Optimal treatment control and bifurcation analysis of a tuberculosis model with effect of multiple re-infections

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Abstract

We derive and analyze a tuberculosis (TB) model including exogenous re-infection and endogenous reactivation, and the re-infection among the treated individuals. The disease-free equilibrium and the existence criterion of endemic equilibrium are investigated. The basic reproduction number \(R_0 \) is derived and it is found that the disease-free equilibrium is stable when \(R_0 <1\), unstable for \(R_0 >1\), and the system undergoes a transcritical bifurcation at the disease-free equilibrium when \(R_0 =1\). Furthermore, for \(R_0 <1\), there are two endemic equilibria, one of which is stable and other one is unstable, indicating the occurrence of backward bifurcation. The local stability analysis of the disease-free and the endemic equilibrium is shown. Also, we studied the sensitivity analysis of the system in refer to some crucial model parameters and the sensitivity indices of \(R_0\) to parameters for the TB model are obtained. Using Pontryagin’s maximum principle, we have discussed about the optimal control of the disease. Various simulation works are given throughout the paper to support our analytical results.

Keywords

Tuberculosis Reproduction number Transcritical bifurcation Backward bifurcation  Optimal control 

Notes

Acknowledgments

Research of T. K. Kar is supported by the Council of Scientific and Industrial Research (CSIR) (Sanction No: 25(0224)/14/EMR-II, dated: 2/12/2014), Human Resource Development Group, India.

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© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of MathematicsShibpur Sri Ramkrishna VidyalayaHowrahIndia
  2. 2.Department of MathematicsIndian Institute of Engineering Science and TechnologyShibpur, HowrahIndia

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