×

zbMATH — the first resource for mathematics

An SIS epidemic model with stage structure and a delay. (English) Zbl 1035.34054
The authors study a SIS epidemic model with stage structure and time delay. Sufficient conditions are derived for the stability of a disease-free equilibrium and an endemic equilibrium. The study shows that the stage structure has no effect on the epidemic model while the Hopf bifurcation occurs as the time delay increases. Some interesting numerical simulations are presented to support the analytical analyses.

MSC:
34D20 Stability of solutions to ordinary differential equations
91A25 Dynamic games
34C60 Qualitative investigation and simulation of ordinary differential equation models
34C23 Bifurcation theory for ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Aiello, W.G. Freedman, H.I. A time delay model of single species growth with stage structure. Math. Biosci., 101: 139–153 (1990) · Zbl 0719.92017 · doi:10.1016/0025-5564(90)90019-U
[2] Aiello, W.G., Freedman, H.I., Wu, J.H. Analysis of a model representing stage-structured population growth with stage-dependent time delay. SIAM J. Appl. Math., 52: 855–869 (1992) · Zbl 0760.92018 · doi:10.1137/0152048
[3] Anderson, R.M., May, R.M. Infectious disease of humans and control. Oxford University Press, Oxford (1991)
[4] Baileg, N.J.T. The mathematical theory of infectious disease and its application. Griffin, London (1975)
[5] Bence, J.R., Nisbet, R.M. Space limited recruitment in open systems: the importance of time delays. Ecology, 70: 1434–1441 (1989) · doi:10.2307/1938202
[6] Brauer, F. Some infectious disease models with population dynamics and general contact rates. Diff. Integ. Equ., 3: 259–278 (1990) · Zbl 0722.92014
[7] Diekmann, O., Heesterbeek, J.A.P., Metz, J.A.J. The legacy of Kermack and Mckendrick. In: D.mollision (Ed.), Epidemic models, Their structure and Relation to Data, Cambridge university Press, Cambridge (1994). · Zbl 0839.92018
[8] Freedman, H.I., Rao, V.scee Hari. The trade-off between mutual interference and time lags in predator-prey-systems. Bull. Math. Biol., 45: 991–1003 (1983) · Zbl 0535.92024
[9] Freedman, H.I., So, J.W.H., Wu, J.H. A model for the growth of a population exhibiting stage structure: cannibalism and cooperation. J. Comput. Appl. Math., 52: 177–198 (1994) · Zbl 0821.34034 · doi:10.1016/0377-0427(94)90356-5
[10] Freedman, H.I., Wu, J.H. Persistence and global asymptotic stability of single species dispersed models with stage structure. Q. Appl. Math., 49: 351–371 (1991) · Zbl 0732.92021
[11] Greenhajgh, D. Some threshold and stability results for epidemic models with density-dependent death rate. Theor. Popul. Biol., 42: 130–151 (1992) · Zbl 0759.92009 · doi:10.1016/0040-5809(92)90009-I
[12] Hale, J. Theory of functional equations. Springer-Verlag, New York (1977) · Zbl 0352.34001
[13] Hethcote, H.W. A thousand and one epidemic models. In: Frontiers in Mathematical Biology, Levin, S.A., Lecture Notes in Biomathe maties 100, Springer-Verlag, Berlin, Heidelberg, New York, 504–515 (1994) · Zbl 0819.92020
[14] Hethcote, H.W., Driessche, van dens P. An SIS epidemic model with variable population size and a delay. J. Math. Biol., 34: 177–194 (1995) · Zbl 0836.92022 · doi:10.1007/BF00178772
[15] Hethcote, H.W., Driessche, van dens P. Some epidemiological models with nonlinear incidence. J. Math. Biol., 29: 271–287 (1991) · Zbl 0722.92015 · doi:10.1007/BF00160539
[16] Kuang, Y. Delay differential equations with application in population dynamics. Academic Press, New York, 119–126 (1993)
[17] Liu, W., Hethcote, H.W., Levin, S.A. Dynamical behavior of epidemiological models with nonlinear incidence rate. J. Math. Biol., 25: 359–380 (1987) · Zbl 0621.92014 · doi:10.1007/BF00277162
[18] Lizana, M., Riveo, J. Multiparametric bifurcations for a model in epidemiology. J. Math. Biol., 35: 21–36 (1996) · Zbl 0868.92024 · doi:10.1007/s002850050040
[19] Wen, X.H., Zhao, S.C. A report on the spread of mumps in some mining area. Chinese J. Epidemiology, 2: 33–34 (2000)
[20] Zhang, X., Chen, L.S. The periodic solution of a class of epidemic models. Comput. Math. Appl., 38: 61–71 (1999) · Zbl 0939.92031 · doi:10.1016/S0898-1221(99)00206-0
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.