In this paper, the existence of a non-trivial, positive and bounded critical traveling wave solution of a diffusive disease model, whose reaction system has infinity many equilibria, is obtained for the first time. This gives an affirmative answer to an open problem left in [X. Wang, H. Wang, J. Wu, Traveling waves of diffusive predator-prey systems: disease outbreak propagation, Discrete Contin. Dyn. Syst. Ser. A 32 (2012) 3303-3324]. Our result shows that the critical traveling wave in this model is a mixed of front and pulse type.

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In a recent paper [K. Zhou, M. Han, Q. Wang, Math. Method. Appl. Sci. 40 (2016) 2772-2783, the authors investigated the traveling wave solutions of a delayed diffusive SIR epidemic model. When the basic reproduction number 1 0  R and the wave speedthey obtained the existence of a non-trivial and non-negative traveling wave solution. When