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Threshold dynamics in a delayed SIS epidemic model. (English) Zbl 0988.92027
Summary: An SIS epidemic model with maturation delay is analysed. It is shown that the disease dies out when the basic reproduction number $R_0<1$, and the disease remains endemic when $R_0>1$ in the sense of uniform persistence. When the disease induced death rate is sufficiently small, the global attractivity of the endemic equilibrium is also proved.

34K60Qualitative investigation and simulation of models
34K25Asymptotic theory of functional-differential equations
Full Text: DOI
[1] Cooke, K.; Den Driessche, P. Van; Zou, X.: Interaction of maturation delay and nonlinear birth in population and epidemic models. J. math. Biol. 39, 332-352 (1999) · Zbl 0945.92016
[2] De Jong, M. C. M.; Diekmann, O.; Heesterbeek, H.: How does transmission of infection depend on population size?. Epidemic models, 84-94 (1995) · Zbl 0850.92042
[3] B. Ermentraut, XPPAUT 4.30--The Differential Tool, 2000.
[4] Freedman, H. I.; Gopalsamy, K.: Global stability in time-delayed single-species dynamics. Bull. math. Biol. 48, 485-492 (1986) · Zbl 0606.92020
[5] Hale, J.: Asymptotic behavior of dissipative systems. Math. surveys and monographs 25 (1988) · Zbl 0642.58013
[6] Hale, J.; Lunel, S. M. Verduyn: Introduction to functional differential equations. (1993) · Zbl 0787.34002
[7] Hale, J.; Waltman, P.: Persistence in infinite-dimensional systems. SIAM J. Math. anal. 20, 388-395 (1995) · Zbl 0692.34053
[8] Mena-Lorca, J.; Hethcote, H. W.: Dynamic models of infectious diseases as regulators of population sizes. J. math. Biol. 30, 693-716 (1992) · Zbl 0748.92012
[9] Nisbet, R. M.; Gurney, W. S. C.: Modelling fluctuating populations. (1982) · Zbl 0593.92013
[10] Smith, H. L.: Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems. Mathematical surveys and monographs 41 (1995) · Zbl 0821.34003
[11] Smith, H. L.; Waltman, P.: Perturbation of a globally stable steady state. Proc. amer. Math. soc. 127, 447-453 (1999) · Zbl 0924.58087
[12] Smith, H. L.; Zhao, X. -Q.: Dynamics of a periodically pulsed bio-reactor model. J. differential equations 155, 368-404 (1999) · Zbl 0930.35085
[13] Smith, H. L.; Zhao, X. -Q.: Microbial growth in a plug flow reactor with wall adherence and cell motility. J. math. Anal. appl. 241, 134-155 (2000) · Zbl 0999.92039
[14] Thieme, H. R.: Convergence results and PoincarĂ©--Bendixson trichotomy for asymptotically autonomous differential equations. J. math. Biol. 30, 755-763 (1992) · Zbl 0761.34039