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

应用MacCluskey（2010）中Lyapunov函数技术解决了延迟SIR流行病模型全局稳定性的开放性问题，我们首先通过摄动技术得到了全局稳定性的充分条件，其次通过扩展MacCluskey的Lyapunov函数技术得到了对应免疫恢复个体小损失率的结果。另一方面，通过Xu和Ma（2010）的改进单调迭代技术，我们改进了免疫恢复个体大损失率的全局稳定性结果。这种单调迭代技术是一种不同于李雅普诺夫函数的技术，我们希望将其应用于其他模型。此外，我们还发现了用后向欧拉方法导出的连续传染病模型的全局稳定性的一些自然推广，从而解决了如何找到保持全局稳定性的连续模型的离散版本的开放问题。

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.