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

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