Stability of Hopf bifurcation of a delayed SIRS epidemic model with stage structure

Nonlinear Analysis: Real World Applications

Volumn 11, Issue 1, 2010, Pages 293-306

  Zhang, Tailei ^a,b   Liu, Junli ^a   Teng, Zhidong ^b  

^a XI'AN JIAOTONG UNIVERSITY   (China)

^b XINJIANG UNIVERSITY   (China)

Author keywords

Epidemiology; Periodic solution; SIRS model; Stability; Stability switch; Stage structure

Indexed keywords

ASYMPTOTICALLY STABLE; BIFURCATING PERIODIC SOLUTIONS; BIFURCATION ANALYSIS; BIFURCATION PARAMETER; CENTER MANIFOLD; CRITICAL VALUE; LOCAL BIFURCATIONS; NORMAL FORM THEORY; NUMERICAL SIMULATION; PERIODIC SOLUTION; POSITIVE EQUILIBRIUM; SIRS EPIDEMIC MODEL; SIRS MODEL; STABILITY SWITCH; STAGE STRUCTURE;

EPIDEMIOLOGY; PERIODIC STRUCTURES; SIMULATORS; STABILITY; TIME DELAY;

HOPF BIFURCATION;

EID: 70350716898     PISSN: 14681218     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.nonrwa.2008.10.059     Document Type: Article

References (28)

1. Kermark M., and Mckendrick A. Contributions to the mathematical theory of epidemics, Part I. Proc. Roy. Soc. A. 115 (1927) 700-721
2. Salemi F., Salone V., and Wang W. Stability of a competition model with two-stage structure. Appl. Math. Comput. 99 (1999) 221-231
3. Beretta E., and Kuang Y. Geometric stability switch criteria in delay differential systems with delay dependent parameters. SIAM J. Math. Anal. 33 (2002) 1144-1165
4. Kuang Y. Delay Differential Equations with Applications in Population Dynamics (1993), Academic Press, New York
5. Mode C.J., and Sleeman C.K. A new design of stochastic partnership models for epidemics of sexually transmitted diseases with stages. Math. Biosci. 156 (1999) 95-122
6. Li X., and Wang W. A discrete epidemic model with stage structure. Chaos Solitions Fractals 26 (2005) 947-958
7. Xiao Y., and Chen L. Analysis of a SIS epidemic model with stage structure and a delay. Commun. Nonlinear Sci. Numer. Simul. 6 (2001) 35-39
8. Cooke K., van den Driessche P., and Zou X. Interaction of maturation delay and nonlinear birth in population and epidemic models. J. Math. Biol. 39 (1999) 332-352
9. Beretta E., and Takeuchi Y. Convergence results in SIR epidemic model with varying population sizes. Nonlinear Anal. 28 (1997) 1909-1921
10. Takeuchi Y., Ma W., and Beretta E. Global asymptotic properties of a delay SIR epidemic model with finite incubation times. Nonlinear Anal. 42 (2000) 931-947
11. Hethcote H.W. Qualitative analyses of communicable disease models. Math. Biosci. 7 (1976) 335-356
12. Zhang T., and Teng Z. Global behavior and permanence of SIRS epidemic model with time delay. Nonlinear Anal. RWA 9 (2008) 1409-1424
13. Greenhalgh D., Khan Q.J.A., and Lewis F.I. Recurrent epidemic cycles in an infectious disease model with a time delay in loss of vaccine immunity. Nonlinear Anal. 63 (2005) 779-788
14. Greenhalgh D., Khan Q.J.A., and Lewis F.I. Hopf bifurcation in two SIRS density dependent epidemic models. Math. Comput. Modelling 39 (2004) 1261-1283
15. Greenhalgh D. Effects of heterogeneity on the spread of HIV/AIDS among intravenous drug users in shooting galleries. Math. Biosci. 136 (1996) 141-186
16. Hethcote H.W., Yi L., and Zhujun J. Hopf bifurcation in models for pertussis epidemiology. Math. Comput. Modelling 30 (1999) 29-45
17. Yan X., and Li W. Hopf bifurcation and global periodic solutions in a delayed predator-prey system. Appl. Math. Comput. 177 (2006) 427-445
18. Yan X. Hopf bifurcation and stability for a delayed tri-neuron network model. J. Comput. Appl. Math. 196 (2006) 579-595
19. Yang H., and Tian Y. Hopf bifurcation in REM algorithm with communication delay. Chaos Solitions Fractals 25 (2005) 1093-1105
20. Song Y., and Wei J. Bifurcation analysis for Chen's system with delayed feedback and its application to control of Chaos. Chaos Solitions Fractals 22 (2004) 75-91
21. Yan X., and Zhang C. Hopf bifurcation in a delayed Lotka-Volterra predator-prey system. Nonlinear Anal. RWA 9 (2008) 114-127
22. Hassard B., Kazarino D., and Wan Y. Theory and Application of Hopf Bifurcation (1981), Cambridge University Press, Cambridge
23. Li X., and Wei J. On the zeros of a fourth degree exponential polynomial with applications to a neural network model with delays. Chaos Solitions Fractals 26 (2005) 519-526
24. Dieuonné J. Foundations of Modern Analysis (1960), Academic
25. Hale J.K., and Lunel S. Introdction to Functional Differential Equations (1993), Springer-Verlag, New York
26. Fine P.E.M., and Clarkson J.A. Measles in England and Wales I. An analysis of factors underlying seasonal patterns. Int. J. Epidemiol. 11 (1982) 5-14
27. Fine P.E.M., and 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 (1982) 15-25
28. Fine P.E.M., and Clarkson J.A. Measles in England and Wales III. Assessing published predictions of the impact of vaccination of incidence. Int. J. Epidemiol. 12 (1983) 332-339