Stability analysis of multi-group deterministic and stochastic epidemic models with vaccination rate

doi:10.1088/1674-1056/23/9/090201

Abstract We discuss in this paper a deterministic multi-group MSIR epidemic model with a vaccination rate, the basic reproduction number R₀, a key parameter in epidemiology, is a threshold which determines the persistence or extinction of the disease. By using Lyapunov function techniques, we show if R₀ is greater than 1 and the deterministic model obeys some conditions, then the disease will prevail, the infective persists and the endemic state is asymptotically stable in a feasible region. If R₀ is less than or equal to 1, then the infective disappear so the disease dies out. In addition, stochastic noises around the endemic equilibrium will be added to the deterministic MSIR model in order that the deterministic model is extended to a system of stochastic ordinary differential equations. In the stochastic version, we carry out a detailed analysis on the asymptotic behavior of the stochastic model. In addition, regarding the value of R₀, when the stochastic system obeys some conditions and R₀ is greater than 1, we deduce the stochastic system is stochastically asymptotically stable. Finally, the deterministic and stochastic model dynamics are illustrated through computer simulations.

Keywords: MSIR epidemic model equilibrium graph theory Brownian motion

Received: 01 January 2014
Revised: 24 February 2014
Accepted manuscript online:

PACS:	02.40.Vh	(Global analysis and analysis on manifolds)
	89.20.Ff	(Computer science and technology)
	02.60.Lj	(Ordinary and partial differential equations; boundary value problems)

Fund: Project supported by the National Natural Science Foundation of China (Grant No. 11326078) and the Project of Science and Technology of Heilongjiang Province of China (Grant No. 12531187).

Corresponding Authors: Fan Xiao-Ming, Han Qi-Xing
E-mail: fanxm093@163.com;hanqixing123@163.com

Cite this article:

Wang Zhi-Gang (王志刚), Gao Rui-Mei (高瑞梅), Fan Xiao-Ming (樊晓明), Han Qi-Xing (韩七星) Stability analysis of multi-group deterministic and stochastic epidemic models with vaccination rate 2014 Chin. Phys. B 23 090201

[1]	Kermack W O and McKendrick A G 1927 Proc. Roy. Soc. London Ser. A 115 700
[2]	Pei W, Chen Z and Yuan Z 2008 Chin. Phys. B 17 373
[3]	Wang W, Liu H, Cai Y and Li Z 2011 Chin. Phys. B 20 074702
[4]	Mishra B K and Jha N 2010 Appl. Math. Model. 34 71
[5]	Lahrouz A, Omari L and Kiouach D 2011 Nonlinear Analysis: Modelling and Control 16 59
[6]	Beretta E, Kolmanovskii V and Shaikhet L 1998 Math. Comput. Simul. 45 269
[7]	Shaikhet L 1998 Stab. Control Theory Appl. 1 3
[8]	Carletti M 2002 Math. Biosci. 175 117
[9]	Sarkar R R and Banerjee S 2005 Math. Biosci. 196 65
[10]	Shaikhet L 2008 Dyn. Systems Appl. 17 235
[11]	Ji C Y, Jiang D Q and Shi N Z 2011 Physica A 390 1747
[12]	Lajmanovich A and Yorke J A 1976 Math. Biosci. 28 221
[13]	Hethcote H W 2000 SIAM Rev. 42 599
[14]	Gong Y, Song Y and Jiang G 2013 Chin. Phys. B 22 040204
[15]	Wang Z and Fan X 2013 Appl. Math. Model. 37 8673
[16]	Fan X, Wang Z and Xu X 2012 Abstr. Appl. Anal. 2012 132095
[17]	Hethcote H W 1978 Theor. Popul. Biol. 14 338
[18]	Guo H, Li M Y and Shuai Z 2008 Proc. Am. Math. Soc. 136 2793
[19]	Li M Y, Shuai Z and Wang C 2010 J. Math. Anal. Appl. 361 38
[20]	Guo H, Li M Y and Shuai Z 2006 Can. Appl. Math. Quaert. 14 259
[21]	Dalal N, Greenhalgh D and Mao X 2007 J. Math. Anal. Appl. 325 36
[22]	Ji C, Jiang D and Shi N 2009 J. Math. Anal. Appl. 359 482
[23]	Ji C, Jiang D and Li X 2011 J. Comput. Appl. Math. 235 1326
[24]	Tornatore E, Buccellato S M and Vetro P 2005 Physica A 354 111
[25]	Yu J, Jiang D and Shi N 2009 J. Math. Anal. Appl. 360 235
[26]	Imhof L and Walcher S 2005 J. Dyn. Diff. Equ. 217 26
[27]	Berman A and Plemmons R J 1979 Nonnegative Matrices in the Mathematical Sciences (New York: Academic Press)
[28]	van den Driessche P and Watmough J 2002 Math. Biosci. 180 29
[29]	Shaikhet L 2011 Lyapunov Functionals and Stability of Stochastic Difference Equations (London, Dordrecht, Heidelberg, New York: Springer-Verlag)
[30]	Shaikhet L 2013 Lyapunov Functionals and Stability of Stochastic Functional Differential Equations (Dordrecht, Heidelberg, New York, London: Springer-Verlag)
[31]	Mao X 1997 Stochastic Differential Equations and Applications (Chichester: Horwood Publication)

