Optimal control of a Tuberculosis model with state and control delays

Cristiana J. Silva; Helmut Maurer; Delfim F. M. Torres; Cristiana J. Silva; Helmut Maurer; Delfim F. M. Torres

doi:10.3934/mbe.2017021

Mathematical Biosciences and Engineering

2017, Volume 14, Issue 1: 321-337. doi: 10.3934/mbe.2017021

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Optimal control of a Tuberculosis model with state and control delays

1.
Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal
2.
Institute of Computational and Applied Mathematics, University of Münster, D-48149 Münster, Germany

Received: 31 October 2015 Accepted: 26 June 2016 Published: 01 January 2017
MSC : Primary: 34D30, 92D30; Secondary: 49M05, 93A30

We introduce delays in a tuberculosis (TB) model, representing the time delay on the diagnosis and commencement of treatment of individuals with active TB infection. The stability of the disease free and endemic equilibriums is investigated for any time delay. Corresponding optimal control problems, with time delays in both state and control variables, are formulated and studied. Although it is well-known that there is a delay between two to eight weeks between TB infection and reaction of body's immune system to tuberculin, delays for the active infected to be detected and treated, and delays on the treatment of persistent latent individuals due to clinical and patient reasons, which clearly justifies the introduction of time delays on state and control measures, our work seems to be the first to consider such time-delays for TB and apply time-delay optimal control to carry out the optimality analysis.
- Tuberculosis,
- time delays,
- stability,
- optimal control
Citation: Cristiana J. Silva, Helmut Maurer, Delfim F. M. Torres. Optimal control of a Tuberculosis model with state and control delays[J]. Mathematical Biosciences and Engineering, 2017, 14(1): 321-337. doi: 10.3934/mbe.2017021

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Abstract

We introduce delays in a tuberculosis (TB) model, representing the time delay on the diagnosis and commencement of treatment of individuals with active TB infection. The stability of the disease free and endemic equilibriums is investigated for any time delay. Corresponding optimal control problems, with time delays in both state and control variables, are formulated and studied. Although it is well-known that there is a delay between two to eight weeks between TB infection and reaction of body's immune system to tuberculin, delays for the active infected to be detected and treated, and delays on the treatment of persistent latent individuals due to clinical and patient reasons, which clearly justifies the introduction of time delays on state and control measures, our work seems to be the first to consider such time-delays for TB and apply time-delay optimal control to carry out the optimality analysis.

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