In this paper, We introduce the relations of uniformly strong continuous、strongly(weakly) differentiable、uniformly strong(weak) differentiable and strongly bounded vareation on an abstract function from -GK- into the Banace space -E- And given the necessary or sufficient conditions of uniformly strong differentiable abstract function.

It is discovered that the inter-infection is crucial to the dynamical behavior of the system, namely, when the disease will die out without the inter-infection for both species, the disease may prevail in both species if the inter-infection is introduced; Moreover, the system is uniformly persistent.

We show a sufficient condition of the zero solution of (1)is uniformly asymptotically stable and every non-negative solution of (1) tends to zero as n→∞ , while we also show another sufficient and necessary condition about (1) is uniformly persistent.

It is shown that model is uniform persistence under some appropriate conditions and sufficient conditions are established the existence of a positive periodic solution which is globally asymptotically stable by differential inequality and lyapunov functional.

The model is discreted via pioncare map and sufficient conditions for uniform persistence of the model are obtained by the theory of infinite dimensional discrete semi dynamical systems.

Based on the technique of Razumikhin,the sufficient conditions for uniform persistence are obtained.