005) · Zbl 1076.37012 · doi:10.1016/j.laa.2005.04.006
[31]Mccluskey, C. C.: Global stability for a class of mass action systems allowing for latency in tuberculosis, J. math. Anal. appl. 338, 518-535 (2008) · Zbl 1131.92042 · doi:10.1016/j.jmaa.2007.05.012
[32]Nelson, P. W.; Gilchrist, M. A.; Coombs, D.; Hyman, J. M.; Perelson, A. S.: An age structured model of HIV infection that allows for variations in the production rate of viral particles and the death rate of productively infected cells, Math. biosci. Eng. 1, 267-288 (2004) · Zbl 1060.92038 · doi:10.3934/mbe.2004.1.267
[33]Nowak, M. A.; May, R. M.: Virus dynamics: mathematical principles of immunology and virology, (2000)
[34]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, 1582-1586 (1996)
[35]Perelson, A. S.; Nelson, P. W.: Mathematical analysis of HIV-1 dynamics in vivo, SIAM rev. 41, 3-44 (1999) · Zbl 1078.92502 · doi:10.1137/S0036144598335107
[36]Perelson, A. S.: Modelling viral and immune system dynamics, Nat. rev. Immunol. 2, 28-36 (2002)
[37]Rong, L.; Gilchrist, M. A.; Feng, Z.; Perelson, A. S.: Modeling within-host HIV-1 dynamics and the evolution of drug resistance: trade-offs between viral enzyme function and drug susceptibility, J. theoret. Biol. 247, 804-818 (2007)
[38]Rong, L.; Feng, Z.; Perelson, A. S.: Mathematical analysis of age-structured HIV-1 dynamics with combination antiretroviral therapy, SIAM J. Appl. math. 67, 731-756 (2007) · Zbl 1121.92043 · doi:10.1137/060663945
[39]Tumwiine, J.; Mugisha, J. Y. T.; Luboobi, L. S.: A host-vector model for malaria with infective immigrants, J. math. Anal. appl. 361, 139-149 (2010) · Zbl 1176.92045 · doi:10.1016/j.jmaa.2009.09.005
[40]Den Driessche, P. Van; Watmough, J.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. biosci. 180, 29-48 (2002) · Zbl 1015.92036 · doi:10.1016/S0025-5564(02)00108-6
[41]Vargas-De-León, C.: Constructions of Lyapunov functions for classic SIS, SIR and SIRS epidemic models with variable population size, Foro-red-mat: revista electrónica de contenido matemático 26 (2009)
[42]C. Vargas-De-León, Analysis of a model for the dynamics of hepatitis B with noncytolytic loss of infected cells, World J. Model. Simul., submitted for publication.
[43]Wang, L.; Li, M. Y.: Mathematical analysis of the global dynamics of a model for HIV infection of CD4+T cells, Math. biosci. 200, 44-57 (2006) · Zbl 1086.92035 · doi:10.1016/j.mbs.2005.12.026