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Stationarity and periodicity of positive solutions to stochastic SEIR epidemic models with distributed delay
1.  School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin 130024, China 
2.  School of Mathematics and Statistics, Guangxi Colleges and Universities Key Laboratory, of Complex System Optimization and Big Data Processing, Yulin Normal University, Yulin, Guangxi 537000, China 
3.  School of Mathematics and Statistics, Key Laboratory of Applied Statistics of MOE, Northeast, Normal University, Changchun, Jilin 130024, China 
4.  Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 121589, Saudi Arabia 
5.  College of Science, China University of Petroleum (East China), Qingdao 266580, China 
6.  School of Mathematics and Statistics, Key Laboratory of Applied Statistics of MOE, Northeast, Normal University, Changchun, Jilin 130024, China 
7.  Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 121589, Saudi Arabia 
8.  Department of Mathematics, QuaidiAzam University 45320, Islamabad 44000, Pakistan 
9.  Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 121589, Saudi Arabia 
In this paper, we consider two SEIR epidemic models with distributed delay in random environments. First of all, by constructing a suitable stochastic Lyapunov function, we obtain the existence of stationarity of the positive solution to the stochastic autonomous system. Then we establish sufficient conditions for extinction of the disease. Finally, by using Khasminskii's theory of periodic solutions, we prove that the stochastic nonautonomous epidemic model admits at least one nontrivial positive Tperiodic solution under a simple condition.
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