一类具有信息变量和等级治愈率的SIR传染病模型的研究

doi:10.6040/j.issn.1671-9352.0.2015.212

Abstract

Abstract: The stability of the SIR epidemic model with information variable, saturated incidence rate and graded cure rate was studied. We obtain the basic reproduction number R₀, The local asymptotical stability of equilibria is verified by analyzing the eigenvalues and using the Routh-Hurwitz criterion,this model exhibits two bifurcations, one is transcritical bifurcation and the other is Hopf bifurcation. We also discuss the global asymptotical stability of the disease free equilibrium by constructing a Lyapunov function and the endemic equilibrium by autonomous convergence theorem. A numerical analysis is given to show the effectiveness of the main results.

Key words: information variable, bifurcation of model, global stability, SIR epidemic model, graded cure rate

CLC Number:

O175.12

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.

References

[1] 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,2012. MA Zhien, ZHOU Yicang, WANG Wendi, et al. The mathematical modeling and research on dynamics of epidemic[M]. Beijing: Science Press, 2012.
[2] DONOFRIO A, MANFREDI P, SALINELLI E. Bifurcation thresholds in a SIR model with information dependent vaccination[J]. Mathematical Modelling of Natural Phenomena, 2007, 2(1):26-43.
[3] DONOFRIO A, MANFREDI P, SALINELLI E. Vaccinating behaviour, information and the dynamics of SIR vaccine preventable diseases[J]. Theor Popul Biol, 2007, 71(3):301-317.
[4] LIN Yuguo, JIANG Daqing, LIU Taihui. Nontrivial periodic solution of a stochastic epidemic model with seasonal variation[J]. Applied Mathematics Letters, 2015, 45(1):103-107.
[5] MUROYA Y, ENATSU Y, KUNIYA T. Global stability of extended multi-group SIR epidemic model with patches through migration and cross patch infection[J]. Acta Math Sci, 2013, 33(2):341-361.
[6] BUONOMO B, DONOFRIO A, LACITIGNOLA D. Global stability of an SIR epidemic model with information dependent vaccination[J]. Mathematical Biosciences, 2008, 216(1):9-16.
[7] PANG Jianhua, CUI Jingan, HUI Jing. Rich dynamics of epidemic model with sub-optimal immunity and nonlinear recovery rate[J]. Mathematical and Computer Modelling, 2011, 54(1/2):440-448.
[8] MUROYA Y, LI H X, KUNIYA T. Complete global analysis of an SIRS epidemic model with graded cure and incomplete recovery rates[J]. J Math Anal Appl, 2014, 410(2):719-732.
[9] HAO Lijie, JIANG Guirong, LIU Suyu, et al. Global dynamics of an SIRS epidemic model with saturation incidence rate[J]. BioSystems, 2013, 114(1):56-63.
[10] ELAIW A M, ALSHAMRANI N H. Global stability of humoral immunity virus dynamics models with nonlinear infection rate and removal[J]. Nonlinear Anal: Real World Appl, 2015, 26:161-190.
[11] ZHANG Xiaobing, HUO Haifeng, XIANG Hong, et al. An SIRS epidemic model with pulse vaccination and non-monotonic incidence rate[J]. Nonlinear Analysis: Hybrid Systems, 2013, 8(5):13-21.
[12] KAR T K, PRASANTA K M. Global dynamics and bifurcation in delayed SIR model[J] , Nonlinear Anal: Real World Appl, 2011, 12(4):2058-2068.
[13] YAN Caijuan, JIA Jianwen. Dynamics of an SIR epidemic model with information variable and limited medical resources revisited[J]. Discrete Dynamics in Nature and Society, 2014, 2014: 861710.1-861710.11.
[14] GUCKENHEIMER J, HOLMES P. Nonlinear oscillations, dynamical systems, and bifurcations of vector fields[M]. New York: Springer-Verlag, 1983: 117-156.
[15] LI M, MULDOWNEY J S. A geometric approach to global-stability problems[J]. SIAM J Math Anal, 1996, 27(4):1070-1083.

Metrics

Viewed

Full text

Abstract

Cited

Shared

Discussed

Comments

Recommended 0

No Suggested Reading articles found!

Study on an SIR epidemic model with information variable and graded cure rate

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 4

Metrics

Comments

Recommended 0

[1]	BAI Bao-li, ZHANG Jian-gang, DU Wen-ju, YAN Hong-ming. Dynamic behavior analysis of a stochastic SIR epidemic model [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(4): 72-82.
[2]	YANG Jun-xian, XU li*. Global stability of a SIQS epidemic model with #br# nonlinear incidence rate and time delay [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2014, 49(05): 67-74.
[3]	JU Pei-jun. Sufficient and necessary conditions for global stability of generalized Lorenz system and its applications to chaos control [J]. J4, 2012, 47(10): 97-101.
[4]	DONG Chun-yan, FU Sheng-mao. The existence of non-constant positive steady state of an epidemic model for pest control [J]. J4, 2011, 46(3): 80-88.