Mathematical Modelling of the Transmission Dynamics of Ebola Virus

Amenaghawon C. Osemwinyen; Aboubakary Diakhaby

doi:doi:10.11648/j.acm.20150404.19

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Mathematical Modelling of the Transmission Dynamics of Ebola Virus

Amenaghawon C. Osemwinyen, Aboubakary Diakhaby

Published in Applied and Computational Mathematics (Volume 4, Issue 4)

Received: 30 May 2015 Accepted: 29 June 2015 Published: 18 July 2015

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Abstract

The study simulated the transmission dynamics of Ebola Zaire virus using two models: a modified SIR model with the understanding that the recovered can become infected again and the infected die at a certain rate and a quarantine model, which ascertained the effects of quarantining the infected. Furthermore, an appropriate system of Ordinary Differential Equations (ODE) was formulated for the transmission and the method of linearized stability approach was used to solve the equations. Stability analysis of both models indicated that, the Disease Free Equilibrium (DFE) states of the models were unstable if they exist. These equilibria states showed that the disease can easily be triggered off, so there is need to be constantly alert and effective preventive measures put in place against its spread. In addition, numerical experiments were carried out with the models' parameters assigned specific hypothetical values and graphs were plotted to investigate the effect of these parameters on the transmission of the disease. The results showed that, with the nature of Ebola Zaire virus, uncontrolled transmittable contacts between the infected and the susceptible can lead to a very serious outbreak with high mortality rate, since no immunity and drugs at moment. However, with effective quarantining structures put in place such situation can be better managed and outbreak controlled.

Published in	Applied and Computational Mathematics (Volume 4, Issue 4)
DOI	10.11648/j.acm.20150404.19
Page(s)	313-320
Creative Commons	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.
Copyright	Copyright © The Author(s), 2015. Published by Science Publishing Group

Keywords

Mathematical Model, Ebola Virus, Transmission Dynamics

References

[1]	Wikipedia, “Ebola virus”, http:// www.en.wikipedia.org/wiki/Ebola_virus, 2014.
[2]	Z. Yarus, “A mathematical look at the Ebola virus”, http://www.home2.fvcc.edu/~dhicketh/DiffEqns/Spring2012Projects/Zach%20Yarus%20-Final%20Project/Final%20Diffy%20Q%20project.pdf, 2012.
[3]	J. Legrand, R.F. Grais, P.Y. Boelle, A.J. Valleron and A. Flahault, “Understanding the dynamics of Ebola epidemics”, Epidemiology and Infection, 135th ed., vol. 4, pp. 610–621. http://www.ncbi.nlm.nih.gov/pubmed/23007439, 2006.
[4]	World Health Organization (WHO), “Ebola virus disease”, http://www.who.int/mediacentre/factsheets/fs103/en/, 2014.
[5]	Wikipedia, “Ebola virus epidemics in West Africa”, http://www.en.m.wikipedia.org/wiki/Ebola_virus_epidemics_in_West_Africa, 2015.
[6]	C. Roy-Macaulay, “Ebola crisis triggers health emergency”, http://www.dddmag.com/news/2014/07/ebola-crisis-triggers-health-emergency, 2014.
[7]	T. Leslie. “Ebola: what is it and how does it spread?” ABC News, http://www.abc.net.au/news/2014-07-30/ebola-virus-explainer/5635028, 2014.
[8]	Wikipedia, “List of Ebola outbreaks”, http://www.en.wikipedia.org/wiki/List_of_Ebola_outbreaks, 2014.
[9]	A. Nossiter, “Fear of Ebola breeds a terror of physicians”, http://www.nytimes.com/2014/07/28/world/africa/ebola-epidemic-west-africa-guinea.html?r=0, 2014.
[10]	D. Logan, A First Course in Differential Equations, Springer Science & Business Media, New York, 2010.
[11]	J.M. Mahaffy, “Epidemic models”, http://www.rohan.sdsu.edu/~jmahaffy/courses/f09/math636/lectures/epidemics/epidemics.pdf, 2009.
[12]	F. Brauer, J. Wu and P.V.D. Driessche, Mathematical Epidemiology, Springer-Verlag, Berlin Heidelberg, 2008.
[13]	C. Yongli, Y. Kangb, W. Wang, and M. Zhaoc. “A Stochastic differential equation SIRS epidemic model with ratio-dependent incidence rate”, http://www.public.asu.edu/_ykang3/_les/CKWZ Stochastic.pdf, 2013.
[14]	J. D. Murray, Mathematical Biology: 1. An Introduction, Springer Science & Business Media, New York, 2002.
[15]	S. G. Deo and V. Raghavendra, Ordinary Differencial Equations and Stability Theory, Tata Mc Graw-Hill Publishing Company Ltd, New York, 1980.

