# American Institute of Mathematical Sciences

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

## Threshold dynamics of a bacillary dysentery model with seasonal fluctuation

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

Received  November 2009 Revised  February 2010 Published  October 2010

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.
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:
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##### References:
 [1] , The Facts about shigella infections and bacillary dysentery,, Association of Medical Microbiologists, (). [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, (). [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,, , (). [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, (). [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.
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