Fan Xiaolin, Wang Lei, Teng Zhidong
1College of Mathematics and System Sciences, Xinjiang University, Urumqi, 830046 People's Republic of China.
2Department of Basic Education, Xinjiang Institute of Engineering, Urumqi, 830091 People's Republic of China.
Adv Differ Equ. 2016;2016(1):123. doi: 10.1186/s13662-016-0846-y. Epub 2016 May 6.
In this paper, a class of discrete SEIRS epidemic models with general nonlinear incidence is investigated. Particularly, a discrete SEIRS epidemic model with standard incidence is also considered. The positivity and boundedness of solutions with positive initial conditions are obtained. It is shown that if the basic reproduction number , then disease-free equilibrium is globally attractive, and if , then the disease is permanent. When the model degenerates into SEIR model, it is proved that if , then the model has a unique endemic equilibrium, which is globally attractive. Furthermore, the numerical examples verify an important open problem that when , the endemic equilibrium of general SEIRS models is also globally attractive.
本文研究了一类具有一般非线性发病率的离散SEIRS传染病模型。特别地，还考虑了具有标准发病率的离散SEIRS传染病模型。得到了具有正初始条件的解的正性和有界性。结果表明，如果基本再生数 ，则无病平衡点是全局吸引的，并且如果 ，则疾病是持久的。当模型退化为SEIR模型时，证明了如果 ，则模型有唯一的地方病平衡点，且该平衡点是全局吸引的。此外，数值例子验证了一个重要的开放性问题，即当 时，一般SEIRS模型的地方病平衡点也是全局吸引的。
