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 of Hopf bifurcation of a delayed SIRS epidemic model with stage structure. (English) Zbl 1195.34130
This paper deals with a SIRS epidemic model with stage structure, formulated as a nonlinear differential system including a discrete time delay. The main goal of this work is to study the effects of the delay on the dynamic behaviour of the system. Taking the delay as a parameter, it is shown that when the delay passes through a critical value, the positive equilibrium of the model loses its stability and a Hopf bifurcation occurs. Previously, existence of such a positive equilibrium is established and its stability is discussed by means of the analysis of the roots of a characteristic equation associated to a linearized equation around the equilibrium. Formulae determining the direction of the Hopf bifurcations and the stability of bifurcating periodic solutions on the center manifold are obtained, by using the normal form theory. The paper ends with some numerical simulations to illustrate the above theoretical results.

34K60Qualitative investigation and simulation of models
34K18Bifurcation theory of functional differential equations
34K13Periodic solutions of functional differential equations
34K20Stability theory of functional-differential equations
34K19Invariant manifolds (functional-differential equations)
34K17Transformation and reduction of functional-differential equations and systems; normal forms
Full Text: DOI
[1] Kermark, M.; Mckendrick, A.: Contributions to the mathematical theory of epidemics, part I, Proc. roy. Soc. A. 115, 700-721 (1927) · Zbl 53.0517.01
[2] Salemi, F.; Salone, V.; Wang, W.: Stability of a competition model with two-stage structure, Appl. math. Comput. 99, 221-231 (1999) · Zbl 0931.92029 · doi:10.1016/S0096-3003(97)10180-1
[3] Beretta, E.; Kuang, Y.: Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. anal. 33, 1144-1165 (2002) · Zbl 1013.92034 · doi:10.1137/S0036141000376086
[4] Kuang, Y.: Delay differential equations with applications in population dynamics, (1993) · Zbl 0777.34002
[5] Mode, Charles J.; Sleeman, Candace K.: A new design of stochastic partnership models for epidemics of sexually transmitted diseases with stages, Math. biosci. 156, 95-122 (1999) · Zbl 0934.92030 · doi:10.1016/S0025-5564(98)10062-7
[6] Li, X.; Wang, W.: A discrete epidemic model with stage structure, Chaos solitions fractals 26, 947-958 (2005) · Zbl 1066.92045 · doi:10.1016/j.chaos.2005.01.063
[7] Xiao, Y.; Chen, L.: Analysis of a SIS epidemic model with stage structure and a delay, Commun. nonlinear sci. Numer. simul. 6, 35-39 (2001) · Zbl 1040.34096 · doi:10.1016/S1007-5704(01)90026-7
[8] Cooke, K.; Den Driessche, P. Van; Zou, X.: Interaction of maturation delay and nonlinear birth in population and epidemic models, J. math. Biol. 39, 332-352 (1999) · Zbl 0945.92016 · doi:10.1007/s002850050194
[9] Beretta, E.; Takeuchi, Y.: Convergence results in SIR epidemic model with varying population sizes, Nonlinear anal. 28, 1909-1921 (1997) · Zbl 0879.34054 · doi:10.1016/S0362-546X(96)00035-1
[10] Takeuchi, Y.; Ma, W.; Beretta, E.: Global asymptotic properties of a delay SIR epidemic model with finite incubation times, Nonlinear anal. 42, 931-947 (2000) · Zbl 0967.34070 · doi:10.1016/S0362-546X(99)00138-8
[11] Hethcote, H. W.: Qualitative analyses of communicable disease models, Math. biosci. 7, 335-356 (1976) · Zbl 0326.92017 · doi:10.1016/0025-5564(76)90132-2
[12] Zhang, T.; Teng, Z.: Global behavior and permanence of SIRS epidemic model with time delay, Nonlinear anal. RWA 9, 1409-1424 (2008) · Zbl 1154.34390 · doi:10.1016/j.nonrwa.2007.03.010
[13] Greenhalgh, D.; Khan, Q. J. A.; Lewis, F. I.: Recurrent epidemic cycles in an infectious disease model with a time delay in loss of vaccine immunity, Nonlinear anal. 63, 779-788 (2005) · Zbl 1222.92055 · doi:10.1016/j.na.2004.12.018
[14] Greenhalgh, D.; Khan, Q. J. A.; Lewis, F. I.: Hopf bifurcation in two SIRS density dependent epidemic models, Math. comput. Modelling 39, 1261-1283 (2004) · Zbl 1065.92042 · doi:10.1016/j.mcm.2004.06.007
[15] Greenhalgh, D.: Effects of heterogeneity on the spread of HIV/AIDS among intravenous drug users in shooting galleries, Math. biosci. 136, 141-186 (1996) · Zbl 0864.92015 · doi:10.1016/0025-5564(96)00063-6
[16] Hethcote, H. W.; Yi, Li; Zhujun, Jing: Hopf bifurcation in models for pertussis epidemiology, Math. comput. Modelling 30, 29-45 (1999) · Zbl 1043.92525 · doi:10.1016/S0895-7177(99)00196-X
[17] Yan, X.; Li, W.: Hopf bifurcation and global periodic solutions in a delayed predator--prey system, Appl. math. Comput. 177, 427-445 (2006) · Zbl 1090.92052 · doi:10.1016/j.amc.2005.11.020
[18] Yan, X.: Hopf bifurcation and stability for a delayed tri-neuron network model, J. comput. Appl. math. 196, 579-595 (2006) · Zbl 1175.37086 · doi:10.1016/j.cam.2005.10.012
[19] Yang, H.; Tian, Y.: Hopf bifurcation in REM algorithm with communication delay, Chaos solitions fractals 25, 1093-1105 (2005) · Zbl 1198.93099 · doi:10.1016/j.chaos.2004.11.085
[20] Song, Y.; Wei, J.: Bifurcation analysis for Chen’s system with delayed feedback and its application to control of chaos, Chaos solitions fractals 22, 75-91 (2004) · Zbl 1112.37303
[21] Yan, X.; Zhang, C.: Hopf bifurcation in a delayed Lotka-Volterra predator--prey system, Nonlinear anal. RWA 9, 114-127 (2008) · Zbl 1149.34048 · doi:10.1016/j.nonrwa.2006.09.007
[22] Hassard, B.; Kazarino, D.; Wan, Y.: Theory and application of Hopf bifurcation, (1981)
[23] Li, X.; Wei, J.: On the zeros of a fourth degree exponential polynomial with applications to a neural network model with delays, Chaos solitions fractals 26, 519-526 (2005) · Zbl 1098.37070 · doi:10.1016/j.chaos.2005.01.019
[24] Dieuonné, J.: Foundations of modern analysis, (1960) · Zbl 0100.04201
[25] Hale, J. K.; Lunel, S.: Introdction to functional differential equations, (1993) · Zbl 0787.34002
[26] Fine, P. E. M.; Clarkson, J. A.: Measles in england and wales I. An analysis of factors underlying seasonal patterns, Int. J. Epidemiol. 11, 5-14 (1982)
[27] Fine, P. E. M.; Clarkson, J. A.: Measles in england and wales II. The impact of the measles vaccination programme on the distribution of immunity in the population, Int. J. Epidemiol. 11, 15-25 (1982)
[28] Fine, P. E. M.; Clarkson, J. A.: Measles in england and wales III. Assessing published predictions of the impact of vaccination of incidence, Int. J. Epidemiol. 12, 332-339 (1983)