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Amenaghawon C. Osemwinyen, Aboubakary Diakhaby. (2015). Mathematical Modelling of the Transmission Dynamics of Ebola Virus. Applied and Computational Mathematics, 4(4), 313-320. https://doi.org/10.11648/j.acm.20150404.19

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ACS Style

Amenaghawon C. Osemwinyen; Aboubakary Diakhaby. Mathematical Modelling of the Transmission Dynamics of Ebola Virus. Appl. Comput. Math. 2015, 4(4), 313-320. doi: 10.11648/j.acm.20150404.19

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AMA Style

Amenaghawon C. Osemwinyen, Aboubakary Diakhaby. Mathematical Modelling of the Transmission Dynamics of Ebola Virus. Appl Comput Math. 2015;4(4):313-320. doi: 10.11648/j.acm.20150404.19

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@article{10.11648/j.acm.20150404.19,
  author = {Amenaghawon C. Osemwinyen and Aboubakary Diakhaby},
  title = {Mathematical Modelling of the Transmission Dynamics of Ebola Virus},
  journal = {Applied and Computational Mathematics},
  volume = {4},
  number = {4},
  pages = {313-320},
  doi = {10.11648/j.acm.20150404.19},
  url = {https://doi.org/10.11648/j.acm.20150404.19},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20150404.19},
  abstract = {The study simulated the transmission dynamics of Ebola Zaire virus using two models: a modified SIR model with the understanding that the recovered can become infected again and the infected die at a certain rate and a quarantine model, which ascertained the effects of quarantining the infected. Furthermore, an appropriate system of Ordinary Differential Equations (ODE) was formulated for the transmission and the method of linearized stability approach was used to solve the equations. Stability analysis of both models indicated that, the Disease Free Equilibrium (DFE) states of the models were unstable if they exist. These equilibria states showed that the disease can easily be triggered off, so there is need to be constantly alert and effective preventive measures put in place against its spread. In addition, numerical experiments were carried out with the models' parameters assigned specific hypothetical values and graphs were plotted to investigate the effect of these parameters on the transmission of the disease. The results showed that, with the nature of Ebola Zaire virus, uncontrolled transmittable contacts between the infected and the susceptible can lead to a very serious outbreak with high mortality rate, since no immunity and drugs at moment. However, with effective quarantining structures put in place such situation can be better managed and outbreak controlled.},
 year = {2015}
}

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TY  - JOUR
T1  - Mathematical Modelling of the Transmission Dynamics of Ebola Virus
AU  - Amenaghawon C. Osemwinyen
AU  - Aboubakary Diakhaby
Y1  - 2015/07/18
PY  - 2015
N1  - https://doi.org/10.11648/j.acm.20150404.19
DO  - 10.11648/j.acm.20150404.19
T2  - Applied and Computational Mathematics
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JO  - Applied and Computational Mathematics
SP  - 313
EP  - 320
PB  - Science Publishing Group
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AB  - The study simulated the transmission dynamics of Ebola Zaire virus using two models: a modified SIR model with the understanding that the recovered can become infected again and the infected die at a certain rate and a quarantine model, which ascertained the effects of quarantining the infected. Furthermore, an appropriate system of Ordinary Differential Equations (ODE) was formulated for the transmission and the method of linearized stability approach was used to solve the equations. Stability analysis of both models indicated that, the Disease Free Equilibrium (DFE) states of the models were unstable if they exist. These equilibria states showed that the disease can easily be triggered off, so there is need to be constantly alert and effective preventive measures put in place against its spread. In addition, numerical experiments were carried out with the models' parameters assigned specific hypothetical values and graphs were plotted to investigate the effect of these parameters on the transmission of the disease. The results showed that, with the nature of Ebola Zaire virus, uncontrolled transmittable contacts between the infected and the susceptible can lead to a very serious outbreak with high mortality rate, since no immunity and drugs at moment. However, with effective quarantining structures put in place such situation can be better managed and outbreak controlled.
VL  - 4
IS  - 4
ER  -

