Stability of an SIR Epidemic Model With 2 Patches and Population Movement

FU Jin-bo; CHEN Lan-sun

doi:10.21656/1000-0887.370087

Volume 38 Issue 4

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FU Jin-bo, CHEN Lan-sun. Stability of an SIR Epidemic Model With 2 Patches and Population Movement[J]. Applied Mathematics and Mechanics, 2017, 38(4): 486-494. doi: 10.21656/1000-0887.370087

Citation:

FU Jin-bo, CHEN Lan-sun. Stability of an SIR Epidemic Model With 2 Patches and Population Movement[J]. Applied Mathematics and Mechanics, 2017, 38(4): 486-494. doi: 10.21656/1000-0887.370087

Citation:

FU Jin-bo, CHEN Lan-sun. Stability of an SIR Epidemic Model With 2 Patches and Population Movement[J]. Applied Mathematics and Mechanics, 2017, 38(4): 486-494. doi: 10.21656/1000-0887.370087

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Stability of an SIR Epidemic Model With 2 Patches and Population Movement

doi: 10.21656/1000-0887.370087

FU Jin-bo¹,
CHEN Lan-sun^1,2

¹Minnan Science and Technology Institute, Fujian Normal University, Quanzhou, Fujian 362332, P.R.China;²Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, P.R.China

Funds: The National Natural Science Foundation of China（11371306）

Received Date: 2016-03-28
Rev Recd Date: 2016-09-18
Publish Date: 2017-04-15

Abstract

Abstract

Based on the epidemic dynamics, in view of the population movement between 2 patches and the nonlinear infection rate, a class of SIR epidemic model with 2 patches and population movement was established. With the qualitative method and the stability method for ordinary differential equations, the permanence of the model and the existence of nonnegative equilibrium points were analyzed. Through construction of proper Lyapunov functions and according to the limit system theory, the sufficient conditions for the global asymptotic stability of the diseasefree equilibrium points and the endemic equilibrium points were obtained. The results show that, the basic reproduction number makes a threshold to determine wether the disease spreads or not. When the basic reproduction number is less than or equal to 1, the infection will gradually disappear, the virus will tend to be extinct; when the dominant regeneration number of the virus is greater than 1 and satisfies the permanence conditions, the infection will persist, and the virus will continue to prevail and become an endemic disease.
- SIR epidemic model,
- equilibrium point,
- basic reproduction number,
- global asymptotic stability

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References(15)

References

[1]	CHEN Liu-juan. Glogal stability of a SEIR epidemic model with nonmonotone incidence rate[J]. Journal of Biomathematics,2009,24(4): 591-598.
[2]	何艳辉, 唐三一. 经典SIR模型辨识和参数估计问题[J]. 应用数学和力学, 2013,34(3): 252-258.(HE Yan-hui, TANG San-yi. Identification and parameter estimation for classical SIR model[J]. Applied Mathematics and Mechanics,2013,34(3): 252-258.(in Chinese))
[3]	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-ying, XIAO Yan-ni. An epidemic model with saturated media/psychological impact[J]. Applied Mathematics and Mechanics,2013,34(4): 399-407.(in Chinese))
[4]	杨亚莉, 李建全, 刘万萌, 等. 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, et al. 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))
[5]	谢英超, 程燕, 贺天宇. 一类具有非线性发生率的时滞传染病模型的全局稳定性[J]. 应用数学和力学, 2015,36(10): 1107-1116.(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.(in Chinese))
[6]	傅金波, 陈兰荪, 程荣福. 具有潜伏期和免疫应答的时滞病毒感染模型的全局稳定性[J]. 高校应用数学学报(A辑), 2015,30(4): 379-388.(FU Jin-bo, CHEN Lan-sun, CHENG Rong-fu. Global stability of a delayed viral infection model with latent period and immune response[J].Applied Mathematics: A Journal of Chinese Universities (Ser A),2015,30(4): 379-388.(in Chinese))
[7]	傅金波, 陈兰荪, 程荣福. 具有Logistic增长和治疗的SIRS传染病模型的后向分支[J]. 吉林大学学报(理学版), 2015,53(6):1166-1170.(FU Jin-bo, CHEN Lan-sun, CHENG Rong-fu. Backward bifurcation of a SIRS epidemic model with Logistic growth and treatment[J]. Journal of Jilin University (Science Edition),2015,53(6): 1166-1170.(in Chinese))
[8]	Yoshida N, Hara T. Global stability of a delayed SIR epidemic model with density dependent birth and death rates[J]. Computational and Applied Mathematics,2007,201(2): 339-347.
[9]	唐晓明, 薛亚奎. 具有饱和治疗函数与密度制约的SIS传染病模型的后向分支[J]. 数学的实践与认识, 2010,40(24): 241-246.(TANG Xiao-ming, XUE Ya-kui. Backward bifurcation of a SIS epidemic model with density dependent birth and death rates and saturated treatment function[J]. Mathematics in Practice and Theory,2010,40(24): 241-246.(in Chinese))
[10]	Brauer F, van den Driessche P. Models for transmission of disease with immigration of infectives[J]. Mathematical Biosciences,2001,171(2): 143-154.
[11]	WANG Wen-di, Mulone G. Threshold of disease transmission in a patch environment[J].Journal of Mathematical Analysis and Applications,2003,285(1): 321-335.
[12]	Takeuchi Y, LIU Xian-ning, CUI Jin-gan. Global dynamics of SIS models with transport-related infection[J]. Journal of Mathematical Analysis and Applications,2007,329(2): 1460-1471.
[13]	李冰, 王辉. 一类在两斑块内人口迁移的传染病模型的研究[J]. 北京工商大学学报(自然科学版),2009,27(1): 56-61.(LI Bing, WANG Hui. Research on an SIS epidemic model with population dispersal in two patches[J]. Journal of Beijing Technology and Business University (Natural Science Edition),2009,27(1): 56-61.(in Chinese))
[14]	马知恩, 周义仓, 王稳地, 等. 传染病动力学的数学建模与研究[M]. 北京: 科学出版社, 2004.(MA Zhi-en, ZHOU Yi-cang, WANG Wen-di, et al. Mathematics Modeling and Research of Infectious Disease Dynamics [M]. Beijing: Science Press, 2004.(in Chinese))
[15]	YANG Kuang. Delay Differential Equation With Application in Population Dynamics [M]. Boston: Academic Press, 1993.