Bulletin of Mathematical Biology

, Volume 72, Issue 4, pp 931–952

A Tuberculosis Model with Seasonality

Original Article

DOI: 10.1007/s11538-009-9477-8

Cite this article as:
Liu, L., Zhao, XQ. & Zhou, Y. Bull. Math. Biol. (2010) 72: 931. doi:10.1007/s11538-009-9477-8


The statistical data of tuberculosis (TB) cases show seasonal fluctuations in many countries. A TB model incorporating seasonality is developed and the basic reproduction ratio R0 is defined. It is shown that the disease-free equilibrium is globally asymptotically stable and the disease eventually disappears if R0<1, and there exists at least one positive periodic solution and the disease is uniformly persistent if R0>1. Numerical simulations indicate that there may be a unique positive periodic solution which is globally asymptotically stable if R0>1. Parameter values of the model are estimated according to demographic and epidemiological data in China. The simulation results are in good accordance with the seasonal variation of the reported cases of active TB in China.


Seasonal patternPeriodic solutionBasic reproduction ratioGlobal stabilityUniform persistence

Copyright information

© Society for Mathematical Biology 2009

Authors and Affiliations

  1. 1.Department of MathematicsXi’an Jiaotong UniversityXi’anChina
  2. 2.Department of Mathematics and StatisticsMemorial University of NewfoundlandSt. John’sCanada