In this paper, we prove the existence of a critical traveling wave solution for a delayed diffusive SIR epidemic model with saturated incidence. Moreover, we establish the nonexistence of traveling wave solutions with nonpositive wave speed for this model. Our results solve some open problems left in the recent paper (Z. Xu in Nonlinear Anal. 111:66–81, 2014).

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In this paper, we propose a nonlocal diffusion infectious disease model with nonlinear incidences and distributed delay to model the transmission of the epidemic. By a fixed point theorem and a limiting argument, we establish the existence of traveling wave solutions for the model. Meanwhile, we obtain the non-existence of traveling wave solutions for the model via two-sided Laplace transform. It is found that the threshold dynamics of traveling wave solutions are entirely determined by the basic reproduction number of the corresponding spatially-homogenous delayed differential system and the minimum wave speed. A typical example is given for supporting our abstract results. Moreover, the effect of the diffusive rate of the infected individuals on the minimum wave speed is discussed. Existence of traveling waves 2.1 PreliminariesLinearizing the second equation in (1.7) at (s 0 , 0) and using g(0) = 0, we have1) Substituting i(z) = e λz into (2.1) yields Θ(λ, c) := d 2 ∞ -∞ J(y) e -λy -1 dycλ + βs 0 g (0) h 0 f (τ )e -λcτ dτγ = 0.(2.2) Lemma 2.1 Assume that R 0 := βs 0 g (0)/γ > 1. Then there exist c * > 0 and λ * > 0 such that

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