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)
Bifurcation analysis of a delayed epidemic model. (English) Zbl 1186.92042
Summary: Hopf bifurcation for a delayed SIS epidemic model with stage structure and nonlinear incidence rate is investigated. Through theoretical analysis, we show the positive equilibrium stability and the conditions that Hopf bifurcations occur. Applying the normal form theory and a center manifold argument, we derive explicit formulas determining the properties of the bifurcating periodic solutions. In addition, we also study the inhibition effect on the properties of the bifurcating periodic solutions. To illustrate our theoretical analysis, some numerical simulations are also included.
MSC:
92D30Epidemiology
34K18Bifurcation theory of functional differential equations
34K13Periodic solutions of functional differential equations
65C60Computational problems in statistics
References:
[1]Wang, W. D.; Ruan, S. G.: Bifurcation in an epidemic model with constant removal rate of the infectives, J. math. Anal. appl. 291, 775-793 (2004) · Zbl 1054.34071 · doi:10.1016/j.jmaa.2003.11.043
[2]Liu, L.; Li, X. G.; Zhuang, K. J.: Bifurcation analysis on a delayed SIS epidemic model with stage structure, Electron. J. Differ. equat. 2007, 1-17 (2007) · Zbl 1140.34427 · doi:emis:journals/EJDE/Volumes/2007/77/abstr.html
[3]Zhang, T. L.; Liu, J. L.; Teng, Z. D.: Bifurcation analysis of a delayed SIS epidemic model with stage structure, Chaos solitons fract. 40, 563-576 (2009) · Zbl 1197.37128 · doi:10.1016/j.chaos.2007.08.004
[4]Zhang, T. L.; Liu, J. L.; Teng, Z. D.: Stability of Hopf bifurcation of a delayed SIRS epidemic model with stage structure, Nonlinear anal. 11, 293-306 (2010) · Zbl 1195.34130 · doi:10.1016/j.nonrwa.2008.10.059
[5]Jia, J. W.; Li, Q. Y.: Qualitative analysis of an SIR epidemic model with stage structure, Appl. math. Comput. 193, 106-115 (2007) · Zbl 1193.34113 · doi:10.1016/j.amc.2007.03.041
[6]Xiao, Y. N.; Chen, L. S.: 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
[7]Cooke, K.; Driessche, P. V. D.; 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
[8]Capasso, V.; Serio, G.: A generalization of the kermack – mckendrick deterministic epidemic model, Math. biosci. 42, 41-61 (1978) · Zbl 0398.92026 · doi:10.1016/0025-5564(78)90006-8
[9]Xiao, D. M.; Ruan, S. G.: Global analysis of an epidemic model with nonmonotone incidence rate, Math. biosci. 208, 419-429 (2007) · Zbl 1119.92042 · doi:10.1016/j.mbs.2006.09.025
[10]Ruan, S. G.; Wang, W. D.: Dynamical behavior of an epidemic model with a nonlinear incidence rate, J. differ. Equat. 188, 135-163 (2003) · Zbl 1028.34046 · doi:10.1016/S0022-0396(02)00089-X
[11]Xu, R.; Ma, Z. E.: Global stability of a SIR epidemic model with nonlinear incidence rate and time delay, Nonlinear anal. 10, 3175-3189 (2009) · Zbl 1183.34131 · doi:10.1016/j.nonrwa.2008.10.013
[12]Zhang, J.; Ma, Z. E.: Global dynamics of an SEIR epidemic model with saturating contact rate, Math. biosci. 185, 15-32 (2003) · Zbl 1021.92040 · doi:10.1016/S0025-5564(03)00087-7
[13]Cooke, K. L.: Stability analysis for a vector disease model, Rocky mountain J. Math. 9, 31-42 (1979) · Zbl 0423.92029 · doi:10.1216/RMJ-1979-9-1-31
[14]Hassard, B. D.; Kazarinoffand, N. D.; Wan, Y. H.: Theory and application of Hopf bifurcation, (1981)
[15]Song, Y. L.; Yuan, S. L.: Bifurcation analysis in a predator – prey system with time delay, Nonlinear anal. 7, 265-284 (2006) · Zbl 1085.92052 · doi:10.1016/j.nonrwa.2005.03.002
[16]Dieudonne, J. A.: Foundations of modern analysis, (1960) · Zbl 0100.04201