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)
Stability and Hopf bifurcation in a viral infection model with nonlinear incidence rate and delayed immune response. (English) Zbl 1239.92071
Summary: A viral infection model with nonlinear incidence rate and delayed immune response is investigated. It is shown that if the basic reproduction ratio of the virus is less than unity, and the infection-free equilibrium is globally asymptotically stable. By analyzing the characteristic equation, the local stability of the chronic infection equilibrium of the system is discussed. Furthermore, the existence of Hopf bifurcations at the chronic infection equilibrium is also studied. By means of an iteration technique, sufficient conditions are obtained for the global attractiveness of the chronic infection equilibrium. Numerical simulations are carried out to illustrate the main results.

92C60Medical epidemiology
34K20Stability theory of functional-differential equations
34K18Bifurcation theory of functional differential equations
65C20Models (numerical methods)
Full Text: DOI
[1] Burić, N.; Mudrinic, M.; Vasović, N.: Time delay in a basic model of the immune response, Chaos solitons fract 12, 483-489 (2001) · Zbl 1026.92015 · doi:10.1016/S0960-0779(99)00205-2
[2] Canabarro, A. A.; Gléria, I. M.; Lyra, M. L.: Periodic solutions and chaos in a non-linear model for the delayed cellular immune response, Physica A 342, 234-241 (2004)
[3] Culshaw, R. V.; Ruan, S.; Webb, G.: A mathematical model of cell-to-cell HIV-1 that include a time delay, J math biol 46, 425-444 (2003) · Zbl 1023.92011 · doi:10.1007/s00285-002-0191-5
[4] Gumel, A. B.; Moghadas, S. M.: HIV control in vivo: dynamical analysis, Commun nonlinear sci numer simul 9, 561-568 (2004) · Zbl 1121.92300 · doi:10.1016/S1007-5704(03)00003-0
[5] Hale, J. K.: Theory of functional differential equations, (1997) · Zbl 1098.34552
[6] Ji, Y.; Min, L. Q.; Zheng, Y.; Su, Y. M.: A viral infection model with periodic immune response and nonlinear CTL response, Math comput simulat 80, 2309-2316 (2010) · Zbl 1195.92037 · doi:10.1016/j.matcom.2010.04.029
[7] Leenheer, P. D.; Smith, H. L.: Virus dynamic: a global analysis, SIAM J appl math 63, 1313-1327 (2003) · Zbl 1035.34045 · doi:10.1137/S0036139902406905
[8] Nelson, P. W.; Perelson, A. S.: Mathematical analysis of a delay differential equation models of HIV-1 infection, Math biosci 179, 73-94 (2002) · Zbl 0992.92035 · doi:10.1016/S0025-5564(02)00099-8
[9] Nowak, M. A.; Bangham, C. R. M.: Population dynamics of immune responses to persistent viruses, Science 272, 74-79 (1996)
[10] Nowak, M. A.; May, R. M.: Virus dynamics: mathematical principles of immunology and virology, (2000) · Zbl 1101.92028
[11] 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
[12] Revilla, T.; Garci-Ramos, G.: Fighting a virus with a virus: a dynamic model for HIV-1 therapy, Math biosci 185, 191-203 (2003) · Zbl 1021.92015 · doi:10.1016/S0025-5564(03)00091-9
[13] Song, X. Y.; Wang, S. L.; Dong, J.: Stability properties and Hopf bifurcation of a delayed viral infection model with lytic immune response, J math anal appl 373, 345-355 (2011) · Zbl 1208.34128 · doi:10.1016/j.jmaa.2010.04.010
[14] Wang, K. F.; Wang, W. D.; Pang, H. Y.; Liu, X. N.: 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
[15] Wodarz, D.: Hepatitis C virus dynamics and pathology: the role of CTL and antibody response, J gen virol 84, 1743-1750 (2003)
[16] Wodarz, D.; Christensen, J. P.; Thomsen, A. R.: The importance of lytic and nonlytic immune responses in viral infections, Trends immunol 23, 194-200 (2002)
[17] Wodarz, D.; Krakauer, D. C.: Defining CTL-induced pathology: implications for HIV, Virology 274, 94-104 (2000)
[18] Xu, R.; Ma, Z. E.: Stability and Hopf bifurcation in a ratio-dependent predator-prey system with stage structure, Chaos solitons fract 38, 669-684 (2008) · Zbl 1146.34323 · doi:10.1016/j.chaos.2007.01.019