×
Jin, Yu; Wang, Wendi; Xiao, Shiwu

An SIRS model with a nonlinear incidence rate. (English) Zbl 1152.34339

Chaos Solitons Fractals 34, No. 5, 1482-1497 (2007).
Summary: The global dynamics of an SIRS model with a nonlinear incidence rate is investigated. We establish a threshold for a disease to be extinct or endemic, analyze the existence and asymptotic stability of equilibria, and verify the existence of bistable states, i.e., a stable disease free equilibrium and a stable endemic equilibrium or a stable limit cycle. In particular, we find that the model admits stability switches as a parameter changes. We also investigate the backward bifurcation, the Hopf bifurcation and Bogdanov-Takens bifurcation and obtain the Hopf bifurcation criteria and Bogdanov-Takens bifurcation curves, which are important for making strategies for controlling a disease.

Cited in 55 Documents

MSC:

34C23 Bifurcation theory for ordinary differential equations
92D30 Epidemiology
37N25 Dynamical systems in biology
PDF BibTeX XML Cite
\textit{Y. Jin} et al., Chaos Solitons Fractals 34, No. 5, 1482--1497 (2007; Zbl 1152.34339)
Full Text: DOI

References:

[1] Alexander, M. E.; Moghadas, S. M., Periodicity in an epidemic model with a generalized non-linear incidence, Math Biosci, 189, 75-96 (2004) · Zbl 1073.92040
[2] Bogdanov, R. I., Bifurcations of a limit cycle for a family of vector fields on the plan, Selecta Math Soviet, 1, 373-388 (1981) · Zbl 0518.58029
[3] Bogdanov, R. I., Versal deformations of a singular point on the plane in the case of zero eigen-values, Selecta Math Soviet, 1, 389-421 (1981) · Zbl 0518.58030
[4] Brauer, F.; van den Driessche, P., Models for translation of disease with immigration of infectives, Math Biosci, 171, 143-154 (2001) · Zbl 0995.92041
[5] Capasso, V.; Serio, G., A generalization of the Kermack-Mckendrick deterministic epidemic model, Math Biosci, 42, 43-61 (1978) · Zbl 0398.92026
[6] Derrick, W. R.; van den Driessche, P., A disease transmission model in a nonconstant population, J Math Biol, 31, 495-512 (1993) · Zbl 0772.92015
[7] Derrick, W. R.; van den Driessche, P., Homoclinic orbits in a disease transmission model with nonlinear incidence and nonconstant population, Discret Contin Dynam Systems Ser B, 3, 299-309 (2003) · Zbl 1126.34337
[8] Glendinning, P.; Perry, L. P., Melnikov analysis of chaos in a simple epidemiological model, J Math Biol, 35, 359-373 (1997) · Zbl 0867.92021
[9] Guckenheimer, J.; Holmes, P., Nonlinear oscillations, dynamical systems, and bifurcations of vector fields (1997), Springer-Verlag: Springer-Verlag Berlin Heidelberg New York, p. 31
[10] Gumel, A. B.; Moghadas, S. M., A qualitative study of a vaccination model with nonlinear incidence, Appl Math Comput, 143, 409-419 (2003) · Zbl 1018.92029
[11] Li, G.; Jin, Z., Global stability of an SEI epidemic model, Chaos, Solitons & Fractals, 21, 4, 925-931 (2004) · Zbl 1045.34025
[12] Li, X.; Wang, W., A discrete epidemic model with stage structure, Chaos, Solitons & Fractals, 26, 3, 947-958 (2005) · Zbl 1066.92045
[13] 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
[14] 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
[15] Liu, X.; Chen, L., Complex dynamics of Holling type II Lotka-Volterra predator-prey system with impulsive perturbations on the predator, Chaos, Solitons & Fractals, 16, 2, 311-320 (2003) · Zbl 1085.34529
[16] Lizana, M.; Rivero, H., Multiparameteric bifurcations for a model in epidemiology, J Math Biol, 35, 21-36 (1996) · Zbl 0868.92024
[17] Margheri, A.; Rebelo, C., Some examples of persistence in epidemiological models, J Math Biol, 46, 564-570 (2003) · Zbl 1023.92027
[18] Moghadas, S. M., Analysis of an epidemic model with bistable equilibria using the Poincare index, Appl Math Comput, 149, 689-702 (2004) · Zbl 1034.92030
[19] Perko, L., Differential equations and dynamical systems (1996), Springer: Springer New York · Zbl 0854.34001
[20] Ruan, S.; Xiao, D., Global analysis in a predator-prey system with nonmonotonic functional response, SIAM J Appl Math, 61, 1445-1472 (2001) · Zbl 0986.34045
[21] Ruan, S.; Wang, W., Dynamical behavior of an epidemic model with a nonlinear incidence rate, J Differn Equat, 188, 135-163 (2003) · Zbl 1028.34046
[22] Takens, F., Forced oscillations and bifurcations, applications of global analysis I, Comm Mathj Inst Rijksuniversitat Utrecht, 3, 1-59 (1974)
[23] van den Driessche, P.; Watmough, J., A simple SIS epidemic model with a backward bifurcation, J Math Biol, 40, 525-540 (2000) · Zbl 0961.92029
[24] van den Driessche, P.; Watmough, J., Epidemic solutions and endemic catastrophes, Fields Inst Commun, 36, 247-257 (2003) · Zbl 1162.92325
[25] Wang, K.; Wang, W.; Liu, X., Viral infection model with periodic lytic immune response, Chaos, Solitons & Fractals, 28, 1, 90-99 (2006) · Zbl 1079.92048
[26] Wang, W.; Zhao, X.-Q., An endemic model in a patchy environment, Math Biosci, 190, 97-112 (2004) · Zbl 1048.92030
[27] Wu, L.; Feng, Z., Homoclinic bifurcation in an SIQR model for childhood disease, J Differ Equat, 168, 150-167 (2000) · Zbl 0969.34042
[28] Zhang, J.; Ma, Z., Global dynamics of an SEIRS epidemic model with saturating contact rate, Math Biosci, 185, 15-32 (2003) · Zbl 1021.92040
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.