Measles is still endemic in many parts of the world including developed nations, despite the availability of the infectious disease vaccine since 1963. Elimination of measles requires maintaining the effective reproduction number by achieving and maintaining low levels of susceptibility R0 <1. In this project, we concentrate on the stochastic modelling of the transmission dynamics of measles with vaccination control. We have obtained the stochastic differential equations model from the deterministic model. Simulation of the stochastic differential equations model have been performed as well as the deterministic model. The stochastic differential equations model has described the transmission dynamics of measles with more information compared to the deterministic counterpart. Mathematical technique used in the simulation of the stochastic differential equations model is Euler-Maruyama numerical scheme and discussions of the model.
| Published in | International Journal of Theoretical and Applied Mathematics (Volume 2, Issue 2) |
| DOI | 10.11648/j.ijtam.20160202.16 |
| Page(s) | 60-73 |
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This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
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Copyright © The Author(s), 2016. Published by Science Publishing Group |
Measles, Vaccination, Immunity, Stochastic Modeling
| [1] | Allen, LJS. An Introduction to Stochastic Processes with Applications to Biology. Prentice Hall, Upper Saddle River, NJ. 2003. |
| [2] | Allen, LJS. Chapter 3: An Introduction to Stochastic Epidemic Models. Mathematical Epidemiology, Lecture Notes in Mathematics. Vol. 1945. pp. 81-130, F. Brauer, P. van den Driessche, and J. Wu (Eds.) Springer. 2008. |
| [3] | Karlin and Taylor. A First Course in Stochastic Processes. 2nd Ed. Acad. Press, NY Taylor. 1975. Kimmel and Axelrod. 2002. Branching Processes in Biology. Springer-Verlag, NY. 2002. |
| [4] | A. Mazer, Sankale, Guide de medicine en Afrique et Ocean Indien, EDICEF, Paris. |
| [5] | J. M. Ochoche and R. I. Gweryina. A Mathematical Model of Measles with Vaccination and Two Phases of Infectiousness. IOSR Journal of Mathematics, Vol 10, pp. 95-105, 2011. |
| [6] | Van Den Driessche, P and Watmough, J (2002). Reproduction numbers and Sub-threshold endemic equilibrium for compartmental models of disease transmission. Mathematical Biosciences, Vol 180, pp. 29-48, 2002. |
| [7] | R. F. Grais, M. J. Ferrari, C. Dubray, O. N. Bjørnstad, B. T. Grenfell, A. Djibo, F. Fermon, P. J. Guerin. Estimating transmission intensity for a measles epidemic in Niamey, Niger: lessons for intervention. Royal society of tropical medicine and hygiene, Vol 100, pp. 867–873. 2006. |
| [8] | O. Diekmann, J. A. P. Heesterbeek and M. G. Roberts. The construction of next-generation matrices for compartmental epidemic models. J. R. Soc. Interface, (doi:10.1098/rsif.2009.0386). (2009). |
| [9] | S. Edward, D. Kuznetsov, S. Mirau. (2014) Modelling and Stability analisis for a Varicella Zoster Virus Model with Vaccination. Science Publishing Group, Vol 3(4), pp 150-162. |
| [10] | Anes Tawhir Bsc. Mathematics. (2012) Modelling and Control of Measles Transmission in Ghana. Master of Plilosophy. Kwame Nkrumah University of Science and Technology” should be [9] Anes Tawhir Bsc. Mathematics. (2012) Modelling and Control of Measles Transmission in Ghana. Master of Philosophy. Kwame Nkrumah University of Science and Technology. |
| [11] | S. Edward, D. Kuznetsov, S. Mirau. (2014) Modelling and Stability analysis for a Varicella Zoster Virus Model with Vaccination. Science Publishing Group, Vol 3(4), pp 150-162. |
| [12] | Y. Yuan, L.J.S. Allen, Stochastic models for virus and immune system dynamics, Math. Biosci. (2011), doi:10.1016/j.mbs.2011.08.007. |
| [13] | Stephen Edward, Kitengeso Raymond E., Kiria Gabriel T., Felician Nestory, Mwema Godfrey G., Mafarasa Arbogast P.. A Mathematical Model for Control and Elimination of the Transmission Dynamics of Measles. Applied and Computational Mathematics. Vol. 4, No. 6, 2015, pp. 396-408. doi: 10.11648/j.acm.20150406.12. |
