JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2019, Vol. 54 ›› Issue (7): 89-99.doi: 10.6040/j.issn.1671-9352.0.2018.588

Previous Articles    

Qualitative analysis of an SIRI epidemic model with stochastic effects

GAO Jian-zhong, ZHANG Tai-lei*   

  1. School of Science, Changan University, Xian 710064, Shaanxi, China
  • Published:2019-06-27

PDF (PC)

413

Abstract: An SIRI bilinear epidemic model with stochastic effects is studied. The global existence, uniqueness and boundedness of its positive solution are proved by using stopping time theory and Lyapunov analysis method. It is also shown that the solution of the stochastic model oscillates around the corresponding deterministic disease-free equilibrium and endemic equilibrium points, and the sufficient conditions for persistence in mean of the solution of the stochastic model and disease extinction are obtained. Finally, numerical simulations are carried out to prove the validity of theoretical results.

Key words: stochastic SIRI model, oscillating behavior, persistence in mean, disease extinction

CLC Number: 

  • O175.1
[1] TUDOR D. A deterministic model for herpes infections in human and animal populations[J]. Siam Review, 1990, 32(1): 136-139.
[2] GUO Peng, YANG Zhichun. The stability of SIRI model with nonliner incidence rate[J]. Journal of Chongqing Normal University: Natural Science, 2011, 28(4): 35-39.
[3] BLOWER S. Modelling the genital herpes epidemic[J]. Herpes, 2004, 11(3): 138-146.
[4] BELOTTO A, LEANES L F, SCHNEIDER M C, et al. Overview of rabies in the Americas[J]. Virus Research, 2005, 111(1): 1-12.
[5] NAZ R, MAHOMED K S, NAEEM I. First integrals and exact solutions of the SIRI and tuberculosis models[J]. Mathematical Methods in the Applied Sciences, 2016, 39(15): 4654-4666.
[6] 雷桥, 杨志春. 具有随机效应的SIRI传染病模型的定性分析[J]. 重庆师范大学学报(自然科学版), 2016, 33(1): 82-85.LEI Qiao, YANG Zhichun. The qualitative analysis of SIRI epidemic model with stochastic effects[J]. Journal of Chongqing Normal University(Natural Science), 2016, 33(1): 82-85.
[7] CAI Yongli, KANG Yun, WANG Weiming. A stochastic SIRS epidemic model with nonlinear incidence rate[J]. Applied Mathematics and Computation, 2017, 305: 221-240.
[8] ZHAO Yanan, JIANG Daqing. The threshold of a stochastic SIRS epidemic model with saturated incidence[J]. Applied Mathematics Letters, 2014, 34(1): 90-93.
[9] LIU Qun, JIANG Daqing, SHI Ningzhong, et al. The threshold of a stochastic SIS epidemic model with imperfect vaccination[J]. Mathematics and Computers in Simulation, 2017, 144: 78-90.
[10] YANG Qingshan, JIANG Daqing, SHI Ningzhong, et al. The ergodicity and extinction of stochastically perturbed SIR and SEIR epidemic models with saturated incidence[J]. Journal of Mathematical Analysis and Applications, 2012, 388(1): 248-271.
[11] 刘杰, 胡志兴, 廖福成. 随机模型SIRS的定性分析[J]. 宁夏大学学报(自然科学版), 2016, 37(1): 1-6. LIU Jie, HU Zhixing, LIAO Fucheng. Qualitative analysis of stochastic SIRS model[J]. Journal of Ningxia University(Natural Science Edition), 2016, 37(1): 1-6.
[12] 张艳宏, 许超群, 原三领. 一类接触率受到噪声干扰的随机SIS流行病模型研究[J]. 上海理工大学学报, 2015, 37(6): 512-516. ZHANG Yanhong, XU Chaoqun, YUAN Sanling. Stochastic SIS epidemic model with contract rate influenced by noise[J]. Journal of University of Shanghai for Science and Technology, 2015, 37(6): 512-516.
[13] LIU Qun, JIANG Daqing, HAYAT T, et al. Stationary distribution and extinction of a stochastic SIRI epidemic model with relapse[J]. Stochastic Analysis and Applications, 2018, 36(1): 138-151.
[14] JI Chunyan, JIANG Daqing, SHI Ningzhong. The behavior of an SIR epidemic model with stochastic perturbation[J]. Stochastic Analysis and Applications, 2014, 30(5): 755-773.
[1] CHEN Yu-jia, YANG He. Existence of periodic solutions of a class of third order delay differential equations in Banach spaces [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(8): 84-94.
[2] CAO Xue-jing, LUO Zhi-xue. Optimal control of forest evolution system in polluted environment [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(7): 15-20.
[3] YANG Dan-dan. Endpoint theorem on existence of solutions for Hadamard-type fractional differential inclusions with nonlocal integral boundary value conditions [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(2): 46-51.
[4] . Existence of periodic solutions for a class of Hamiltonian systems with p-Laplace [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2016, 51(12): 42-46.
[5] SHI Xue-wei, JIA Jian-wen. Study on an SIR epidemic model with information variable and graded cure rate [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2016, 51(3): 51-59.
[6] SUN Guo-wei, MAI A-li. Multiple homoclinic solutions for second order nonlinear difference equations [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2015, 50(05): 51-54.
[7] FAN Jin-jun, LU Xiao-dong. Existence of nonoscillatory solutions to second order forced neutral dynamic equations with time delay on time scales [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2015, 50(05): 45-50.
[8] CHEN Wen, YAO Jing-sun, YANG Xue-jie. A nonlinear mixed boundary value problem for singularly perturbed forth-order differential equation [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2015, 50(03): 67-72.
[9] ZHANG Zhi-Reng. The uniqueness of the limit cycle and the bifurcation of singular  points for a class of polynomial systems [J]. J4, 2009, 44(9): 90-92.
[10] . Successive iteration of positive solutions for some SturmLiouvillelike boundary value problems with pLaplacian on infinite intervals [J]. J4, 2009, 44(5): 86-90.
[11] ZHANG Xing-qiu,ZHONG Qiu-yan .

Positive solutions for singular m-point boundary value problems in a Banach space

[J]. J4, 2008, 43(9): 51-56 .
[12] CUI Yu-jun,ZOU Yu-mei . Topological degree computation and its applications to three-point boundary value problems in Banach space [J]. J4, 2008, 43(3): 84-86 .
[13] LI Zong-cheng, . Distribution of limit cycles for a class of higherdegree degenerate planar polynomial systems of codimension two [J]. J4, 2007, 42(2): 19-27 .
[14] TIAN Jia-cai and GAO Li . Multiple solutions for a class of second order differential equations [J]. J4, 2007, 42(6): 55-60 .
[15] LIU Xin-min,CUI Yu-jun* . Existence of solutions to differential system on the non-cylindrical domain in Banach spaces [J]. J4, 2008, 43(4): 1-05 .
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
No Suggested Reading articles found!
Copyright 2018 © Editorial office of JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)
Supported by Beijing Magtech Co.Ltd