Optimal Control Measures for Tuberculosis in a Population Affected with Insurgency
In this chapter, optimal control theory is applied to a mathematical model describing the population dynamics of tuberculosis (TB) with variability in susceptibility due to difference in awareness level. Seeking to minimize the number of high-risk suscept
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Abstract In this chapter, optimal control theory is applied to a mathematical model describing the population dynamics of tuberculosis (TB) with variability in susceptibility due to difference in awareness level. Seeking to minimize the number of high-risk susceptible individuals with low level of TB awareness and to maximize the number of isolated actively-infected individuals placed under Directly Observed Treatment Short-Course (DOTS), we incorporated time-dependent control functions that represent educational campaign programs in the midst of insurgency, and case finding techniques for chronic TB cases, as they affect the dynamics of TB in a population. A particular case of the TB model without controls is presented and analyzed. Furthermore, the optimal controls are characterized in terms of the optimality systems, which are solved numerically for several scenarios using an iterative method with Runge-Kutta fourth order scheme. Numerical simulations were performed for various setting to illustrate the effect of the controls on the population dynamics of the disease in a given population. Keywords Tuberculosis · Mathematical model · Awareness campaign · Case finding techniques · Insurgency · Optimal control theory · Numerical simulations
1 Introduction Tuberculosis (TB) is an infectious disease caused by Mycobacterium tuberculosis baccillus, and it remains one of the world’s deadliest diseases [47]. According to the 2017 World Health Organization (WHO) Global TB Report, there was an estimated 10.4 million persons who fell ill with TB in 2016: 90% were adults, 65% were male, 10% were people living with HIV (74% in Africa) and about 56% from five A. O. Egonmwan · D. Okuonghae (B) Department of Mathematics, University of Benin, P.M.B. 1154, Benin City, Nigeria e-mail: [email protected]; [email protected] A. O. Egonmwan e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. T. Smith et al. (eds.), Mathematics Applied to Engineering, Modelling, and Social Issues, Studies in Systems, Decision and Control 200, https://doi.org/10.1007/978-3-030-12232-4_19
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countries: India, Indonesia, China, the Philippines and Pakistan [50]. In 2016, there were approximately 1.3 million TB deaths among HIV-negative people as well as an additional 374,000 deaths among HIV-positive people [50]. Drug-resistant TB has continued to be a major problem in the effective management and control of TB in a population. In 2016 alone, there were an estimated 600,000 new TB cases with resistance to rifampicin (RR-TB), of which 490,000 had cases of multidrug-resistant TB (MDR-TB) [50]. A global control strategy adopted by the WHO in the mid 1970s to help reduce the burden resulting from pulmonary TB and to help promote and support proper treatment of patients with latent and active TB is the Directly Observed Treatment Short-Course (DOTS). The DOTS strategy makes it compulsory for actively-infected TB patients to complete their treatment regime by making
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