Abstract
A dynamical model of TB for two age groups that incorporate vaccination of children at birth, behavior change in adult population, treatment of infectious children and adults is formulated and analyzed. Three types of control measures (vaccination, behavior change and anti-TB treatment strategies) are applied with separate rate for children and adults to analyze the solution of the controlled system by using the concept of optimal control theory. It is indicated that vaccination at birth and treatment for both age groups have impact in reducing the value of the reproduction number (\(\mathcal{R}_{o}\)) whereas behavior modification does not have any impact on \(\mathcal{R}_{o}\). Pontryagin’s Minimum Principle has been used to characterize the optimal level of controls applied on the model. It is shown that the optimal combination strategy of vaccination, behavior change and treatment for the two age groups can help to reduce the disease epidemic with minimum cost of interventions, in shorter possible time.
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References
Barbu V., Precupanu T.: Convexity and Optimization in Banach Spaces, 4th edn. Springer, Dordrecht (2010)
Bekele B.T.: Modeling tuberculosis dynamics in children and adults in the presence of vaccination. M.Sc. thesis, Stellenbosch University (2010)
Central Statistical Agency Ethiopia, and ICF International: Ethiopia Demographic and Health Survey 2011. Addis Ababa, Ethiopia and Calverton, Maryland, USA: Central Statistical Agency and ICF International (2012)
Coddington, E.A.: An Introduction to Ordinary Differential Equations. Prentice-Hall, Englewood Cliffs (1961)
Coddington, E.A., Levinson, N.: Theory of Ordinary Differential Equations. McGraw Hill, New York (1955)
Federal Democratic Republic of Ethiopia Ministry of Health: Health and Health Related Indicators (2011)
Fleming, W.H., Rishel R.W.: Deterministic and Stochastic Optimal Control Applications of Mathematics, vol. 1. Springer, New York (1975)
Gaff, H., Schaefer, E.: Optimal control applied to vaccination and treatment strategies for various epidemiological models. Math. Biosci. Eng. 6(3), 469492 (2009)
Grass, D., Caulkins, J.P., Feichtinger, G., Tragler, G., Behrens, D.A.: Optimal Control of Nonlinear Processes, with Applications in Drugs, Corruption, and Terror. Springer, Berlin (2008)
Kassa, S.M., Ouhinou, A.: Epidemiological models with prevalence dependent endogenous self-protection measure. Math. Biosci. 229, 41–49 (2011)
Kassa, S.M., Ouhinou, A.: The impact of self-protective measures in the optimal interventions for controlling infectious diseases of human population. J. Math. Biol. (2014). doi:10.1007/s00285-014-0761-3
Kernodle, D.: Decrease in the effectiveness of Bacille Calmette-Guerin vaccine against pulmonary tuberculosis. Clin. Infect. Dis. 51(2), 177–184 (2010)
Lietman, T., Blower, S.: Potential impact of tuberculosis vaccine as epidemic control agents. Clin. Infect. Dis. 30(3), 316–322 (2000)
Ma, S., Xia, Y.: Mathematical Understanding of Infectious Disease Dynamics, vol. 16. World Scientific, Singapore (2009)
Silva, C.J., Torres, D.F.M.: Optimal control strategies for tuberculosis treatment: a case study in Angola. Center for Research and Development in Mathematics and Applications, Department of Mathematics, Portugal; Numer. Algebra Control Optim. 2(3), 601–617 (2012)
Van den Driessche, P., Watmough, J.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180, 29–48 (2002)
World Health organization: Reported estimates of BCG coverage. http://apps.who.int/immunization_monitoring/en/globalsummary/timeseries/tscoveragebcg.htm (2013). Accessed 20 Apr 2013
World Bank: http://www.worldbank.org/indicator/SP.DYN.CBRT.IN (2013). Accssed on 19 Apr 2013
Trading Economics: http://www.tradingeconomics.com/sub-saharan-africa/death-rate-crude-per-1-000-people-wb-data.html (2013). Accessed 16 Aug 2013
Index Mundi: http://www.indexmundi.com/g/g.aspx?c=et&v=30 (2013). Accssed 21 April 2013
Acknowledgements
This work is in part supported by the Swedish International Science Program (ISP) at the Department of Mathematics, Addis Ababa University.
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Aweke, T.D., Kassa, S.M. (2015). Impacts of Vaccination and Behavior Change in the Optimal Intervention Strategy for Controlling the Transmission of Tuberculosis. In: Bourguignon, JP., Jeltsch, R., Pinto, A., Viana, M. (eds) Mathematics of Energy and Climate Change. CIM Series in Mathematical Sciences, vol 2. Springer, Cham. https://doi.org/10.1007/978-3-319-16121-1_2
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DOI: https://doi.org/10.1007/978-3-319-16121-1_2
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