# American Institute of Mathematical Sciences

May  2016, 21(3): 1009-1022. doi: 10.3934/dcdsb.2016.21.1009

## Global stability and optimal control for a tuberculosis model with vaccination and treatment

 1 College of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710062, China 2 College of Mathematics and Information Science, Shaanxi Normal University, Xi'an, 710062 3 Science College, Air Force Engineering University, Xi'an 710051, China, China, China

Received  February 2015 Revised  July 2015 Published  January 2016

We formulate a mathematical model to explore the impact of vaccination and treatment on the transmission dynamics of tuberculosis (TB). We develop a technique to prove that the basic reproduction number is the threshold of global stability of the disease-free and endemic equilibria. We then incorporate a control term and evaluate the cost of control strategies, and then perform an optimal control analysis by Pontryagin's maximum principle. Our numerical simulations suggest that the maximum vaccination strategy should be enforced regardless of its efficacy.
Citation: Yali Yang, Sanyi Tang, Xiaohong Ren, Huiwen Zhao, Chenping Guo. Global stability and optimal control for a tuberculosis model with vaccination and treatment. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 1009-1022. doi: 10.3934/dcdsb.2016.21.1009
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