Threshold dynamics of a bacillary dysentery model with seasonal fluctuation

Previous Article
A nonlocal and fully nonlinear degenerate parabolic system from strain-gradient plasticity
DCDS-B Home
This Issue
Next Article

January 2011, 15(1): 1-14. doi: 10.3934/dcdsb.2011.15.1

Threshold dynamics of a bacillary dysentery model with seasonal fluctuation

Zhenguo Bai ^1, and Yicang Zhou ^1,

1.	Department of Mathematics, Xi’an Jiaotong University, Xi’an, 710049, China

Received November 2009 Revised February 2010 Published October 2010

Abstract
Related Papers

A bacillary dysentery model with seasonal fluctuation is formulated and studied. The basic reproductive number $\mathcal {R}_0$ is introduced to investigate the disease dynamics in seasonal fluctuation environments. It is shown that there exists only the disease-free periodic solution which is globally asymptotically stable if $\mathcal {R}_0<1$, and there exists a positive periodic solution if $\mathcal {R}_0>1$. $\mathcal {R}_0$ is a threshold parameter, its magnitude determines the extinction or the persistence of the disease. Parameters in the model are estimated on the basis of bacillary dysentery epidemic data. Numerical simulations have been carried out to describe the transmission process of bacillary dysentery in China.

Keywords: Bacillary dysentery, global asymptotic stability., basic reproductive number, seasonal fluctuation, periodic solution.

Mathematics Subject Classification: Primary: 92D30, 34K13; Secondary: 37N2.

Citation: Zhenguo Bai, Yicang Zhou. Threshold dynamics of a bacillary dysentery model with seasonal fluctuation. Discrete and Continuous Dynamical Systems - B, 2011, 15 (1) : 1-14. doi: 10.3934/dcdsb.2011.15.1

References:

[1]	, The Facts about shigella infections and bacillary dysentery, Association of Medical Microbiologists, http://www.amm.co.uk/files/factsabout/fa_shig.htm.
[2]	, Disease Control and Public Health,, China's Health Statistical Yearbook 2009, (2009).
[3]	, Main Population Data in 2008, China, China Population and Development Research Center, http://www.cpirc.org.cn/en-cpdrc/en-file/endata/en-data-10.html.
[4]	, Birth rate, Death Rate and Natural Growth Rate of Population,, China Statistical Yearbook 2008, (2008).
[5]	, Population and Its Composition,, China Statistical Yearbook 2008, (2008).
[6]	, http://baike.baidu.com/view/1161053.htm, http://baike.baidu.com/view/1161053.htm.
[7]	Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Academic Press, Boston, 1993.
[8]	V. Lakshmikantham, S. Leela and A. A. Martynyuk, "Stability Analysis of Nonlinear Systems," Marcell Dekker, Inc., New York, Basel, 1989.
[9]	L. Liu, X. -Q. Zhao and Y. Zhou, A tuberculosis model with seasonality, Bulletin of Mathematical Biology, 72 (2010), 931-952. doi: doi:10.1007/s11538-009-9477-8.
[10]	, National report of notifiable diseases, 2005-2008, Ministry of Health of the People's Republic of China, http://www.moh.gov.cn/publicfiles//business/htmlfiles/zwgkzt/pyq/list.htm.
[11]	Z. Teng and L. Chen, The positive periodic solutions for high dimensional periodic Kolmogorov-type systems with delays, Acta Mathematicae Applicatae Sinica (Chinese Series), 22 (1999), 446-456.
[12]	X. Wang, F. Tao, D. Xiao, Lee H, Deen J, Gong J, et al., Trend and disease burden of bacillary dysentery in China (1991-2000), Bull World Health Organ., 84 (2006), 561-568.
[13]	W. Wang, X. -Q. Zhao, Threshold Dynamics for Compartmental Epidemic Models in Periodic Environments, J. Dyn. Diff. Equat., 20 (2008), 699-717.
[14]	X. -Q. Zhao, "Dynamical Systems in Population Biology," Springer-Verlag, New York, 2003.

