Optimal control using state-dependent Riccati equation of lost of sight in a tuberculosis model
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Optimal control using state-dependent Riccati equation of lost of sight in a tuberculosis model Yves Emvudu · Ramses Djidjou Demasse · Dany Djeudeu
Received: 1 July 2011 / Revised: 22 October 2012 / Accepted: 31 January 2013 / Published online: 16 April 2013 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2013
Abstract This paper deals with the problem of optimal control for the transmission dynamics of tuberculosis (TB). A TB model which considers the existence of a new class (mainly in the African context) is considered: the lost of sight individuals. Based on the model formulated and studied in Tewa and Bowong (Commun Nonlinear Sci Numer Simul 14:4010–4021, 2009), the TB control is formulated and solved as an optimal control theory problem using state-dependent Riccati equation. This control strategy indicates how the control of the class of lost of sight can considerably influence the basic reproduction ratio so as to reduce their number. Numerical results illustrate the performance of the optimization strategy. Keywords Nonlinear dynamical systems · Tuberculosis models · Basic reproduction ratio · Optimal control · Lost of sight Mathematics Subject Classification
92D25 · 49K15 · 34D23 · 92D30 · 49J15
Communicated by Ruben Spies. Y. Emvudu (B) · R. D. Demasse · D. Djeudeu Laboratory of Applied Mathematics, Department of Mathematics, Faculty of Science, University of Yaounde 1, P.O.Box 812, Yaounde, Cameroon e-mail: [email protected] D. Djeudeu e-mail: [email protected] R. D. Demasse UMI 209 IRD& UPMC UMMISCO, Bondy, France e-mail: [email protected] R. D. Demasse LIRIMA, project team GRIMCAPE, Yaounde, Cameroon
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1 Introduction Tuberculosis (TB) is a disease caused by infection with Mycobacterium tuberculosis, which most frequently affects the lungs (pulmonary TB). At present, about 95 % of the estimated 8 million new cases of TB occurring each year are in developing countries, where 80 % occur among people between the ages of 15 and 59 Dye et al. (1999). In Sub-Saharan Africa, TB is the leading cause of mortality, and in developing countries it accounts for an estimated 2 million deaths which accounts for a quarter of avoidable adult deaths Raviglione et al. (1997). TB was assumed to be on its way out in developed countries until the number of TB cases began to increase in the 1980s. With this return, we face the paradox of a wellknown bacteria, fully treatable with efficient and affordable drugs according to internationally recommended guidelines, which yet causes increasing human suffering and death. As the world is experiencing the devastating effect of HIV/AIDS epidemic, it is now necessary to ask why we have so far failed to control TB and define the limits of the global TB control programs Raviglione (2002). Currently, half of the people living with HIV are TB co-infected and three quarters of all dually infected people live in Sub-Saharan Africa. In Cameroon for example, it is estimated that in the absence of effective epidemiology statistics, there
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