This paper is concerned with the propagation dynamics of a nonlocal dispersal vaccination model with general nonlinear incidence rate and spatio-temporal delay. We first apply Schauder’s fixed point theorem together with the upper-lower solutions to prove the existence of traveling wave solutions, when the wave speed \(c>c^{*}\) (a critical speed) and the basic reproduction number \(\mathcal {R}_0>1\) . The upper-lower solutions imply that the traveling wave solutions connect the disease-free equilibrium at negative infinity. Then, we investigate the asymptotic behavior of traveling wave solutions at positive infinity by constructing an appropriate Lyapunov function. Finally, by employing the method of two-sided Laplace transform and the contradictory approach, we further prove the nonexistence of traveling wave solutions when the wave speed \(c<c^{*}\) and \(\mathcal {R}_0>1\) , or \(c>0\) and \(\mathcal {R}_0\le 1\) . Our results show that the factors, such as nonlocal spatial dispersal, nonlocal spatio-temporal delays and vaccination do not affect the existence of traveling wave solutions, but they do have an impact on the critical wave speed \(c^*\) .
