Stationarity and periodicity of positive solutions to stochastic SEIR epidemic models with distributed delay

Qun Liu; Daqing Jiang; Ningzhong Shi; Tasawar Hayat; Ahmed Alsaedi

doi:10.3934/dcdsb.2017127

Article Contents

2017, Volume 22, Issue 6: 2479-2500. Doi: 10.3934/dcdsb.2017127

This issue Previous Article Chiellini integrability condition, planar isochronous systems and Hamiltonian structures of Liénard equation Next Article Quasi-periodic solutions of generalized Boussinesq equation with quasi-periodic forcing

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, Quaid-i-Azam 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

Author Bio: E-mail address: liuqun151608@163.com; E-mail address: shinz@nenu.edu.cn; E-mail address: tahaksag@yahoo.com; E-mail address: aalsaedi@hotmail.com
Daqing Jiang, E-mail: daqingjiang2010@hotmail.com, Tel.: +86 43185099589; fax: +86 43185098237
Daqing Jiang, E-mail: daqingjiang2010@hotmail.com, Tel.: +86 43185099589; fax: +86 43185098237

Received: May 31, 2016

Revised: February 28, 2017

Published: May 2017

The first author was supported by NSFC of China (No: 11561069), 2016GXNSFBA380006 and KY2016YB370, the second author was supported by NSFC of China (No: 11371085) and the Fundamental Research Funds for the Central Universities (No.15CX08011A).

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Abstract

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 T-periodic solution under a simple condition.

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Mathematics Subject Classification: Primary:92B05, 92D30;Secondary:34E10, 60H10.

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