show all references

References:

[1]	, The Facts about shigella infections and bacillary dysentery, Association of Medical Microbiologists, http://www.amm.co.uk/files/factsabout/fa_shig.htm.
[2]	, Disease Control and Public Health,, China's Health Statistical Yearbook 2009, (2009).
[3]	, Main Population Data in 2008, China, China Population and Development Research Center, http://www.cpirc.org.cn/en-cpdrc/en-file/endata/en-data-10.html.
[4]	, Birth rate, Death Rate and Natural Growth Rate of Population,, China Statistical Yearbook 2008, (2008).
[5]	, Population and Its Composition,, China Statistical Yearbook 2008, (2008).
[6]	, http://baike.baidu.com/view/1161053.htm, http://baike.baidu.com/view/1161053.htm.
[7]	Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Academic Press, Boston, 1993.
[8]	V. Lakshmikantham, S. Leela and A. A. Martynyuk, "Stability Analysis of Nonlinear Systems," Marcell Dekker, Inc., New York, Basel, 1989.
[9]	L. Liu, X. -Q. Zhao and Y. Zhou, A tuberculosis model with seasonality, Bulletin of Mathematical Biology, 72 (2010), 931-952. doi: doi:10.1007/s11538-009-9477-8.
[10]	, National report of notifiable diseases, 2005-2008, Ministry of Health of the People's Republic of China, http://www.moh.gov.cn/publicfiles//business/htmlfiles/zwgkzt/pyq/list.htm.
[11]	Z. Teng and L. Chen, The positive periodic solutions for high dimensional periodic Kolmogorov-type systems with delays, Acta Mathematicae Applicatae Sinica (Chinese Series), 22 (1999), 446-456.
[12]	X. Wang, F. Tao, D. Xiao, Lee H, Deen J, Gong J, et al., Trend and disease burden of bacillary dysentery in China (1991-2000), Bull World Health Organ., 84 (2006), 561-568.
[13]	W. Wang, X. -Q. Zhao, Threshold Dynamics for Compartmental Epidemic Models in Periodic Environments, J. Dyn. Diff. Equat., 20 (2008), 699-717.
[14]	X. -Q. Zhao, "Dynamical Systems in Population Biology," Springer-Verlag, New York, 2003.

