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Zbl 0967.34070
Takeuchi, Yasuhiro; Ma, Wanbiao; Beretta, Edoardo
Global asymptotic properties of a delay SIR epidemic model with finite incubation times. (English)
[J] Nonlinear Anal., Theory Methods Appl. 42, No.6, A, 931-947 (2000). ISSN 0362-546X

Here, the SIR model with distributed delays $$\align {ds(t)\over dt} & = -\beta s(t) \int^h_0 f(\tau)i(t-\tau) d\tau- \mu_1s(t)+ b,\\ {di(t)\over dt} & =\beta s(t) \int^h_0 f(\tau)i(t- \tau) d\tau- \mu_2i(t)- \lambda i(t),\\ {dr(t)\over dt} & =\lambda i(t)- \mu_3 r(t),\endalign$$ is analyzed. By the Lyapunov-Lasalle invariance principle, the authors show that the disease free equilibrium is globally attractive whenever ${b\over\mu_1}\le s^*= {\mu_2+\lambda\over \beta}$. Sufficient conditions are given to ensure the global asymptotic stability of the endemic equilibrium based on some difference inequality and the construction of Lyapunov functionals.
[Chen Lan Sun (Beijing)]

MSC 2000:: *34K20 Stability theory of functional-differential equations
92D30 Epidemiology
34D23 Global stability

Keywords: epidemic model; time delay; global asymptotic stability; endemic equilibrium

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