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 of a delayed SIRS epidemic model with saturation incidence and temporary immunity. (English) Zbl 1193.34115
Summary: In this paper, a delayed SIRS epidemic model with saturation incidence and temporary immunity is investigated. The immunity gained by experiencing a disease is temporary, whenever infected the diseased individuals will return to the susceptible class after a fixed period. By analyzing the corresponding characteristic equations, the local stability of an endemic equilibrium and a disease-free equilibrium is discussed. By comparison arguments, it is proved that if the basic reproduction number is less than unity, the disease-free equilibrium is globally asymptotically stable. If the basic reproduction number is greater than unity, by means of an iteration technique, sufficient conditions are obtained for the global asymptotic stability of the endemic equilibrium. Numerical simulations are carried out to illustrate the main theoretical results.
MSC:
34D23Global stability of ODE
92D30Epidemiology
34K20Stability theory of functional-differential equations
WorldCat.org
Full Text: DOI
References:
[1] Beretta, E.; Hara, T.; Ma, W.; Takeuchi, Y.: Global asymptotic stability of an SIR epidemic model with distributed time delay, Nonlinear anal. 47, 4107-4115 (2001) · Zbl 1042.34585 · doi:10.1016/S0362-546X(01)00528-4
[2] Beretta, E.; Takeuchi, Y.: Global stability of an SIR epidemic model with time delays, J. math. Biol. 33, 250-260 (1995) · Zbl 0811.92019 · doi:10.1007/BF00169563
[3] 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
[4] 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
[5] Gao, S.; Chen, L.; Nieto, J.; Torres, A.: Analysis of a delayed epidemic model with pulse vaccination and saturation incidence, Vaccine 24, 6037-6045 (2006)
[6] Kyrychkoa, Y. N.; Blyuss, K. B.: Global properties of a delayed SIR model with temporary immunity and nonlinear incidence rate, Nonlinear anal. Real world appl. 6, 495-507 (2005) · Zbl 1144.34374 · doi:10.1016/j.nonrwa.2004.10.001
[7] Ma, W.; Song, M.; Takeuchi, Y.: Global stability of an SIR epidemic model with time delay, Appl. math. Lett. 17, 1141-1145 (2004) · Zbl 1071.34082 · doi:10.1016/j.aml.2003.11.005
[8] Ma, W.; Takeuchi, Y.; Hara, T.; Beretta, E.: Permanence of an SIR epidemic model with distributed time delays, Tohoku math. J. 54, 581-591 (2002) · Zbl 1014.92033 · doi:10.2748/tmj/1113247650
[9] Takeuchi, Y.; Ma, W.: Stability analysis on a delayed SIR epidemic model with density dependent birth process, Dyn. contin. Discrete impuls. Syst. 5, 171-184 (1999) · Zbl 0937.92026
[10] Takeuchi, Y.; Ma, W.; Beretta, E.: Global asymptotic properties of a SIR epidemic model with finite incubation time, Nonlinear anal. 42, 931-947 (2000) · Zbl 0967.34070 · doi:10.1016/S0362-546X(99)00138-8
[11] Wen, L.; Yang, X.: Global stability of a delayed SIRS model with temporary immunity, Chaos solitons fractals 38, 221-226 (2008) · Zbl 1142.34354 · doi:10.1016/j.chaos.2006.11.010
[12] Zhang, T.; Teng, Z.: Global behavior and permanence of SIRS epidemic model with time delay, Nonlinear anal. Real world appl. 9, 1409-1424 (2008) · Zbl 1154.34390 · doi:10.1016/j.nonrwa.2007.03.010
[13] Brauer, F.; Den Driessche, P. Van; Wang, L.: Oscillations in a patchy environment disease model, Math. biosci. 215, 1-10 (2008) · Zbl 1176.34098 · doi:10.1016/j.mbs.2008.05.001
[14] 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
[15] Liu, W. M.; Hethcote, H. W.; Levin, S. A.: Dynamical behavior of epidemiological models with nonlinear incidence rates, J. math. Biol. 25, 359-380 (1987) · Zbl 0621.92014 · doi:10.1007/BF00277162
[16] Liu, W. M.; Levin, S. A.; Iwasa, Y.: Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J. math. Biol. 23, 187-204 (1986) · Zbl 0582.92023 · doi:10.1007/BF00276956
[17] Hale, J.: Theory of functional differential equations, (1977) · Zbl 0352.34001
[18] 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