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We introduce a diffusive SEIR model with nonlocal delayed transmission between the infected subpopulation and the susceptible subpopulation with a general nonlinear incidence. We show that our results on existence and nonexistence of traveling wave solutions are determined by the basic reproduction number    $R_{0}=\partial _{I}F(S_{0},0)/\gamma $  R 0 = ∂ I F ( S 0 , 0 ) / γ of the corresponding ordinary differential equations and the minimal wave speed    $c^{*}$  c ∗ . The main difficulties lie in the fact that the semiflow generated here does not admit the order-preserving property. In the present paper, we overcome these difficulties to obtain the threshold dynamics. In view of the numerical simulations, we also obtain that the minimal wave speed is explicitly determined by the time delay and nonlocality in disease transmission and by the spatial movement pattern of the exposed and infected individuals.

Wave propagation in a diffusive SEIR epidemic model with nonlocal transmission and a general nonlinear incidence rate