Optimal control for a tuberculosis model with reinfection and post-exposure interventions.

We apply optimal control theory to a tuberculosis model given by a system of ordinary differential equations. Optimal control strategies are proposed to minimize the cost of interventions, considering reinfection and post-exposure interventions. They depend on the parameters of the model and reduce effectively the number of active infectious and persistent latent individuals. The time that the optimal controls are at the upper bound increase with the transmission coefficient. A general explicit expression for the basic reproduction number is obtained and its sensitivity with respect to the model parameters is discussed. Numerical results show the usefulness of the optimization strategies.

[1]  Giulia Marchetti,et al.  Molecular Epidemiology Study of Exogenous Reinfection in an Area with a Low Incidence of Tuberculosis , 2001, Journal of Clinical Microbiology.

[2]  O Afonso,et al.  Exogenous reinfection with tuberculosis on a European island with a moderate incidence of disease. , 2001, American journal of respiratory and critical care medicine.

[3]  C. Castillo-Chavez,et al.  To treat or not to treat: the case of tuberculosis , 1997, Journal of mathematical biology.

[4]  W. Fleming,et al.  Deterministic and Stochastic Optimal Control , 1975 .

[5]  M. Thakur,et al.  Global tuberculosis control report. , 2001, The National medical journal of India.

[6]  Suzanne Lenhart,et al.  Optimal control of treatments in a two-strain tuberculosis model , 2002 .

[7]  P D van Helden,et al.  Exogenous reinfection as a cause of recurrent tuberculosis after curative treatment. , 1999, The New England journal of medicine.

[8]  Global Tuberculosis Programme.  Global tuberculosis control : WHO report , 1997 .

[9]  P. Small,et al.  Management of tuberculosis in the United States. , 2001, The New England journal of medicine.

[10]  Marc Lipsitch,et al.  Beneficial and perverse effects of isoniazid preventive therapy for latent tuberculosis infection in HIV-tuberculosis coinfected populations. , 2006, Proceedings of the National Academy of Sciences of the United States of America.

[11]  J. Hyman,et al.  Determining Important Parameters in the Spread of Malaria Through the Sensitivity Analysis of a Mathematical Model , 2008, Bulletin of mathematical biology.

[12]  Dany Djeudeu,et al.  Optimal Control of the Lost to Follow Up in a Tuberculosis Model , 2011, Comput. Math. Methods Medicine.

[13]  Khalid Hattaf,et al.  Optimal Control of Tuberculosis with Exogenous Reinfection , 2009 .

[14]  Delfim F. M. Torres,et al.  Dengue disease, basic reproduction number and control , 2011, Int. J. Comput. Math..

[15]  Marion Muehlen,et al.  Implications of partial immunity on the prospects for tuberculosis control by post-exposure interventions. , 2007, Journal of theoretical biology.

[16]  Zi Sang,et al.  Optimal control of a vector-host epidemics model , 2011 .

[17]  Carlos Castillo-Chavez,et al.  A Model for Tuberculosis w Exogenous Reinfection , 2000 .

[18]  Trina M Girimont,et al.  Dengue disease. , 2010, AAOHN journal : official journal of the American Association of Occupational Health Nurses.

[19]  J. Ball.  OPTIMIZATION—THEORY AND APPLICATIONS Problems with Ordinary Differential Equations (Applications of Mathematics, 17) , 1984 .

[20]  Samuel Bowong,et al.  Optimal control of the transmission dynamics of tuberculosis , 2010 .

[21]  Lamberto Cesari,et al.  Optimization-Theory And Applications , 1983 .

[22]  Margaret E Kruk,et al.  Timing of default from tuberculosis treatment: a systematic review , 2008, Tropical medicine & international health : TM & IH.

[23]  M Elizabeth Halloran,et al.  Epidemiological benefits of more-effective tuberculosis vaccines, drugs, and diagnostics , 2009, Proceedings of the National Academy of Sciences.

[24]  Delfim F. M. Torres,et al.  Dynamics of Dengue epidemics when using optimal control , 2010, Math. Comput. Model..

[25]  John T. Workman,et al.  Optimal Control Applied to Biological Models , 2007 .

[26]  M. L. Chambers.  The Mathematical Theory of Optimal Processes , 1965 .

[27]  L. S. Pontryagin,et al.  Mathematical Theory of Optimal Processes , 1962 .