Advertisement

Optimal Control Strategies of a Tuberculosis Model with Exogenous Reinfection

  • Yali Yang
  • Xiuchao Song
  • Yuzhou Wang
  • Guoyun Luo
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7389)

Abstract

For the tuberculosis (TB) model with exogenous reinfection, we study the impact of two control strategies: chemoprophylaxis and treatment. We focus primarily on controlling the disease using an objective function based on a combination of minimizing the number of TB infections and minimizing the cost of control strategies. By using Pontryagin’s Maximum Principle, we derive the optimal levels of the two controls. Numerical simulations of the optimal system indicate that the strategies should be improved as the exogenous reinfection transmission coefficient increasing.

Keywords

tuberculosis model exogenous reinfection optimal control pontryagin’s Maximum Principle 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    WHO: Global Tuberculosis Control, WHO Report 2011, Switzerland (2011)Google Scholar
  2. 2.
    WHO: Global Tuberculosis Control, WHO Report 2006, Geneva (2006)Google Scholar
  3. 3.
    Blower, S.M., Small, P.M., Howell, P.C.: Control Strategies for Tuberculosis Epidemics: New Models for Old Problems. Science 273, 497–500 (1996)CrossRefGoogle Scholar
  4. 4.
    Feng, Z.L., Castillo-Chavez, C., Capurro, A.F.: A Model for Tuberculosis with Exogenous Reinfection. Theor. Popul. Biol. 57, 235–247 (2000)zbMATHCrossRefGoogle Scholar
  5. 5.
    Russell, D.G., Barry, C.E., Flynn, J.L.: Tuberculosis: What We Don’t Know Can, and Does, Hurt Us. Science 328, 852–856 (2010)CrossRefGoogle Scholar
  6. 6.
    Ziv, E., Daley, C.L., Blower, S.M.: Early Therapy for Latent Tuberculosis Infection. Am. J. Epidemiol. 153, 381–385 (2001)CrossRefGoogle Scholar
  7. 7.
    Tchuenche, J., Khamis, S., Agusto, F., et al.: Optimal Control and Sensitivity Analysis of an Influenza Model with Treatment and Vaccination. Acta Biotheor. (2010)Google Scholar
  8. 8.
    Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., et al.: The Mathematical Theory of Optimal Processes. Wiley, New York (1962)zbMATHGoogle Scholar
  9. 9.
    Fleming, W.H., Rishel, R.W.: Deterministic and Stochastic Optimal Control. Springer, New York (1975)zbMATHGoogle Scholar
  10. 10.
    Castillo-chavez, C., Song, B.J.: Dynamical Models of Tuberculosis and Their Applications. Math. Biosci. Eng. 1, 361–404 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Blower, S.M., McLean, A.R., Porco, T.C., et al.: The Intrinsic Transmission Dynamics of Tuberculosis Epidemics. Nature Med. 1, 815–821 (1995)CrossRefGoogle Scholar
  12. 12.
    Bhunu, C., Garira, W.: Tuberculosis Transmission Model with Chemoprophylaxis and Treatment. Bulletin of Mathematical Biology 70, 1163–1191 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Rodriguesa, P., Gomes, M.G.M., Rebelo, C.: Drug Resistance in Tuberculosis-a Reinfection Model. Theor. Pop. Biol. 71, 196–212 (2007)CrossRefGoogle Scholar
  14. 14.
    Ted, C., Megan, M.: Modeling Epidemics of Multidrug-Resisitant M.tuberculosis of Heterogeneous Fitness. Nature Medicine 10, 1117–1121 (2004)CrossRefGoogle Scholar
  15. 15.
    Sanchez, M.A., Blower, S.M.: Uncertainty and Sensitivity Analysis of the Basic Reproductive Rate: Tuberculosis as an Example. American Journal of Epidemiology 145, 1127–1137 (1997)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Yali Yang
    • 1
    • 2
  • Xiuchao Song
    • 2
  • Yuzhou Wang
    • 2
  • Guoyun Luo
    • 3
  1. 1.College of Mathematics and Information ScienceShaanxi Normal UniversityXi’anChina
  2. 2.College of ScienceAir Force Engineering UniversityXi’anChina
  3. 3.Unit 94170 of the PLAXi’anChina

Personalised recommendations