Abstract
This paper is concerned with the existence and exponential decay of traveling waves with the critical speed for a periodic and diffusive Kermack-McKendrick epidemic model. By a delicate analysis of periodic traveling waves with super-critical speeds and strong maximum principle of parabolic equations, we obtain the existence of positive periodic traveling waves with the critical speed. Thus, the open problem [29] is solved completely. In addition, this approach is also valid for the model in [30] without any limitation condition. Besides the existence, the main part of the paper is devoted to the exponential decay of the infected component when the critical periodic traveling waves approach the initial disease-free steady state.
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