The paper is devoted to the study of the asymptotic speed of spread and traveling wave solutions for a time-periodic reaction–diffusion SI epidemic model which lacks the comparison principle. By using the basic reproduction number \(R_0\) of the corresponding periodic ordinary differential system and the minimal wave speed \(c^*\), the spreading properties of the corresponding solution of the model are established. More precisely, if \(R_{0} \leqslant 1\), then the solution of the system converges to the disease-free equilibrium as \(t \rightarrow \infty \) and if \(R_0 > 1\), the disease is persistent behind the front and extinct ahead the front. On the basis of it, we then analyze the full information about the existence and nonexistence of traveling wave solutions of the system involved with \(R_0\) and \(c^*\).

本文致力于研究缺乏比较原理的时间周期反应-扩散 SI 流行病模型的传播和行波解的渐近速度。利用相应周期常微分系统的基本再生数\(R_0\)和最小波速\(c^*\)，建立了模型相应解的扩展性质。更准确地说，如果\(R_{0} \leqslant 1\)，则系统的解收敛到无病平衡为\(t \rightarrow \infty \)并且如果\(R_0 > 1\)，此病前身后存，前身绝迹。在此基础上，分析了\(R_0\)和\(c^*\)所涉及的系统行波解存在与不存在的全部信息。