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Author Information

Amenaghawon C. Osemwinyen

Department of Mathematics, FCT-College of Education, Zuba-Abuja, Nigeria
Aboubakary Diakhaby

Department of Mathematics, Gaston Berger University, Saint-Louis, Senegal

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Amenaghawon C. Osemwinyen, Aboubakary Diakhaby. (2015). Mathematical Modelling of the Transmission Dynamics of Ebola Virus. Applied and Computational Mathematics, 4(4), 313-320. https://doi.org/10.11648/j.acm.20150404.19

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ACS Style

Amenaghawon C. Osemwinyen; Aboubakary Diakhaby. Mathematical Modelling of the Transmission Dynamics of Ebola Virus. Appl. Comput. Math. 2015, 4(4), 313-320. doi: 10.11648/j.acm.20150404.19

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AMA Style

Amenaghawon C. Osemwinyen, Aboubakary Diakhaby. Mathematical Modelling of the Transmission Dynamics of Ebola Virus. Appl Comput Math. 2015;4(4):313-320. doi: 10.11648/j.acm.20150404.19

Copy | Download

@article{10.11648/j.acm.20150404.19,
  author = {Amenaghawon C. Osemwinyen and Aboubakary Diakhaby},
  title = {Mathematical Modelling of the Transmission Dynamics of Ebola Virus},
  journal = {Applied and Computational Mathematics},
  volume = {4},
  number = {4},
  pages = {313-320},
  doi = {10.11648/j.acm.20150404.19},
  url = {https://doi.org/10.11648/j.acm.20150404.19},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20150404.19},
  abstract = {The study simulated the transmission dynamics of Ebola Zaire virus using two models: a modified SIR model with the understanding that the recovered can become infected again and the infected die at a certain rate and a quarantine model, which ascertained the effects of quarantining the infected. Furthermore, an appropriate system of Ordinary Differential Equations (ODE) was formulated for the transmission and the method of linearized stability approach was used to solve the equations. Stability analysis of both models indicated that, the Disease Free Equilibrium (DFE) states of the models were unstable if they exist. These equilibria states showed that the disease can easily be triggered off, so there is need to be constantly alert and effective preventive measures put in place against its spread. In addition, numerical experiments were carried out with the models' parameters assigned specific hypothetical values and graphs were plotted to investigate the effect of these parameters on the transmission of the disease. The results showed that, with the nature of Ebola Zaire virus, uncontrolled transmittable contacts between the infected and the susceptible can lead to a very serious outbreak with high mortality rate, since no immunity and drugs at moment. However, with effective quarantining structures put in place such situation can be better managed and outbreak controlled.},
 year = {2015}
}

Copy | Download

TY  - JOUR
T1  - Mathematical Modelling of the Transmission Dynamics of Ebola Virus
AU  - Amenaghawon C. Osemwinyen
AU  - Aboubakary Diakhaby
Y1  - 2015/07/18
PY  - 2015
N1  - https://doi.org/10.11648/j.acm.20150404.19
DO  - 10.11648/j.acm.20150404.19
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 313
EP  - 320
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20150404.19
AB  - The study simulated the transmission dynamics of Ebola Zaire virus using two models: a modified SIR model with the understanding that the recovered can become infected again and the infected die at a certain rate and a quarantine model, which ascertained the effects of quarantining the infected. Furthermore, an appropriate system of Ordinary Differential Equations (ODE) was formulated for the transmission and the method of linearized stability approach was used to solve the equations. Stability analysis of both models indicated that, the Disease Free Equilibrium (DFE) states of the models were unstable if they exist. These equilibria states showed that the disease can easily be triggered off, so there is need to be constantly alert and effective preventive measures put in place against its spread. In addition, numerical experiments were carried out with the models' parameters assigned specific hypothetical values and graphs were plotted to investigate the effect of these parameters on the transmission of the disease. The results showed that, with the nature of Ebola Zaire virus, uncontrolled transmittable contacts between the infected and the susceptible can lead to a very serious outbreak with high mortality rate, since no immunity and drugs at moment. However, with effective quarantining structures put in place such situation can be better managed and outbreak controlled.
VL  - 4
IS  - 4
ER  -

Copy | Download