具有时滞和脉冲接种的非线性发生率的流行病模型分析

Abstract

Abstract:

The aim of this study is to introduce an impulsive SEIQRS epidemic model with time delays, quarantine measure and nonlinear incidence rate. The total population size is varied. The global attractivity of an `infection-free' periodic solution, the existence, and the permanence of an endemic periodic solution are investigated. We obtain a sufficient condition for the permanence of the epidemic model with pulse vaccination. We show that time delay, pulse vaccination can bring different effects on the dynamic behavior of the model by numerical analysis. Our results also show a smaller pulse vaccination rate or a shorter latent period of the disease or a shorter immunity period of the recovered could cause global attractive `infection-free' periodic solution to lose and epidemic disease to be permanent. The main feature of this study is to introduce three time delays, nonlinear incidence rates and impulses into the SEIQRS epidemic model and give pulse vaccination strategies.

Key words: Nonlinear incidence rate, Time delay, Pulse vaccination, Quarantine, Permanence

CLC Number:

34D23

JIANG Yu, MEI Li-Quan, SONG Xin-Yu, TIAN Wei-Jun, DING Xue-Mei. Analysis of an Impulsive Epidemic Model with Time Delays and Nonlinear Incidence Rate[J].Acta mathematica scientia,Series A, 2012, 32(4): 670-684.

Trendmd

References

[1] Li J Q, Ma Z E, Zhou Y C. Global analysis of SIS epidemic model with a simple vaccination and multiple endemic equilibria.
Acta Mathematica Scientia, 2006, 26B(1): 83--93

[2] Li M, Smith H, Wang L. Global dynamics of an SEIR epidemic model with vertical transmission. SIAM J Appl Math, 2001, 62: 58--69

[3] Jin Z, Ma Z E, Han M A. Global stability of an SIRS epidemic model with delays. Acta Mathematica Scientia, 2006, 26B(2): 291--306

[4] Shulgin B, Stone L, Agur Z. Theoretical examination of pulse vaccination policy in the sir epidemic model. Math Comp Mod, 2000, 31(4/5): 207--215

[5] Nokes D J, Swinton J. The control of childhood viral infection by pulse vaccination. IMA J Math Appl Med Biol, 1995, 12: 29--53

[6] Ramsay M, Gay N, Miller E. The epidemiology of measles in England and Wales: rationale for 1994 national vaccination campaign.
Commun Dis Rep, 1994, 4: 141--146

[7] Sabin A B. Measles, killer of millions in developing countries: strategies of elimination and continuing control. Eur J Epidemiol, 1991, 7: 1--22

[8] Shulgin B, Stone L, Agur Z. Pulse vaccination strategy in the SIR epidemic mode. Bull Math Biol, 1998, 60: 1123--1148

[9] Pei Y Z, Liu S Y, Gao S J, et al. A delayed SEIQR epidemic model with pulse vaccination and the quarantine measure. Computers and Mathematics with Applications, 2009, 58: 135--145

[10] Sattenspiel L, Herring D A. Simulating the effect of quarantine on the spread of the 1918--1919 flu in central Canada. Bull Math Biol, 2003, 65: 1--26

[11] Bainov D D, Simeonov P S. Impulsive Differential Equations: Periodic Solutions and Applications. New York: Longman Scientific and Technical, 1993

[12] Cooke K, Van Den Driessche P. Analysis of an SEIRS epidemic model with two delays. J Math Biol, 1996, 35: 240--260

[13] Kuang Y. Delay Differential Equation with Application in Population Dynamics. New York: Academic Press, 1993: 67--70

[14] Lakshmikantham V, Bainov D D, Simeonov P S. Theory of Impulsive Differential Equations. Singapore: World Scientific, 1989

[15] Zhang J, Li J Q, Ma Z E. Global dynamics of an SEIR epidemic model with immigration of different compartments. Acta Mathematica Scientia, 2006, 26B(3): 551--567

[16] Korobeinikov A, Wake G C. Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models. Appl Math Lett, 2002, 15: 955--960

[17] Song X Y, Jiang Y, Wei H M. Analysis of a saturation incidence SVEIRS epidemic model with pulse and two time delays. Appl Math Comp, 2009, 214: 381--390

[18] Wang W, Ruan S. Bifurcations in an epidemic model with constant removal rate of the infectives. J Math Anal Appl, 2004, 291: 774--793

