Global asymptotic properties of a delay SIR epidemic model with finite incubation times. (English) Zbl 0967.34070

Takeuchi, Yasuhiro; Ma, Wanbiao; Beretta, Edoardo

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 42, No. 6, 931-947 (2000).

Here, the SIR model with distributed delays \[ \begin{aligned} {ds(t)\over dt} & = -\beta s(t) \int^h_0 f(\tau)i(t-\tau) d\tau- \mu_1s(t)+ b,\\ {di(t)\over dt} & =\beta s(t) \int^h_0 f(\tau)i(t- \tau) d\tau- \mu_2i(t)- \lambda i(t),\\ {dr(t)\over dt} & =\lambda i(t)- \mu_3 r(t),\end{aligned} \] is analyzed. By the Lyapunov-Lasalle invariance principle, the authors show that the disease free equilibrium is globally attractive whenever \({b\over\mu_1}\leq s^*= {\mu_2+\lambda\over \beta}\). 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.

Reviewer: Chen Lan Sun (Beijing)

MSC:

34K20	Stability theory of functional-differential equations
92D30	Epidemiology
34D23	Global stability of solutions to ordinary differential equations

Keywords:

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

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