Optimal control of the transmission dynamics of tuberculosis

This paper deals with the problem of optimal control for the transmission dynamics of tuberculosis (TB). A tuberculosis model which incorporates the essential biological and epidemiological features of the disease such as exogenous reinfection and chemoprophylaxis of latently infected individuals, and treatment of the infectious is developed and rigorously analyzed. Based on this continuous model, the tuberculosis control is formulated and solved as an optimal control theory problem, indicating how a control term on the chemoprophylaxis should be introduced in the population to reduce the number of individuals with active TB. The feedback control law has been proved to be capable of reducing the number of individuals with active TB. An advantage is that the proposed scheme accounts for the energy wasted by the controller and the closed-loop performance on tracking. Numerical results show the performance of the optimization strategy.

[1]  F. M. Moukam Kakmeni,et al.  Chaos synchronization and duration time of a class of uncertain chaotic systems , 2006, Math. Comput. Simul..

[2]  Richard Bellman,et al.  Introduction to the mathematical theory of control processes , 1967 .

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

[4]  John M. Neuberger,et al.  A minmax principle, index of the critical point, and existence of sign-changing solutions to elliptic boundary value problems , 1998 .

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

[6]  Christopher Dye,et al.  Global Burden of Tuberculosis: Estimated Incidence, Prevalence, and Mortality by Country , 1999 .

[7]  P. Davies,et al.  Multi-Drug-Resistant Tuberculosis , 2004 .

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

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

[10]  J Li,et al.  Relapse in persons treated for drug-susceptible tuberculosis in a population with high coinfection with human immunodeficiency virus in New York City. , 2001, Clinical infectious diseases : an official publication of the Infectious Diseases Society of America.

[11]  Carlos Castillo-Chavez,et al.  Markers of disease evolution: the case of tuberculosis. , 2002, Journal of theoretical biology.

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

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

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

[15]  P E Fine,et al.  The long-term dynamics of tuberculosis and other diseases with long serial intervals: implications of and for changing reproduction numbers , 1998, Epidemiology and Infection.

[16]  J. Chin,et al.  Control of Communicable Diseases Manual , 2002 .

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

[18]  G. Renard,et al.  Le C.H.R. 10.000 calculateur hybride a reseaux a commutation electronique , 1969 .

[19]  Christopher Dye,et al.  Global burden of tuberculosis , 1999 .

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

[21]  C. Castillo-Chavez,et al.  Global stability of an age-structure model for TB and its applications to optimal vaccination strategies. , 1998, Mathematical biosciences.

[22]  Miller Bess,et al.  Preventive therapy for tuberculosis. , 1993, The Medical clinics of North America.

[23]  De Cock Km,et al.  Will DOTS do it? A reappraisal of tuberculosis control in countries with high rates of HIV infection. , 1999 .

[24]  R. May.  Uses and Abuses of Mathematics in Biology , 2004, Science.

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

[26]  Gieri Simonett,et al.  Mathematical models in medical and health science , 1998 .

[27]  William L. Brogan,et al.  Modern control theory (3rd ed.) , 1991 .

[28]  Rafael Castro-Linares,et al.  Trajectory tracking for non-holonomic cars: A linear approach to controlled leader-follower formation , 2010, 49th IEEE Conference on Decision and Control (CDC).

[29]  J.A.M. Felippe de Souza,et al.  Optimal control theory applied to the anti-viral treatment of AIDS , 2000, Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187).

[30]  A D Harries,et al.  Relapse and recurrent tuberculosis in the context of a national tuberculosis control programme. , 2000, Transactions of the Royal Society of Tropical Medicine and Hygiene.

[31]  K. Schmitt,et al.  Permanence and the dynamics of biological systems. , 1992, Mathematical biosciences.

[32]  S. Blower,et al.  The intrinsic transmission dynamics of tuberculosis epidemics , 1995, Nature Medicine.

[33]  J. C. Burkill,et al.  Ordinary Differential Equations , 1964 .

[34]  D. Kirschner,et al.  Optimal control of the chemotherapy of HIV , 1997, Journal of mathematical biology.

[35]  José Manoel Balthazar,et al.  On control and synchronization in chaotic and hyperchaotic systems via linear feedback control , 2008 .

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

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

[38]  C. Dye,et al.  Consensus statement. Global burden of tuberculosis: estimated incidence, prevalence, and mortality by country. WHO Global Surveillance and Monitoring Project. , 1999, JAMA.

[39]  C. Bhunu,et al.  Modelling the effects of pre-exposure and post-exposure vaccines in tuberculosis control. , 2008, Journal of theoretical biology.

[40]  Xingfu Zou,et al.  Modeling diseases with latency and relapse. , 2007, Mathematical biosciences and engineering : MBE.

[41]  Marat Rafikov,et al.  Simulating malaria model for different treatment intensities in a variable environment , 2007 .

[42]  Xiefei Yan,et al.  Optimal quarantine and isolation strategies in epidemics control , 2007 .

[43]  Suzanne Lenhart,et al.  Optimal control for degenerate parabolic equations with logistic growth , 1995 .

[44]  Folashade B. Agusto,et al.  Optimal chemoprophylaxis and treatment control strategies of a tuberculosis transmission model , 2009 .

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

[46]  J. Gerberding,et al.  Understanding, predicting and controlling the emergence of drug-resistant tuberculosis: a theoretical framework , 1998, Journal of Molecular Medicine.

[47]  S. Lenhart,et al.  OPTIMIZING CHEMOTHERAPY IN AN HIV MODEL , 1998 .

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

[49]  S. Blower,et al.  Control Strategies for Tuberculosis Epidemics: New Models for Old Problems , 1996, Science.

[50]  W. Brogan.  Modern Control Theory , 1971 .

[51]  S. Blower,et al.  Quantifying the intrinsic transmission dynamics of tuberculosis. , 1998, Theoretical population biology.

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

[53]  鍋谷 清治,et al.  The Mathematical Theory of Optimal Processes by L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mischenko , 1964 .

[54]  G W Comstock,et al.  Epidemiology of tuberculosis. , 1982, The American review of respiratory disease.

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

[56]  V. Lakshmikantham,et al.  Stability Analysis of Nonlinear Systems , 1988 .

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

[58]  Zhong-Ping Jiang,et al.  Design of Robust Adaptive Controllers for Nonlinear Systems with Dynamic Uncertainties , 1998, Autom..

[59]  C. McCluskey,et al.  Lyapunov functions for tuberculosis models with fast and slow progression. , 2006, Mathematical biosciences and engineering : MBE.