American Journal of Applied Mathematics
Volume 7, Issue 2, April 2019, Pages: 37-48
Received: Apr. 13, 2019;
Accepted: May 28, 2019;
Published: Jun. 26, 2019
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Temesgen Debas Awoke, Department of Mathematics, College of Natural and Computational Science, Kotebe Metropolitan University, Addis Ababa, Ethiopia
The author developed a deterministic mathematical model for Typhoid fever disease dynamics that accounts for Vaccination and relapse of treatment. Three control strategies (vaccination, treatment of infection, screening and treatment of carriers) are applied to investigate the optimal intervention strategy of controlling Typhoid disease transmission. The aim of this study is to determine the optimal combination strategy of vaccination, treatment of infection, screening and treatment of carriers that will minimize the cost of those strategies and the number of Infective and Carriers. The author used Pontryagin’s maximum principle to characterize the optimal level of those three strategies. The result is simulated numerically using Runge-Kutta fourth order method through MATLAB software. Numerical results showed that implementation of all controls or a combination of vaccination, treatment of invectives as well as screening and treatment of carriers is the best strategy to eradicate the disease at an optimal level with minimum cost of interventions.
Temesgen Debas Awoke,
Optimal Control Strategy for the Transmission Dynamics of Typhoid Fever, American Journal of Applied Mathematics.
Vol. 7, No. 2,
2019, pp. 37-48.
Roumagnac P. (2006), Evolutionary history of Salmonella typhi, Science, 314, 1301-304.
WHO (2011), Guidelines for the Management of Typhoid Fever.
WHO (2007), Background paper on vaccination against typhoid fever using New Generation Vaccines presented at the SAGE.
Anwar E, Goldberg E, Fraser A, Acosta CJ, Paul M. and Leibovici L. (2014), Vaccination for preventing typhoid fever. The Cochrane Database of Systematic Reviews.
Gaff H. and Schaefer E. (2009), Optimal control applied to vaccination and treatment Strategies for various epidemiological models, mathematical bio-sciences and engineering, Volume 6, Number 3, 469492.
Hethcote H. (2000), The mathematics of infectious diseases. SIAM Review. 42 (4): 599-53.
Kassa, S. M., Ouhinou, A. (2014), The impact of self-protective measures in the optimal Interventions for controlling infectious diseases of human population. J. Math. Biol.
Lakshmikantham V., Leela S, and Martynyuk A. (1989), Stability Analysis of NonlinearSystems. Marcel Dekker, Inc., New York and Basel.
Ma S. and Xia Y. (2009), Mathematical understanding of infectious disease dynamics, World Scientific Publishing Co. Pte. Ltd. vol. 16.
Awoke, T. D.; Kassa, S. M. Optimal Control Strategy for TB-HIV/AIDS Co-Infection Model in the Presence of Behaviour Modification. Processes 2018, 6, 48.
Aweke T. D. and Kassa S. M. (2015), Impacts of vaccination and behavior change in the optimal intervention strategy fo1r controlling the transmission of Tuberculosis, Springer International Publishing Switzerland 2015, CIM Series in Mathematical Sciences, Vol. 2, pp. 15-32.
Sethi S. P. and Thompson G. L. (2000), Optimal control theory: Applications to management science and economics, Kluwer, Boston, second edition.
Tunde Tajudeen Yusuf and Francis Benyah (2012), Optimal control of vaccination and treatment for an SIR epidemiological model, World Journal of Modelling and Simulation, Vol. 8 No. 3, pp. 194-204.
Cook JH. (2010), Are Cholera and typhoid vaccines a good investment for slums in Kolkata, India. Pediatr. Infect. Dis. J; 9 (5): 485-496.
Mushayabasa S. (2011), Impacts of Vaccines on controlling Typhoid Fever in Kassena Nankana district of upper east region of Ghana: Insight from a mathematical model, Journalof modern mathematics and statistics, 5 (2): 54-59.
Pitzer VE, Bowles CC, Baker S, Kang G, Balaji V, Farrar J. (2014), is predicting the Impactof vaccination on the transmission dynamics of typhoid in South Asia: A Mathematicalmodeling study. PLoS Negl Dis, 8 (1): 1-12.
Kalajdzievska D. and Li M. (2011), Modeling the effects of carriers on transmission dynamics of infectious disease, Mathematical Bioscience and Engineering, Num. 3, Vol. 8, 711-722.
Mushayabasa S.(2014)., Modeling the impact of optimal screening on typhoid dynamics, Int. J. Dynam. Control, Springer-Verlag Berlin Heidelberg.
Kariuki S. (2008), Typhoid fever in sub-saharan Africa; Challenges of diagnosis and management of infections. Journal of infection in Developing Countries. 2 (6): 443-447.
Kgosimore M. and G. R. Kelatlhegile G. (2016), Mathematical analysis of Typhoid Infectionwith treatment, Journal of Mathematical Science: Advances and applications, Vol. 40, 75-91.
Rachel B. Slayton, Kashmira A. Date3 and Eric D. Mintz (2013), Vaccination for typhoid fever in Sub-Saharan Africa, Human Vaccines & Immunotherapeutics 9: 4, 903–906.
Van den Driessche, P. and Watmough J. (2002), Reproduction numbers and sub-thresholdendemic equilibria for compartmental models of disease transmission. Mathematical Bio-sciences, Volume 180, pp: 29-48.
Waaler H., Gese A., and Anderson S. (1962), The use of mathematical models in the study of the epidemiology of tuberculosis, Am. J. Publ. Health, 52, pp. 1002-1013.
Coddington, E. A. (1961), An Introduction to Ordinary Differential Equations. Prentice-Hall, Englewood Cliffs.
Cvjetanovic B., Grab B. and Uemura K. (1971), Epidemiological model of typhoid fever and its use in the planning and evaluation of antityphoid immunization and sanitation Programs, Bull. Org. Mond. Sante (45), 53-75.
Ghosh M., Chandra P., Sinha P. and Shukla J. B (2004),, Modelling the spread of Carrierdependent infectious diseases with an environmental effect, Appl. Math. Comp.