In this paper, we proved existence and nonexistence of traveling wave solution for a diffusive simple epidemic model with a free boundary in the case where the diffusion coefficient $ d $ of susceptible population is zero and the basic reproduction number is greater than 1. We obtained a curve in the parameter plane which is the boundary between the regions of existence and nonexistence of traveling wave. We numerically observed that in the region where the traveling wave exists the disease successfully propagate like traveling wave but in the region of no traveling wave disease stops to invade. We also numerically observed that as $ d $ increases the speed of propagation slows down and the parameter region of propagation narrows down.

中文翻译：

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具有自由边界的扩散简单传染病模型的行波解

在本文中，我们证明了在易感人群的扩散系数$ d $为零且基本复制数大于1的情况下，具有自由边界的扩散简单流行病模型的行波解的存在和不存在。参数平面中的一条曲线，是行波存在和不存在区域之间的边界。我们从数值上观察到，在行波存在的区域，疾病像行波一样成功地传播，但在没有行波的区域中，疾病停止了入侵。我们还从数值上观察到，随着$ d $的增加，传播速度变慢，并且传播的参数区域变窄。