Global Stability of a Class of Delayed Epidemic Models With Nonlinear Incidence Rates

XIE Ying-chao; CHENG Yan; HE Tian-yu

doi:10.3879/j.issn.1000-0887.2015.10.010

Volume 36 Issue 10

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XIE Ying-chao, CHENG Yan, HE Tian-yu. Global Stability of a Class of Delayed Epidemic Models With Nonlinear Incidence Rates[J]. Applied Mathematics and Mechanics, 2015, 36(10): 1107-1116. doi: 10.3879/j.issn.1000-0887.2015.10.010

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Global Stability of a Class of Delayed Epidemic Models With Nonlinear Incidence Rates

doi: 10.3879/j.issn.1000-0887.2015.10.010

Army Officer Academy of PLA, Hefei 230031, P.R.China

Funds: The National Natural Science Foundation of China(11202106)

Received Date: 2015-01-27
Rev Recd Date: 2015-05-21
Publish Date: 2015-10-15

Abstract

Abstract

In view of the demographic effects, the latent period and the complexity of disease spread, the dynamic behavior of a class of delayed SIRS epidemic models with nonlinear incidence rates was investigated. The characteristic equation of the corresponding linearized approximation system was analyzed to prove the local stability of the disease-free equilibrium. By means of the Lyapunov-LaSalle invariant set principle, it was proved that the disease-free equilibrium was globally asymptotically stable when the basic reproduction number was less than 1; and the sufficient conditions were obtained for the global asymptotic stability of the endemic equilibrium when the basic reproduction number was greater than 1. Consequently, the conclusions provide a theoretical reference for the effective prevention and control of the spread of communicable diseases.
- SIRS epidemic model,
- nonlinear incidence rate,
- time delay,
- equilibrium,
- stability

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References(18)

References

[1]	温朝晖, 莫嘉琪. 一类流行性传染病生态模型的摄动解[J]. 武汉大学学报(理学版), 2011,57(2): 105-108.(WEN Zhao-hui, MO Jia-qi. Perturbed solution of a class of epidemic contagion ecological model[J]. Journal of Wuhan University (Natural Science Edition),2011,57(2): 105-108.(in Chinese))
[2]	陈兰荪, 孟新柱, 焦建军. 生物动力学[M]. 北京: 科学出版社, 2009: 252-340.(CHEN Lan-sun, MENG Xin-zhu, JIAO Jian-jun. Biological Dynamics [M]. Beijing: Science Press, 2009: 252-340.(in Chinese))
[3]	RUAN Shi-gui, WANG Wen-di. Dynamical behavior of an epidemic model with a nonlinear incidence rate[J]. Journal of Differential Equations,2003,188(1): 135-163.
[4]	Oliveira R D S, Rezende A C. Global phase portraits of a SIS model[J]. Applied Mathematics and Computation,2013,219(9): 4924-4930.
[5]	GUO Li-na, PEI Yong-zhen, LIU Yuan. An infectious disease model with a time delay in loss of vaccine immunity and vertical transmission[J]. Journal of Biomathematics,2013,28(3): 385-389.
[6]	刘玉英, 肖燕妮. 一类受媒体影响的传染病模型的研究[J]. 应用数学和力学, 2013,34(4): 399-407.(LIU Yu-yan, XIAO Yan-ni. An epidemic model with saturated media/psychological impact[J]. Applied Mathematics and Mechanics,2013,34(4): 399-407.(in Chinese))
[7]	宫兆刚, 杨柳, 李浏兰. 具有常数输入率的SIRS传染病模型的稳定性分析[J]. 应用数学, 2013,26(3): 477-481.(GONG Zhao-gang, YANG Liu, LI Liu-lan. Global stability of an SIRS epidemic model with constant removal rate[J]. Mathematica Applicata,2013,26(3): 477-481.(in Chinese))
[8]	王拉娣, 李建全. 一类带有非线性传染率的SEIS传染病模型的定性分析[J]. 应用数学和力学, 2006,27(5): 591-596.(WANG La-di, LI Jian-quan. Qualitative analysis of an SEIS epidemic model with nonlinear incidence rate[J]. Applied Mathematics and Mechanics,2006,27(5): 591-596.(in Chinese))
[9]	杨洪, 魏俊杰. 一类带有非线性接触率的SIR传染病模型的稳定性[J]. 高校应用数学学报, 2014,29(1): 11-16.(YANG Hong, WEI Jun-jie. Stability of SIR epidemical model with a class of nonlinear incidence rates[J]. Applied Mathematics—A Journal of Chinese Universities,2014,29(1): 11-16.(in Chinese))
[10]	杨亚莉, 李建全, 刘万萌, 唐三一. 一类具有分布时滞和非线性发生率的媒介传染病模型的全局稳定性[J]. 应用数学和力学, 2013,34(12): 1291-1299.(YANG Ya-li, LI Jian-quan, LIU Wan-meng, TANG San-yi. Global stability of a vector-borne epidemic model with distributed delay and nonlinear incidence[J]. Applied Mathematics and Mechanics,2013,34(12): 1291-1299.(in Chinese))
[11]	周艳丽, 张卫国. 一类具有非单调传染率的SEIRS时滞传染病模型的全局稳定性[J]. 上海理工大学学报, 2014,36(2): 103-109.(ZHOU Yan-li, ZHANG Wei-guo. Global stability of a delayed SEIRS epidemic model with non-monotone incidence rate[J]. Journal of University of Shanghai for Science and Technology,2014,36(2): 103-109.(in Chinese))
[12]	HUANG Gang, Takeuchi Y, MA Wan-biao, WEI Dai-jun. Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate[J]. Bulletin of Mathematical Biology,2010,72(5): 1192-1207.
[13]	Enatsu Y, Nakata Y, Muroya Y. Lyapunov functional techniques for the global stability analysis of a delayed SIRS epidemic model[J]. Nonlinear Analysis: Real World Applications,2012,13(5): 2120-2133.
[14]	XU Rui, MA Zhi-en. Global stability of a delayed SEIRS epidemic model with saturation incidence rate[J]. Nonlinear Dynamics,2010,61(1): 229-239.
[15]	SONG Mei, LIU Guang-chen, GUO Hong-xia. The local asymptotic stability of an SIR epidemic model with nonlinear incidence rate and time delay[J]. Journal of Biomathematics,2011,26(2): 255-262.
[16]	李鸣明, 刘贤宁, 吴凡. 一个具有非线性发生率的时滞SIR传染病模型的稳定性[J]. 西南大学学报(自然科学版), 2014,36(5): 61-66.(LI Ming-ming, LIU Xian-ning, WU Fan. The stability of a delayed SIR epidemic model with nonlinear incidence rate[J]. Journal of Southwest University(Natural Science Edition),2014,36(5): 61-66.(in Chinese))
[17]	MA Zhi-en, ZHOU Yi-cang, WU Jian-hong. Modeling and Dynamics of Infectious Diseases[M]. Beijing: Higher Education Press, 2009: 1-35, 289-314.
[18]	YANG Kuang. Delay Differential Equations With Applications in Population Dynamics [M]. San Diego: Academic Press, 1993.