[1]	Timothy C. Reluga, Jan Medlock, Alison Galvani. The discounted reproductive number for epidemiology. Mathematical Biosciences & Engineering, 2009, 6 (2) : 377-393. doi: 10.3934/mbe.2009.6.377
[2]	Hui Cao, Yicang Zhou. The basic reproduction number of discrete SIR and SEIS models with periodic parameters. Discrete and Continuous Dynamical Systems - B, 2013, 18 (1) : 37-56. doi: 10.3934/dcdsb.2013.18.37
[3]	Ariel Cintrón-Arias, Carlos Castillo-Chávez, Luís M. A. Bettencourt, Alun L. Lloyd, H. T. Banks. The estimation of the effective reproductive number from disease outbreak data. Mathematical Biosciences & Engineering, 2009, 6 (2) : 261-282. doi: 10.3934/mbe.2009.6.261
[4]	Dominika Pilarczyk. Asymptotic stability of singular solution to nonlinear heat equation. Discrete and Continuous Dynamical Systems, 2009, 25 (3) : 991-1001. doi: 10.3934/dcds.2009.25.991
[5]	Lakmi Niwanthi Wadippuli, Ivan Gudoshnikov, Oleg Makarenkov. Global asymptotic stability of nonconvex sweeping processes. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 1129-1139. doi: 10.3934/dcdsb.2019212
[6]	Christian Lax, Sebastian Walcher. A note on global asymptotic stability of nonautonomous master equations. Discrete and Continuous Dynamical Systems - B, 2013, 18 (8) : 2143-2149. doi: 10.3934/dcdsb.2013.18.2143
[7]	Rafael Luís, Sandra Mendonça. A note on global stability in the periodic logistic map. Discrete and Continuous Dynamical Systems - B, 2020, 25 (11) : 4211-4220. doi: 10.3934/dcdsb.2020094
[8]	Mi-Young Kim. Uniqueness and stability of positive periodic numerical solution of an epidemic model. Discrete and Continuous Dynamical Systems - B, 2007, 7 (2) : 365-375. doi: 10.3934/dcdsb.2007.7.365
[9]	Qiang Li, Mei Wei. Existence and asymptotic stability of periodic solutions for neutral evolution equations with delay. Evolution Equations and Control Theory, 2020, 9 (3) : 753-772. doi: 10.3934/eect.2020032
[10]	Kentarou Fujie. Global asymptotic stability in a chemotaxis-growth model for tumor invasion. Discrete and Continuous Dynamical Systems - S, 2020, 13 (2) : 203-209. doi: 10.3934/dcdss.2020011
[11]	Fabien Crauste. Global Asymptotic Stability and Hopf Bifurcation for a Blood Cell Production Model. Mathematical Biosciences & Engineering, 2006, 3 (2) : 325-346. doi: 10.3934/mbe.2006.3.325
[12]	Shiwang Ma, Xiao-Qiang Zhao. Global asymptotic stability of minimal fronts in monostable lattice equations. Discrete and Continuous Dynamical Systems, 2008, 21 (1) : 259-275. doi: 10.3934/dcds.2008.21.259
[13]	Anatoli F. Ivanov, Musa A. Mammadov. Global asymptotic stability in a class of nonlinear differential delay equations. Conference Publications, 2011, 2011 (Special) : 727-736. doi: 10.3934/proc.2011.2011.727
[14]	Tianhui Yang, Lei Zhang. Remarks on basic reproduction ratios for periodic abstract functional differential equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6771-6782. doi: 10.3934/dcdsb.2019166
[15]	Dongfeng Zhang, Junxiang Xu. On the reducibility of analytic quasi-periodic systems with Liouvillean basic frequencies. Communications on Pure and Applied Analysis, 2022, 21 (4) : 1417-1445. doi: 10.3934/cpaa.2022024
[16]	Zigen Ouyang, Chunhua Ou. Global stability and convergence rate of traveling waves for a nonlocal model in periodic media. Discrete and Continuous Dynamical Systems - B, 2012, 17 (3) : 993-1007. doi: 10.3934/dcdsb.2012.17.993
[17]	Nicolas Bacaër, Xamxinur Abdurahman, Jianli Ye, Pierre Auger. On the basic reproduction number $R_0$ in sexual activity models for HIV/AIDS epidemics: Example from Yunnan, China. Mathematical Biosciences & Engineering, 2007, 4 (4) : 595-607. doi: 10.3934/mbe.2007.4.595
[18]	Nitu Kumari, Sumit Kumar, Sandeep Sharma, Fateh Singh, Rana Parshad. Basic reproduction number estimation and forecasting of COVID-19: A case study of India, Brazil and Peru. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2021170
[19]	Zhijun Zhang. Optimal global asymptotic behavior of the solution to a singular monge-ampère equation. Communications on Pure and Applied Analysis, 2020, 19 (2) : 1129-1145. doi: 10.3934/cpaa.2020053
[20]	Jing Li, Boling Guo, Lan Zeng, Yitong Pei. Global weak solution and smooth solution of the periodic initial value problem for the generalized Landau-Lifshitz-Bloch equation in high dimensions. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1345-1360. doi: 10.3934/dcdsb.2019230

2021 Impact Factor: 1.497

Tools

Metrics

Other articles
by authors

on AIMS
Zhenguo Bai Yicang Zhou
on Google Scholar
Zhenguo Bai Yicang Zhou

[Back to Top]