Accurateness and stability of delayed integral and differential equations and their discrete versions.

利用Macrobinson key（2010）关于时滞SIR传染病模型全局稳定性的公开问题的李雅普诺夫函数技巧，首先利用扰动技巧，然后扩展Macrobinson key的李雅普诺夫函数技巧，得到了免疫恢复个体损失率小时全局稳定的充分条件.另一方面，通过对Xu和Ma（2010）的单调迭代技巧的改进，我们改进了免疫恢复个体大损失率下的全局稳定性结果，这种单调迭代技巧是一种不同于李雅普诺夫函数的技巧，我们希望将其应用于其他模型。我们发现连续传染病模型的全局稳定性到由向后欧拉方法导出的离散模型的一些自然推广，解决了如何找到连续模型的离散形式并保持全局稳定性的公开问题。

Applying the Lyapunov function techniques in MacCluskey(2010) which solve the open question on the global stability for delayed SIR epidemic models, we obtain results on sufficient condition of global stability, first by perturbation techniques, and second by extending the Lyapunov function techniques of MacCluskey which are corresponding to small loss rate of immunity recovery individuals. On the other hand, by modified monotone iterative techniques in Xu and Ma(2010), we improve results on the global stability for large loss rate of immunity recovery individuals.This monotone iterative techniques is a different techniques than that of Lyapunov function and we may hope to their applications to other models.Moreover, we find some natural extensions on global stability of continuous epidemic models to discrete models derived by the backward Euler method, by which we solve the open problem how to find the discrete version of the continuous model which keeps the global stability property.