Optimal treatment control and bifurcation analysis of a tuberculosis model with effect of multiple re-infections
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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.
KeywordsTuberculosis Reproduction number Transcritical bifurcation Backward bifurcation Optimal control
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.
- 2.Diekmann O, Heesterbeek JAP (2000) Mathematical epidemiology of infectious diseases. In: Model building, analysis and interpretation, Wiley Series in Mathematical and Computational Biology, John Wiley and Sons, ChichesterGoogle Scholar
- 25.Okosun KO, Ouifki R, Marcus N (2011) Optimal control analysis of a malaria disease transmission model that includes treatment and vaccination with waning immunity. Biosystems 106:136–145. doi: 10.1016/j.biosystems.2011.07.006
- 33.Kar TK, Batabyal A (2011) Stability analysis and optimal control of an SIR epidemic model with vaccination. Biosystems 104:127–135Google Scholar
- 36.Lukes DL (1982) Differential equations: classical to controlled. In: Mathematics in science and Engineering, vol. 162, Academic Press, New York, p. 182Google Scholar
- 39.Pontryagin LS, Boltyanskii VG, Gamkrelidze RV, Mishchenko EF (1962) The mathematical theory of optimal processes. Wiley, New YorkGoogle Scholar
- 40.Zaman G, Kang YH, Jung IH (2007) Optimal vaccination and treatment in the SIR epidemic model. Proc KSIAM 3:31–33Google Scholar