中文摘要：

考虑具有非线性发病率及分布时滞^ｈ∑ｋ＝０Pkf（Sn＋1,In-k）的离散SIRS模型,利用差分不等式理论得到模型持久性的充分条件。当f（x,y）=βxG（y）时,对应模型持久的充分条件为：G（y）在[0,∞）连续单增,G（0）=0,函数G（y）/y在（0,∞）单减有界。该结论改进了[生物数学学报,2013,38（2）：274-259]中的相关结论。当易感者输入率等于死亡率时,本文结论是[Appl Math Comput,2012,39：15-34]中定理4.1的离散化形式。

英文摘要：

This paper considers the discrete SIRS model with nonlinear incidence rates and distributed delay f（x,y）=βxG（y） k=OThe sufficient conditions for the persistence of the model are obtained by using differ-ence inequality. When f（x,y）=flxG（y）,the corresponding sufficient conditions are that G（y） is continuous and nondeereasing on [0,∞）,G（y）/y is bounded and nonincreasing on （0,∞） and G（O）=0. This re-sult extends and improves the corresponding result on [Journal of Biomathematics,2013,28（2）：247-259]. If the birth rate of susceptible is equal to the death rate of susceptible,then the conclusion of this paper is discrete form of theorem 4.1 of [Appl Math Comput,2012,39：15-34].