[1]	How graph features decipher the soot assisted pigmental energy transport in leaves? A laser-assisted thermal lens study in nanobiophotonics S Sankararaman. Chin. Phys. B, 2022, 31(8): 088201.
[2]	Physical aspects of magnetized Jeffrey nanomaterial flow with irreversibility analysis Fazal Haq, Muhammad Ijaz Khan, Sami Ullah Khan, Khadijah M Abualnaja, and M A El-Shorbagy. Chin. Phys. B, 2022, 31(8): 084703.
[3]	Effect of crystallographic orientations on transport properties of methylthiol-terminated permethyloligosilane molecular junction Ming-Lang Wang(王明郎), Bo-Han Zhang(张博涵), Wen-Fei Zhang(张雯斐), Xin-Yue Tian(田馨月), Guang-Ping Zhang(张广平), and Chuan-Kui Wang(王传奎). Chin. Phys. B, 2022, 31(7): 077303.
[4]	A rational quantum state sharing protocol with semi-off-line dealer Hua-Li Zhang(张花丽), Bi-Chen Che(车碧琛), Zhao Dou(窦钊), Yu Yang(杨榆), and Xiu-Bo Chen(陈秀波). Chin. Phys. B, 2022, 31(5): 050309.
[5]	Ratchet transport of self-propelled chimeras in an asymmetric periodic structure Wei-Jing Zhu(朱薇静) and Bao-Quan Ai(艾保全). Chin. Phys. B, 2022, 31(4): 040503.
[6]	Theoretical design of thermal spin molecular logic gates by using a combinational molecular junction Yi Guo(郭逸), Peng Zhao(赵朋), and Gang Chen(陈刚). Chin. Phys. B, 2022, 31(4): 047202.
[7]	Detection of multi-spin interaction of a quenched XY chain by the average work and the relative entropy Xiu-Xing Zhang(张修兴), Fang-Jv Li(李芳菊), Kai Wang(王凯), Jing Xue(薛晶), Guang-Wen Huo(霍广文), Ai-Ping Fang(方爱平), and Hong-Rong Li(李宏荣). Chin. Phys. B, 2021, 30(9): 090504.
[8]	Passivation and dissociation of P_b-type defects at a-SiO₂/Si interface Xue-Hua Liu(刘雪华), Wei-Feng Xie(谢伟锋), Yang Liu(刘杨), and Xu Zuo(左旭). Chin. Phys. B, 2021, 30(9): 097101.
[9]	Device design based on the covalent homocouplingof porphine molecules Minghui Qu(曲明慧), Jiayi He(贺家怡), Kexin Liu(刘可心), Liemao Cao(曹烈茂), Yipeng Zhao(赵宜鹏), Jing Zeng(曾晶), and Guanghui Zhou(周光辉). Chin. Phys. B, 2021, 30(9): 098504.
[10]	Origin of anomalous enhancement of the absorption coefficient in a PN junction Xiansheng Tang(唐先胜), Baoan Sun(孙保安), Chen Yue(岳琛), Xinxin Li(李欣欣), Junyang Zhang(张珺玚), Zhen Deng(邓震), Chunhua Du(杜春花), Wenxin Wang(王文新), Haiqiang Jia(贾海强), Yang Jiang(江洋), Weihua Wang(汪卫华), and Hong Chen(陈弘). Chin. Phys. B, 2021, 30(9): 097804.
[11]	Numerical simulation of anode heat transfer of nitrogen arc utilizing two-temperature chemical non-equilibrium model Chong Niu(牛冲), Surong Sun(孙素蓉), Jianghong Sun(孙江宏), and Haixing Wang(王海兴). Chin. Phys. B, 2021, 30(9): 095206.
[12]	Nonequilibrium free energy and information flow of a double quantum-dot system with Coulomb coupling Zhiyuan Lin(林智远), Tong Fu(付彤), Juying Xiao(肖菊英), Shanhe Su(苏山河), Jincan Chen(陈金灿), and Yanchao Zhang(张艳超). Chin. Phys. B, 2021, 30(8): 080501.
[13]	Impact of counter-rotating-wave term on quantum heat transfer and phonon statistics in nonequilibrium qubit-phonon hybrid system Chen Wang(王晨), Lu-Qin Wang(王鲁钦), and Jie Ren(任捷). Chin. Phys. B, 2021, 30(3): 030506.
[14]	Exploring how hydrogen at gold-sulfur interface affects spin transport in single-molecule junction Jing Zeng(曾晶), Ke-Qiu Chen(陈克求), Yanhong Zhou(周艳红). Chin. Phys. B, 2020, 29(8): 088503.
[15]	A polaron theory of quantum thermal transistor in nonequilibrium three-level systems Chen Wang(王晨), Da-Zhi Xu(徐大智). Chin. Phys. B, 2020, 29(8): 080504.

No Suggested Reading articles found!

Viewed

Full text

Abstract

Cited

Metrics
Related Articles

Stability analysis of multi-group deterministic and stochastic epidemic models with vaccination rate

Cite this article:

Online attention