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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

ds(t) dt=-βs(t) 0 h f(τ)i(t-τ)dτ-μ 1 s(t)+b,di(t) dt=βs(t) 0 h f(τ)i(t-τ)dτ-μ 2 i(t)-λi(t),dr(t) dt=λi(t)-μ 3 r(t),

is analyzed. By the Lyapunov-Lasalle invariance principle, the authors show that the disease free equilibrium is globally attractive whenever b μ 1 s * =μ 2 +λ β. 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.

34K20Stability theory of functional-differential equations
34D23Global stability of ODE