[19] Wei H M, Jiang Y, Song X Y, et al. Global attractivity and permanence of a SVEIR epidemic model with pulse vaccination and time delay.
J Comp Appl Math, 2009, 229: 302--312

[20] Jiang Y, Wei H M, Song X Y, et al. Global attractivity and permanence of a delayed SVEIR epidemic model with pulse vaccination and saturation incidence. Appl Math Comp, 2009, 213: 312--321

[21] Dong L Z, Chen L S, Sun L H. Optimal harvesting policy for inshore-offshore fishery model with impulsive diffusion. Acta Mathematica Scientia, 2007, 27: 405--412

[22] Lu Zhonghua, Gao Shujing, Chen Lansun. Analysis of an SI epidemic model with nonlinear transmission and stage structure. Acta Mathematica Scientia, 2003, 23B(4): 440--446

[23] Xu R, Chen L S, Chaplain M A J. Global asymptotic stability in n-species nonautonomous Lotka-Volterra competitive systems with delays. Acta Mathematica Scientia, 2003, 23: 208--218

Analysis of an Impulsive Epidemic Model with Time Delays and Nonlinear Incidence Rate

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 13

Metrics

Comments

Recommended 10

[1]	Zheng Lifei, Guo Jie, Wu Meihua, Wang Xiaorui, Wan Aying. The Research on a Class of the Predator-Prey-Mutualist System with Delays [J]. Acta mathematica scientia,Series A, 2018, 38(5): 1001-1013.
[2]	Li Zhouhong, Zhang Fengshuo, Cao Jinde, Alsaedi Ahmed, Alsaadi Fuad E. Almost Periodic Solution for a Non-Autonomous Two Species Competitive System with Feedback Controls on Time Scales [J]. Acta mathematica scientia,Series A, 2017, 37(4): 730-750.
[3]	Luo Ricai, Xu Honglei, Wang Wusheng. Globally Asymptotic Stability of A New Class of Neutral Neural Networks with Time-Varying Delays [J]. Acta mathematica scientia,Series A, 2015, 35(3): 634-640.
[4]	Zhang Shuwen. A Stochastic Predator-Prey System with Time Delays and Prey Dispersal [J]. Acta mathematica scientia,Series A, 2015, 35(3): 592-603.
[5]	ZHONG Min-Ling, LIU Xiu-Xiang. Dynamical Analysis of a Predator-prey System with Hassell-Varley-Holling Functional Response [J]. Acta mathematica scientia,Series A, 2011, 31(5): 1295-1310.
[6]	Wu Shiliang ;Li Wantong. Stability and Traveling Fronts in Lotka-Volterra Cooperation Model with Stage Structure [J]. Acta mathematica scientia,Series A, 2008, 28(3): 454-464.
[7]	Gui Zhanji; Jia Jing; Ge Weigao. Global Stability of Single-species Diffusion Models with Time Delay [J]. Acta mathematica scientia,Series A, 2007, 27(3): 496-505.
[8]	Gao Shujing ;Chen Lansun. ermanence and Global Stability for Single-species Model with Three Life Stages and Time Delay [J]. Acta mathematica scientia,Series A, 2006, 26(4): 527-533.
[9]	Xu Rui; Hao Feilong ;Chen Lansun. A Stage-Structured Predator-Prey Model with Time Delays [J]. Acta mathematica scientia,Series A, 2006, 26(3): 387-395.
[10]	DIAO Li-Chun, ZHANG Qiang-Ling, YANG Qi-Chang. The Induction Control of a Single Species Model with Stage Structure [J]. Acta mathematica scientia,Series A, 2005, 25(5): 710-717.
[11]	YUAN San-Ling, MA Zhi-En, HAN Mao-An. Global Stability on an SIS Epidemic Model with Time Delays [J]. Acta mathematica scientia,Series A, 2005, 25(3): 349-356.
[12]	CHEN You-Peng, XIE Chun-Hong. Quenching for a Nonlinear Degenerate Parabolic Equation with Time Delay [J]. Acta mathematica scientia,Series A, 2004, 24(3): 265-274.
[13]	SONG Xin-Yu, CHEN Lan-Sun. Periodic Solution of \|a \|Delay Differential Equation of \|Plankton Allelopathy [J]. Acta mathematica scientia,Series A, 2003, 23(1): 8-13.