中文摘要：

建立一类非线性微分方程系统来描述带有交叉免疫和完全接种两菌株传染病动力学行为．我们得到每种菌株的基本再生数R1（φ）,R2（φ）和侵入再生数R2^1,R1^2表达式，讨论模型中菌株的共存和稳定性．通过数学分析和数值模拟，证明了交叉免疫和完全接种在疾病传播过程中发挥着重要作用．

英文摘要：

A nonlinear system of different equations that models the dynamics of a two-strain disease with cross immunity and perfect vaccination is formulated. We establish the criteria for determining the coexistence and stability of strains equilibria according to the strain basic reproductive numbers .R1 （φ） and R2 （φ）, and the invasion reproduction numbers R2^1 and R1. Mathematical analysis and simulations show that cross immunity and perfect vaccination play a role in the dynamical behavior of the disease transmission.