| [14] | Allen, E. J. (2007). Modelling with Ito stochastic differential equations. Springer, Dirchrecht, The Netherlands. |
| [15] | Britton, T. (2010). Stochastic epidemic models: a survey Mathematical biosciences 225(1):24-35. |
| [16] | A. Mazer, Sankale, Guide de medicine en Afrique et Ocean Indien, EDICEF, Paris (1988). |
| [17] | Llyoyd, A. L. (2004). Estimating variability in models for recurrent epidemics: assessing the use of moment closure technigues. Theoretical population biology 65(1):49-65. |
APA Style
Kitengeso Raymond E. (2016). Stochastic Modelling of the Transmission Dynamics of Measles with Vaccination Control. International Journal of Theoretical and Applied Mathematics, 2(2), 60-73. https://doi.org/10.11648/j.ijtam.20160202.16
ACS Style
Kitengeso Raymond E. Stochastic Modelling of the Transmission Dynamics of Measles with Vaccination Control. Int. J. Theor. Appl. Math. 2016, 2(2), 60-73. doi: 10.11648/j.ijtam.20160202.16
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Download@article{10.11648/j.ijtam.20160202.16,
author = {Kitengeso Raymond E.},
title = {Stochastic Modelling of the Transmission Dynamics of Measles with Vaccination Control},
journal = {International Journal of Theoretical and Applied Mathematics},
volume = {2},
number = {2},
pages = {60-73},
doi = {10.11648/j.ijtam.20160202.16},
url = {https://doi.org/10.11648/j.ijtam.20160202.16},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20160202.16},
abstract = {Measles is still endemic in many parts of the world including developed nations, despite the availability of the infectious disease vaccine since 1963. Elimination of measles requires maintaining the effective reproduction number by achieving and maintaining low levels of susceptibility R0 <1. In this project, we concentrate on the stochastic modelling of the transmission dynamics of measles with vaccination control. We have obtained the stochastic differential equations model from the deterministic model. Simulation of the stochastic differential equations model have been performed as well as the deterministic model. The stochastic differential equations model has described the transmission dynamics of measles with more information compared to the deterministic counterpart. Mathematical technique used in the simulation of the stochastic differential equations model is Euler-Maruyama numerical scheme and discussions of the model.},
year = {2016}
}
TY - JOUR T1 - Stochastic Modelling of the Transmission Dynamics of Measles with Vaccination Control AU - Kitengeso Raymond E. Y1 - 2016/12/05 PY - 2016 N1 - https://doi.org/10.11648/j.ijtam.20160202.16 DO - 10.11648/j.ijtam.20160202.16 T2 - International Journal of Theoretical and Applied Mathematics JF - International Journal of Theoretical and Applied Mathematics JO - International Journal of Theoretical and Applied Mathematics SP - 60 EP - 73 PB - Science Publishing Group SN - 2575-5080 UR - https://doi.org/10.11648/j.ijtam.20160202.16 AB - Measles is still endemic in many parts of the world including developed nations, despite the availability of the infectious disease vaccine since 1963. Elimination of measles requires maintaining the effective reproduction number by achieving and maintaining low levels of susceptibility R0 <1. In this project, we concentrate on the stochastic modelling of the transmission dynamics of measles with vaccination control. We have obtained the stochastic differential equations model from the deterministic model. Simulation of the stochastic differential equations model have been performed as well as the deterministic model. The stochastic differential equations model has described the transmission dynamics of measles with more information compared to the deterministic counterpart. Mathematical technique used in the simulation of the stochastic differential equations model is Euler-Maruyama numerical scheme and discussions of the model. VL - 2 IS - 2 ER -
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