dc.contributor.advisor Witbooi, Peter J. dc.contributor.author Nemaranzhe, Lutendo dc.contributor.other Dept. of Mathematics dc.date.accessioned 2014-03-28T10:29:04Z dc.date.available 2013/03/20 dc.date.available 2013/03/20 12:16 dc.date.available 2014-03-28T10:29:04Z dc.date.issued 2010 dc.identifier.uri http://hdl.handle.net/11394/3065 dc.description Magister Scientiae - MSc en_US dc.description.abstract We review a number of compartmental models in epidemiology which leads to a nonlinear system of ordinary differential equations. We focus an SIR, SEIR and SIS epidemic models with and without vaccination. A threshold parameter R0 is identified which governs the spread of diseases, and this parameter is known as the basic reproductive number. The models have at least two equilibria, an endemic equilibrium and the disease-free equilibrium. We demonstrate that the disease will die out, if the basic reproductive number R0 < 1. This is the case of a disease-free state, with no infection in the population. Otherwise the disease may become endemic if the basic reproductive number R0 is bigger than unity. Furthermore, stability analysis for both endemic and disease-free steady states are investigated and we also give some numerical simulations. The second part of this dissertation deals with optimal vaccination strategy in epidemiology. We use optimal control technique on vaccination to minimize the impact of the disease. Hereby we mean minimizing the spread of the disease in the population, while also minimizing the effort on vaccination roll-out. We do this optimization for the cases of SIR and SEIR models, and show how optimal strategies can be obtained which minimize the damage caused by the infectious disease. Finally, we describe the numerical simulations using the fourth-order Runge-Kutta method. These are the most useful references: [G. Zaman, Y.H Kang, II. H. Jung. BioSystems 93, (2008), 240 − 249], [K. Hattaf, N. Yousfi. The Journal of Advanced Studies in Biology, Vol. 1(8), (2008), 383 − 390.], [Lenhart, J.T. Workman. Optimal Control and Applied to Biological Models. Chapman and Hall/CRC, (2007).], [P. Van den Driessche, J. Watmough. Math. Biosci., 7, (2005)], and [J. Wu, G. R¨ost. Mathematical Biosciences and Engineering, Vol 5(2), (2008), 389 − 391]. en_US dc.language.iso en en_US dc.publisher University of the Western Cape en_US dc.subject Basic reproductive number en_US dc.subject Disease-free equilibrium en_US dc.subject Epidemiology en_US dc.subject Endemic model en_US dc.subject Numerical simulation en_US dc.subject Population model en_US dc.subject Optimization en_US dc.subject Vaccination en_US dc.title A mathematical modeling of optimal vaccination strategies in epidemiology en_US dc.rights.holder Copyright: University of the Western Cape en_US dc.description.country South Africa
﻿