Optimal intervention strategies for tuberculosis

Abstract This paper deals with the problem of optimal control of a deterministic model of tuberculosis (abbreviated as TB for tubercle bacillus). We first present and analyze an uncontrolled tuberculosis model which incorporates the essential biological and epidemiological features of the disease. The model is shown to exhibit the phenomenon of backward bifurcation, where a stable disease-free equilibrium co-exists with one or more stable endemic equilibria when the associated basic reproduction number is less than the unity. Based on this continuous model, the tuberculosis control is formulated and solved as an optimal control problem, indicating how control terms on the chemoprophylaxis and detection should be introduced in the population to reduce the number of individuals with active TB. Results provide a framework for designing the cost-effective strategies for TB with two intervention methods.

[1]  Samuel Bowong,et al.  Mathematical analysis of a tuberculosis model with differential infectivity , 2009 .

[2]  Christopher Dye,et al.  Assessment of worldwide tuberculosis control , 1997, The Lancet.

[3]  Hem Raj Joshi,et al.  Optimal control of an HIV immunology model , 2002 .

[4]  B. Anderson,et al.  Optimal control: linear quadratic methods , 1990 .

[5]  Julien Arino,et al.  Global Results for an Epidemic Model with Vaccination that Exhibits Backward Bifurcation , 2003, SIAM J. Appl. Math..

[6]  Xiao-Qiang Zhao,et al.  A Tuberculosis Model with Seasonality , 2010, Bulletin of mathematical biology.

[7]  S. Levin.  Lectu re Notes in Biomathematics , 1983 .

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

[9]  Shigui Ruan,et al.  Simulating the SARS outbreak in Beijing with limited data , 2003, Journal of Theoretical Biology.

[10]  R. May,et al.  Population biology of infectious diseases: Part II , 1979, Nature.

[11]  Fred Brauer,et al.  Backward bifurcations in simple vaccination models , 2004 .

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

[13]  Carlos Castillo-Chavez,et al.  Backwards bifurcations and catastrophe in simple models of fatal diseases , 1998, Journal of mathematical biology.

[14]  C. Bhunu,et al.  Tuberculosis Transmission Model with Chemoprophylaxis and Treatment , 2008, Bulletin of mathematical biology.

[15]  C. Castillo-Chavez,et al.  A model for tuberculosis with exogenous reinfection. , 2000, Theoretical population biology.

[16]  Herbert W. Hethcote,et al.  The Mathematics of Infectious Diseases , 2000, SIAM Rev..

[17]  MC Raviglione,et al.  Evolution of WHO policies for tuberculosis control, 1948–2001 , 2002, The Lancet.

[18]  L. Riley,et al.  Exogenous reinfection in tuberculosis. , 2005, The Lancet. Infectious diseases.

[19]  J. Watmough,et al.  Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. , 2002, Mathematical biosciences.

[20]  Thomas R Frieden,et al.  Tuberculosis control: past 10 years and future progress. , 2003, Tuberculosis.

[21]  C. Dye,et al.  The Population Dynamics and Control of Tuberculosis , 2010, Science.

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

[23]  Samuel Bowong,et al.  Global analysis of a dynamical model for transmission of tuberculosis with a general contact rate , 2010 .

[24]  Denise Kirschner,et al.  On treatment of tuberculosis in heterogeneous populations. , 2003, Journal of theoretical biology.

[25]  Rachid Ouifki,et al.  Modeling the joint epidemics of TB and HIV in a South African township , 2008, Journal of mathematical biology.

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

[27]  Carlos Castillo-Chavez,et al.  Dynamical models of tuberculosis and their applications. , 2004, Mathematical biosciences and engineering : MBE.

[28]  James Watmough,et al.  Role of incidence function in vaccine-induced backward bifurcation in some HIV models. , 2007, Mathematical biosciences.

[29]  S. Lenhart,et al.  Modeling Optimal Intervention Strategies for Cholera , 2010, Bulletin of mathematical biology.

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

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