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

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

doi:10.3934/dcdsb.2017127

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

Qun Liu
, Daqing Jiang
, Ningzhong Shi
, Tasawar Hayat
, Ahmed Alsaedi
, Xiaoying Han

Northeast Normal University
Yulin Normal University
Faculty of Sciences, King Abdulaziz University
China University of Petroleum (East China)
Quaid-I-Azam University

Research output: Contribution to journal › Article › peer-review

27 Scopus citations

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.

Original language	English
Pages (from-to)	2479-2500
Number of pages	22
Journal	Discrete and Continuous Dynamical Systems - Series B
Volume	22
Issue number	6
DOIs	https://doi.org/10.3934/dcdsb.2017127
State	Published - Aug 2017
Externally published	Yes

UN SDGs

This output contributes to the following UN Sustainable Development Goals (SDGs)

SDG 3 Good Health and Well-being

Keywords

Extinction
Lyapunov function
Periodic solution
Stationary solution
Stochastic SEIR epidemic model

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10.3934/dcdsb.2017127

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title = "Stationarity and periodicity of positive solutions to stochastic seir epidemic models with distributed delay",

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.",

keywords = "Extinction, Lyapunov function, Periodic solution, Stationary solution, Stochastic SEIR epidemic model",

author = "Qun Liu and Daqing Jiang and Ningzhong Shi and Tasawar Hayat and Ahmed Alsaedi and Xiaoying Han",

year = "2017",

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doi = "10.3934/dcdsb.2017127",

language = "English",

volume = "22",

pages = "2479--2500",

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publisher = "American Institute of Mathematical Sciences",

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TY - JOUR

T1 - Stationarity and periodicity of positive solutions to stochastic seir epidemic models with distributed delay

AU - Liu, Qun

AU - Jiang, Daqing

AU - Shi, Ningzhong

AU - Hayat, Tasawar

AU - Alsaedi, Ahmed

AU - Han, Xiaoying

PY - 2017/8

Y1 - 2017/8

N2 - 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.

AB - 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.

KW - Extinction

KW - Lyapunov function

KW - Periodic solution

KW - Stationary solution

KW - Stochastic SEIR epidemic model

UR - https://www.scopus.com/pages/publications/85018699442

U2 - 10.3934/dcdsb.2017127

DO - 10.3934/dcdsb.2017127

M3 - Article

AN - SCOPUS:85018699442

SN - 1531-3492

VL - 22

SP - 2479

EP - 2500

JO - Discrete and Continuous Dynamical Systems - Series B

JF - Discrete and Continuous Dynamical Systems - Series B

IS - 6

ER -

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

Abstract

UN SDGs

Keywords

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