中文摘要：

针对媒体宣传教育对人们行为方式和生活习惯的影响，本文考虑了由于媒体影响而导致易感性不同的一个SEI传染病模型．分析了模型可能出现的后项分支及其平衡点的稳定性和持久性．结果表明，当基本再生数小于1时，模型的无病平衡点全局稳定；当基本再生数大于1时，地方病平衡点一致持久．同时，利用控制理论，本文也研究了媒体的宣传作用对易感者进行影响和教育的最优控制措施，给出了使目标函数值最小的最优控制，并用数值模拟显示了模型解的动力学性态及控制措施对防止疾病蔓延所起的作用．

英文摘要：

In view of the impact of media on people＇s behavior and daily life, an SEI epidemic model with different susceptibility has been studied, in which the stabilities and persistence are analyzed in addition to the backward bifurcation. It is proved that the disease-free equilibrium is globally stable when the basic reproductive number is less than one, and the endemic equilibrium is uniformly persistent when it is greater than one. The impact of media on the susceptible is also formulated and solved as an optimal control problem. The optimal solution of the model is found, it can make the objective function achieve its minimum value. Numerical results show the dynamics of the solutions and the performance of the optimization strategy.