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2012, 9(1): 111-122. doi: 10.3934/mbe.2012.9.111

## Threshold dynamics for a Tuberculosis model with seasonality

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

Received  February 2011 Revised  March 2011 Published  December 2011

In this paper, we investigate a SEILR tuberculosis model incorporating the effect of seasonal fluctuation, where the loss of sight class is considered. The basic reproduction number $R_{0}$ is defined. It is shown that the disease-free equilibrium is globally asymptotically stable and the disease eventually disappears if $R_{0}<1$, and there exists at least one positive periodic solution and the disease is uniformly persistent if $R_{0}>1$. Numerical simulations are provided to illustrate analytical results.
Citation: Xinli Hu. Threshold dynamics for a Tuberculosis model with seasonality. Mathematical Biosciences & Engineering, 2012, 9 (1) : 111-122. doi: 10.3934/mbe.2012.9.111
##### References:
 [1] D. Bleed, C. Watt and C. Dye, World health report 2001: Global tuberculosis control, Technical report, World Health Organization, WHO/CDS/TB/2001.287., Available from: \url{http://whqlibdoc.who.int/hq/2001/WHO_CDS_TB_2001.287.pdf}., 2001 ().   Google Scholar [2] S. M. Blower, The intrinsic transmission dynamics of tuberculosis epidemics, Nature Medicine, 1 (1995), 815-821. doi: 10.1038/nm0895-815.  Google Scholar [3] Samuel Bowong and Jean Jules Tewa, Mathematical analysis of a tuberculosis model with differential infectivity, Commun. Nonlinear Sci. Numer. Simul., 14 (2009), 4010-4021.  Google Scholar [4] B. Song, C. Castillo-Chavez and J. P. Aparicio, Tuberculosis models with fast and slow dynamics: The role of close and casual contacts, Mathematical Biosciences, 180 (2002), 187-205. doi: 10.1016/S0025-5564(02)00112-8.  Google Scholar [5] C. C. McCluskey, Global stability for a class of mass action systems allowing for latency in tuberculosis, Journal of Mathematical Analysis and Application, 338 (2008), 518-535. doi: 10.1016/j.jmaa.2007.05.012.  Google Scholar [6] C. C. McCluskey, Lyapunov functions for tuberculosis models with fast and slow progression, Mathematical Biosciences and Engineering, 3 (2006), 603-614. doi: 10.3934/mbe.2006.3.603.  Google Scholar [7] Z. Feng, W. Huang and C. Castillo-Chavez, On the role of variable latent periods in mathematical models for tuberculosis, Journal of Dynamics and Differential Equations, 13 (2001), 425-452. doi: 10.1023/A:1016688209771.  Google Scholar [8] S. M. Blower, P. M. Small and P. C. Hopewell, Control strategies for tuberculosis epidemics: New models for old problems, Science, 273 (1996), 497-500. doi: 10.1126/science.273.5274.497.  Google Scholar [9] L. Liu, X.-Q. Zhao and Y. Zhou, A tuberculosis model with seasonality, Bull. Math. Biol., 72 (2010), 931-952. doi: 10.1007/s11538-009-9477-8.  Google Scholar [10] O. Sharomi, C. N. Podder, A. B. Gumel and B. Song, Mathematical analysis of the transmission dynamics of HIV/TB coinfection in the presence of treatment, Math. Biosci. Eng., 5 (2008), 145-174. doi: 10.3934/mbe.2008.5.145.  Google Scholar [11] Y. Zhou, K. Khan, Z. Feng and J. Wu, Projection of tuberculosis incidence with increasing immigration trends, J. Theor. Biol., 254 (2008), 215-228. doi: 10.1016/j.jtbi.2008.05.026.  Google Scholar [12] H. L. Smith, Subharmonic bifurcation in a S-I-R epidemic model, J. Math. Biol., 17 (1983), 163-177. doi: 10.1007/BF00305757.  Google Scholar [13] C. J. Duncan, S. R. Duncan and S. Scott, Oscillatory dynamics of small-pox and the impact of vaccination, J. Theor. Biol., 183 (1996), 447-454. doi: 10.1006/jtbi.1996.0234.  Google Scholar [14] H. L. Smith and P. Waltman, "The Theory of the Chemostat. Dynamics of Microbial Competition," Cembridge Studies in Mathematical Biology, 13, Cambridge Univ. Press, Cambridge, 1995.  Google Scholar [15] H. L. Smith, "Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems," Mathematical Surveys and Monographs, 41, American Mathematical Society, Providence, RI, 1995.  Google Scholar [16] W. Wang and X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dynam. Differential Equations, 20 (2008), 699-717. doi: 10.1007/s10884-008-9111-8.  Google Scholar [17] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar [18] H. R. Thieme, Convergence results and a Poincaré-Bendixson trichotomy for asymptotical autonomous differential equations, J. Math. Biol., 30 (1992), 755-763.  Google Scholar [19] F. Zhang and X.-Q. Zhao, A periodic epidemic model in a patchy environment, J. Math. Anal. Appl., 325 (2007), 496-516. doi: 10.1016/j.jmaa.2006.01.085.  Google Scholar [20] X.-Q. Zhao, "Dynamical Systems in Population Biology," CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 16, Springer-Verlag, New York, 2003.  Google Scholar

