Abstract: An SI model with epidemic in the time delays predator and prey is proposed.Using the method of latent root the equilibrium points are obtained.Through analysis，the result is achieved that the predator and the prey with epidemic will be extinct when the infection percentage of susceptible is smaller than a threshold.The predator will be extinct when the infection percentage of the susceptible prey is larger than a threshold and the conversion coefficient of the susceptible predator is smaller than a threshold.The boundary equilibrium is locally asymptotically stable when the time delay of prey is suitable small，while a loss of stability by a Hopf bifurcation can occur as the delay increases.The time delay of predator has no influence on the stability of the equilibrium.

Key words: SI model, asymptotically stable, threshold, time delay, Hopf bifurcation

摘要： 建立并分析了一个带有多时滞的捕食者和食饵都染病的SI模型，用特征根的方法求得了它的非负平衡点。通过分析得到当易感食饵的感染率小于某一阈值时，染病食饵和捕食者将最终灭绝。当易感食饵的感染率大于某一阈值且易感捕食者的转化系数小于某一阈值时，捕食者将最终灭绝。边界平衡点是局部渐近稳定的，随着食饵时滞的增加该平衡点由稳定变为不稳定，系统在该平衡点附近发生Hopf分支。捕食者的时滞对该平衡点的稳定性不产生影响。

关键词: SI模型, 渐近稳定性, 阈值, 时滞, Hopf分支