A tuberculosis model with seasonality.

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 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 indicate that there may be a unique positive periodic solution which is globally asymptotically stable if R(0)>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.