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 analysis of a delayed SIR epidemic model with stage structure and nonlinear incidence. (English) Zbl 1183.34130
Summary: We investigate the stability of an SIR epidemic model with stage structure and time delay. By analyzing the eigenvalues of the corresponding characteristic equation, the local stability of each feasible equilibrium of the model is established. By using comparison arguments, we prove that if the basic reproduction number is less than 1, the disease free equilibrium is globally asymptotically stable. When the basic reproduction number is greater than 1, sufficient conditions are derived for the global stability of an endemic equilibrium of the model. Numerical simulations are carried out to illustrate the theoretical results.

34K60Qualitative investigation and simulation of models
34K20Stability theory of functional-differential equations
34K21Stationary solutions of functional-differential equations
Full Text: DOI EuDML
[1] Z. Ma, Y. Zhou, W. Wang, and Z. Jin, Mathematical Modeling and Research of Epidemic Dynamical System, Science Press, Beijing, China, 2004.
[2] E. Beretta, T. Hara, W. Ma, and Y. Takeuchi, “Global asymptotic stability of an SIR epidemic model with distributed time delay,” Nonlinear Analysis: Theory, Methods & Applications, vol. 47, no. 6, pp. 4107-4115, 2001. · Zbl 1042.34585 · doi:10.1016/S0362-546X(01)00528-4
[3] Z. Jin, Z. E. Ma, and S. L. Yuan, “A SIR epidemic model with varying population size,” Journal of Engineering Mathematics, vol. 20, no. 3, pp. 93-98, 2003. · Zbl 1162.92321
[4] G. Pang and L. Chen, “A delayed SIRS epidemic model with pulse vaccination,” Chaos, Solitons & Fractals, vol. 34, no. 5, pp. 1629-1635, 2007. · Zbl 1152.34379 · doi:10.1016/j.chaos.2006.04.061
[5] S. Ruan and W. Wang, “Dynamical behavior of an epidemic model with a nonlinear incidence rate,” Journal of Differential Equations, vol. 188, no. 1, pp. 135-163, 2003. · Zbl 1028.34046 · doi:10.1016/S0022-0396(02)00089-X
[6] Y. Takeuchi, W. Ma, and E. Beretta, “Global asymptotic properties of a delay SIR epidemic model with finite incubation times,” Nonlinear Analysis: Theory, Methods & Applications, vol. 42, no. 6, pp. 931-947, 2000. · Zbl 0967.34070 · doi:10.1016/S0362-546X(99)00138-8
[7] W. Wang, G. Mulone, F. Salemi, and V. Salone, “Permanence and stability of a stage-structured predator-prey model,” Journal of Mathematical Analysis and Applications, vol. 262, no. 2, pp. 499-528, 2001. · Zbl 0997.34069 · doi:10.1006/jmaa.2001.7543
[8] Y. Xiao and L. Chen, “Modeling and analysis of a predator-prey model with disease in the prey,” Mathematical Biosciences, vol. 171, no. 1, pp. 59-82, 2001. · Zbl 0978.92031 · doi:10.1016/S0025-5564(01)00049-9
[9] Y. Xiao, L. Chen, and F. ven den Bosch, “Dynamical behavior for a stage-structured SIR infectious disease model,” Nonlinear Analysis: Real World Applications, vol. 3, no. 2, pp. 175-190, 2002. · Zbl 1007.92032 · doi:10.1016/S1468-1218(01)00021-9
[10] C. D. Yuan and B. A. Hu, “A SI epidemic model with two-stage structure,” Acta Mathematicae Applicatae Sinica, vol. 25, no. 2, pp. 193-203, 2002. · Zbl 1001.92042
[11] T. Zhang and Z. Teng, “Global asymptotic stability of a delayed SEIRS epidemic model with saturation incidence,” Chaos, Solitons & Fractals, vol. 37, no. 5, pp. 1456-1468, 2008. · Zbl 1142.34384 · doi:10.1016/j.chaos.2006.10.041
[12] V. Capasso and G. Serio, “A generalization of the Kermack-McKendrick deterministic epidemic model,” Mathematical Biosciences, vol. 42, no. 1-2, pp. 43-61, 1978. · Zbl 0398.92026 · doi:10.1016/0025-5564(78)90006-8
[13] D. Xiao and S. Ruan, “Global analysis of an epidemic model with nonmonotone incidence rate,” Mathematical Biosciences, vol. 208, no. 2, pp. 419-429, 2007. · Zbl 1119.92042 · doi:10.1016/j.mbs.2006.09.025
[14] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, vol. 191 of Mathematics in Science and Engineering, Academic Press, Boston, Mass, USA, 1993. · Zbl 0777.34002
[15] R. Xu and Z. Ma, “The effect of dispersal on the permanence of a predator-prey system with time delay,” Nonlinear Analysis: Real World Applications, vol. 9, no. 2, pp. 354-369, 2008. · Zbl 1142.34055 · doi:10.1016/j.nonrwa.2006.11.004
[16] X. Song and L. Chen, “Optimal harvesting and stability for a two-species competitive system with stage structure,” Mathematical Biosciences, vol. 170, no. 2, pp. 173-186, 2001. · Zbl 1028.34049 · doi:10.1016/S0025-5564(00)00068-7