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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.

##### MSC:
 92D30 Epidemiology 34K60 Qualitative investigation and simulation of models 34K25 Asymptotic theory of functional-differential equations
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##### References:
 [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 Poincaré--Bendixson trichotomy for asymptotically autonomous differential equations. J. math. Biol. 30, 755-763 (1992) · Zbl 0761.34039