×

Global analysis of an epidemic model with nonmonotone incidence rate. (English) Zbl 1119.92042

Summary: We study an epidemic model with nonmonotonic incidence rate, which describes the psychological effect of certain serious diseases on the community when the number of infectives is getting larger. By carrying out a global analysis of the model and studying the stability of the disease-free equilibrium and the endemic equilibrium, we show that either the number of infective individuals tends to zero as time evolves or the disease persists.

MSC:

92C60 Medical epidemiology
37N25 Dynamical systems in biology
34C60 Qualitative investigation and simulation of ordinary differential equation models
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Alexander, M. E.; Moghadas, S. M., Periodicity in an epidemic model with a generalized non-linear incidence, Math. Biosci., 189, 75 (2004) · Zbl 1073.92040
[2] Capasso, V.; Serio, G., A generalization of the Kermack-Mckendrick deterministic epidemic model, Math. Biosci., 42, 43 (1978) · Zbl 0398.92026
[3] Derrick, W. R.; van den Driessche, P., A disease transmission model in a nonconstant population, J. Math. Biol., 31, 495 (1993) · Zbl 0772.92015
[4] Gumel, A. B., Modelling strategies for controlling SARS outbreaks, Proc. R. Soc. Lond. B, 271, 2223 (2004)
[5] Hethcote, H. W., The mathematics of infectious disease, SIAM Rev., 42, 599 (2000) · Zbl 0993.92033
[6] Hethcote, H. W.; Levin, S. A., Periodicity in epidemiological models, (Gross, L.; Hallam, T. G.; Levin, S. A., Applied Mathematical Ecology (1989), Springer-Verlag: Springer-Verlag Berlin), 193
[7] Hethcote, H. W.; van den Driessche, P., Some epidemiological models with nonlinear incidence, J. Math. Biol., 29, 271 (1991) · Zbl 0722.92015
[8] Leung, G. M., The impact of community psychological response on outbreak control for severe acute respiratory syndrome in Hong Kong, J. Epidemiol. Commun. Health, 57, 857 (2003)
[9] Liu, W. M.; Hethcote, H. W.; Levin, S. A., Dynamical behavior of epidemiological models with nonlinear incidence rates, J. Math. Biol., 25, 359 (1987) · Zbl 0621.92014
[10] 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 (1986) · Zbl 0582.92023
[11] Perko, L., Differential Equations and Dynamical Systems (1996), Springer-Verlag: Springer-Verlag New York · Zbl 0854.34001
[12] Ruan, S.; Wang, W., Dynamical behavior of an epidemic model with a nonlinear incidence rate, J. Differ. Equations, 188, 135 (2003) · Zbl 1028.34046
[13] Wang, W.; Ruan, S., Simulating the SARS outbreak in Beijing with limited data, J. Theoret. Biol., 227, 369 (2004) · Zbl 1439.92185
[14] Z.-F. Zhang, T.-R. Ding, W.-Z. Huang, Z.-X. Dong, Qualitative theory of differential equations, Transl. Math. Monogr. vol. 101, Amer. Math. Soc., Providence, 1992.; Z.-F. Zhang, T.-R. Ding, W.-Z. Huang, Z.-X. Dong, Qualitative theory of differential equations, Transl. Math. Monogr. vol. 101, Amer. Math. Soc., Providence, 1992.
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.