Nonlinear Dynamics

, Volume 97, Issue 4, pp 2643–2659 | Cite as

Optimal control strategies for tuberculosis dynamics with exogenous reinfections in case of treatment at home and treatment in hospital

  • Tuğba Akman YıldızEmail author
  • Esra Karaoğlu
Original paper


We propose an extended mathematical model for tuberculosis dynamics including five groups as susceptible individuals, S(t); latent, infected with TB but not infectious individuals, E(t); individuals who have active TB and treated at home, \(I_1(t)\); individuals who have active TB and treated in hospital \(I_2(t)\) and recovered individuals R(t). We incorporate exogenous reinfection and infection of recovered individuals to extend the model. Since healthcare facilities are limited and treatment in hospital might hard to afford for the people living in resource-poor countries, being treated at home or hospital are two important cases affecting the transmission and eradication of the disease. Therefore, an optimal control problem (OCP) is constructed governed by the extended TB model where the importance of being treated at home and in hospital can be revealed and the success of the treatment can be measured mathematically in case of limited number of healthcare facilities and treatment expenses. The aim of applying optimal control strategy to this new SEIR model is to minimize the number of latent and infectious individuals as well as the cost of applying three controls, namely distancing control, \(u_1(t)\); effort that prevents the failure of treatment at home for active TB infectious individuals, \(u_2(t)\); and effort that prevents the failure of treatment in hospital for active TB infectious individuals, \(u_3(t)\). We set two different controls \(u_2(t)\) and \(u_3(t)\) special to the subgroups \(I_1(t)\) and \(I_2(t)\) so that we can catch the contribution of being treated in hospital or being treated at home. While constructing the OCP, the form of the reproduction number is used so that we can construct the OCP in a way to decrease the reproduction number. In addition, local asymptotic stability of the disease-free equilibrium (DFE) point is proven with the use of the reproduction number; but, global asymptotic stability of the DFE point may not be achieved due to exogenous reinfection. It means that the disease might become persistent even if there is a small infectious group of people in the population. Therefore, the need for optimal control strategy is revealed even for a small infectious subgroup. At the end, we present some numerical results to obtain the optimal intervention strategies in case the transition rate from/to home to/from hospital is large.


Optimal control Epidemiological models Tuberculosis Exogenous reinfection Basic reproduction number 

Mathematics Subject Classification

49J15 92D30 37N25 34D20 



The authors would like to thank the reviewers for their comments and suggestions that helped to improve the manuscript.

Author Contributions

All authors read and approved the final manuscript.


There is no funding for this research.

Compliance with ethical standards

Conflict of interest

The authors have no conflict of interest concerning the publication of this manuscript.


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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Logistics ManagementUniversity of Turkish Aeronautical AssociationAnkaraTurkey
  2. 2.Department of Electrical and Electronical EngineeringUniversity of Turkish Aeronautical AssociationAnkaraTurkey

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