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##### References:
 [1] D. Bleed, C. Watt and C. Dye, World health report 2001: Global tuberculosis control, Technical report, World Health Organization, WHO/CDS/TB/2001.287., Available from: \url{http://whqlibdoc.who.int/hq/2001/WHO_CDS_TB_2001.287.pdf}., 2001 ().   Google Scholar [2] S. M. Blower, The intrinsic transmission dynamics of tuberculosis epidemics, Nature Medicine, 1 (1995), 815-821. doi: 10.1038/nm0895-815.  Google Scholar [3] Samuel Bowong and Jean Jules Tewa, Mathematical analysis of a tuberculosis model with differential infectivity, Commun. Nonlinear Sci. Numer. Simul., 14 (2009), 4010-4021.  Google Scholar [4] B. Song, C. Castillo-Chavez and J. P. Aparicio, Tuberculosis models with fast and slow dynamics: The role of close and casual contacts, Mathematical Biosciences, 180 (2002), 187-205. doi: 10.1016/S0025-5564(02)00112-8.  Google Scholar [5] C. C. McCluskey, Global stability for a class of mass action systems allowing for latency in tuberculosis, Journal of Mathematical Analysis and Application, 338 (2008), 518-535. doi: 10.1016/j.jmaa.2007.05.012.  Google Scholar [6] C. C. McCluskey, Lyapunov functions for tuberculosis models with fast and slow progression, Mathematical Biosciences and Engineering, 3 (2006), 603-614. doi: 10.3934/mbe.2006.3.603.  Google Scholar [7] Z. Feng, W. Huang and C. Castillo-Chavez, On the role of variable latent periods in mathematical models for tuberculosis, Journal of Dynamics and Differential Equations, 13 (2001), 425-452. doi: 10.1023/A:1016688209771.  Google Scholar [8] S. M. Blower, P. M. Small and P. C. Hopewell, Control strategies for tuberculosis epidemics: New models for old problems, Science, 273 (1996), 497-500. doi: 10.1126/science.273.5274.497.  Google Scholar [9] L. Liu, X.-Q. Zhao and Y. Zhou, A tuberculosis model with seasonality, Bull. Math. Biol., 72 (2010), 931-952. doi: 10.1007/s11538-009-9477-8.  Google Scholar [10] O. Sharomi, C. N. Podder, A. B. Gumel and B. Song, Mathematical analysis of the transmission dynamics of HIV/TB coinfection in the presence of treatment, Math. Biosci. Eng., 5 (2008), 145-174. doi: 10.3934/mbe.2008.5.145.  Google Scholar [11] Y. Zhou, K. Khan, Z. Feng and J. Wu, Projection of tuberculosis incidence with increasing immigration trends, J. Theor. Biol., 254 (2008), 215-228. doi: 10.1016/j.jtbi.2008.05.026.  Google Scholar [12] H. L. Smith, Subharmonic bifurcation in a S-I-R epidemic model, J. Math. Biol., 17 (1983), 163-177. doi: 10.1007/BF00305757.  Google Scholar [13] C. J. Duncan, S. R. Duncan and S. Scott, Oscillatory dynamics of small-pox and the impact of vaccination, J. Theor. Biol., 183 (1996), 447-454. doi: 10.1006/jtbi.1996.0234.  Google Scholar [14] H. L. Smith and P. Waltman, "The Theory of the Chemostat. Dynamics of Microbial Competition," Cembridge Studies in Mathematical Biology, 13, Cambridge Univ. Press, Cambridge, 1995.  Google Scholar [15] H. L. Smith, "Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems," Mathematical Surveys and Monographs, 41, American Mathematical Society, Providence, RI, 1995.  Google Scholar [16] W. Wang and X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dynam. Differential Equations, 20 (2008), 699-717. doi: 10.1007/s10884-008-9111-8.  Google Scholar [17] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar [18] H. R. Thieme, Convergence results and a Poincaré-Bendixson trichotomy for asymptotical autonomous differential equations, J. Math. Biol., 30 (1992), 755-763.  Google Scholar [19] F. Zhang and X.-Q. Zhao, A periodic epidemic model in a patchy environment, J. Math. Anal. Appl., 325 (2007), 496-516. doi: 10.1016/j.jmaa.2006.01.085.  Google Scholar [20] X.-Q. Zhao, "Dynamical Systems in Population Biology," CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 16, Springer-Verlag, New York, 2003.  Google Scholar
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