一类传染病模型无病平衡点的全局稳定性

过去的半个多世纪，传染病模型在数学生态学领域已受广泛重视.研究了一个具时滞和扩散的传染病模型，重点讨论了该模型解的定性性质和稳态解的渐近行为；利用线性化和特征值方法讨论了正稳态解的局部稳定性，通过构造单调迭代序列，给出了正稳态解的全局稳定性. 最后给出了数值模拟和讨论，当接触率充分小时，问题的无病平衡点是全局渐近稳定的.

The global stability of a class of infectious disease model for the disease free equilibrium

Over the past half century, infectious disease model has attracted great attention in mathematical ecology. People use mathematical tools to research the causes of disease, the development process of the disease, and provide the theoretical basis and quantitative basis for the decision of the prevention and treatment. Focusing on the qualitative properties of solutions of the model, an infectious disease model with time delay and diffusion is given. We mainly discuss the asymptotic behavior of the solution： Linearization and eigenvalue methods are used to discuss the local stability of the positive steady-state solutions; Through constructing monotone iterative sequences, the global stability of positive steady-state solutions are given. Numerical simulation and some discussions are given to emphasize our results： Free equilibrium is globally asymptotically stable when the contact rate of the disease is small.