Queueing Based Compartmental Models for Ebola Virus Disease Analysis

Ikeme John Dike^1, and Chinyere Ogochukwu Dike²

¹Department of Statistics & Operations Research, Modibbo Adama University of Technology, P.M.B 2076, Yola, Adamawa State, Nigeria

²Federal College of Education P.M.B 2042, Yola, Adamawa State, Nigeria

Cite this paper

Ikeme John Dike and Chinyere Ogochukwu Dike. Queueing Based Compartmental Models for Ebola Virus Disease Analysis. American Journal of Applied Mathematics and Statistics. 2018; 6(3):96-106. doi: 10.12691/AJAMS-6-3-2

Abstract

Ebola Virus Disease (EVD) is a complex and unprecedented epidemic killer disease. Recently, the disease has caused serious loss of life, waste of economic and material resources in West African nations. The most prevalent countries are Guinea, Liberia and Sierra Leone. Compartmental models are traditional epidemiological mdels that try to explain epidemic problems through the use of specific compartments that are subsets of a given population. Analysis using the developed queueing based compartmental models for Ebola Virus Disease (EVD) resulted in estimates of R₀= (2.2550, 3.5264, 2.2325) for Guinea, Liberia and Sierra Leone. R₀ > 1 for each of the countries, implying that the transmission and control of the epidemic was unstable and needed urgent intervention. The developed SEI_LI_CDR model outperformed the existing SEIR model by 13.10%, 91.76%, and 83.14%, respectively on the basis of their RMSE. Finally, analysis using queueing in SEI_LI_CDR compartmental models led to the discovery that, at a probability of 0.4 in each compartment, the transmission of EVD can be controlled.

Keywords

Ebola Virus Disease, epidemic, comparmental model, transmission, control, queueing

Copyright

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

References

[1]	Bashar, S., Percy, M., Singhai, R.: Predicting the 2014 Ebola outbreak in West Africa using network analysis. News.updatephotos.com/ebola_breakout_in_africa(2015). Retrieved May, 2017.
[2]	Kublik R.A., Chopp, D.L.: A locally adaptive time stepping algorithm for the solution to reaction diffusion equations on branched structures. Advan. Comput. Math. 42, 621-649 (2016).
[3]	Chowell, G., Hengartner, N.W., Castillo-Chavez, C., Fenimore, P.W., Hyman, J.M.: The basic reproduction number of Ebola and the effects of public health measures: the cases of Congo and Uganda. J. of Theoret. Bio. 229(1), 119-26 (2004).
[4]	Hernandez-Suarez C. M., Castillo-Chavez C., Lopez O. M., and Hernandez-Cuevas K.: An application of queueing theory to SIS and SEIS epidemic models. Math. Biosci. and Eng. 7(4), 809-823 (2010).
[5]	Siettos, C., Anastassopoulou, C., Russo, L., Grigoras, C., Mylonakis, E.: Modeling the 2014 Ebola virus epidemic-agent-based simulations, temporal analysis and future predictions for Liberia and Sierra Leone. Public Library of Science Current Outbreaks (2015).
[6]	Legrand, J., Grais, R.F., Bolle, P.Y., Valleron, A.J., Flahault, A.: Understanding the dynamics of Ebola epidemics. Epidemio. and Infect. 135 (4), 610-621 (2007).
[7]	Dike, C.O., Zainuddin, Z.M., Ikeme John Dike, I.J.: Susceptible Infected Removed Epidemic Model Extension for Efficient Analysis of Ebola Virus Disease Transmission. American Scientific Publishers. Advanced Science Letters. 23(9), 9107-9114 (2017).
[8]	Sztrik, J.: Basic Queueing Theory. http://irh.inf.unideb.hu/user/jsztrik (2012). Retrieved September, 2015.
[9]	Diekmann, O., Heesterbeek, J.A.P. Metz, J.A.J.: On the definition and the computation of the basic reproduction ratio Ro in models for infectious diseases in heterogeneous populations. J. of Math. Bio. 28, 365 (1990).
[10]	Hethcote, H.W., Ark, J.W.V.: Epidemiological models for heterogenous population: proportionate mixing, parameter estimation, and immunization programs. Math. Biosci. 84, 85 (1987).
[11]	Cadtillo-Chavaz, C., Feng, Z.: To treat or not to treat: The case of Tuberculosis. J. of Math. Bio. 35: 629 (1997).
[12]	Blower, S.M., Small, P.M., Hopewell, P.C.: Control strategies for Tuberculosis epidemics: New Models for old problems. Science. 273: 497 (1996).
[13]	Driessche, P.V.D., Watmough, J.: Reproduction numbers and sub-threshhold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180, 29-48 (2002).
[14]	Cooper, R.B.: Introduction to queueing theory (2nd Edition). Elsevier North Holland, Inc. New York: Oxford (1981).
[15]	Ross, S.M. Introduction to probability models. (9th ed.). Academic Press, Elsevier Inc.: USA (2007).
[16]	Winston, W.L.: Operations Research (applications and algorithms). (4th ed.). Thomson Books: USA (2004).
[17]	CNN: Ebola: Patient zero was a toddler in Guinea - CNN.com. http://edition.cnn.com/2014/10/28/health/ebola-patient-zero/index.html (2014). Retrieved 9 October, 2017.
[18]	International Business Times (IBT): Ebola patient zero: Emile Ouamouno of Guinea first to contract disease. http://www.ibtimes.com/ebola-patient-zero-emile-ouamouno-guinea-first-contract-disease-1714698 (2014). Retrieved 9 October, 2017.
[19]	CDC: (2014). Ebola hemorrhagic fever signs and symptoms. http://www.cdc.gov/vhf/ebola/symptoms/index.html (2014). Retrieved 28 May, 2017.
[20]	Ebola Virus Disease in Guinea (Situation as of 25 March 2014). http://www.afro.who.int/en/clusters-a-programmes/dpc/epidemic-a-pandemic-alert-and-response/outbreak-news/4065-ebola-virus-disease-in-guinea-25-march-2014.html. Retrieved 9 October, 2017.
[21]	Guinea: Ebola epidemic declared. MSF UK. http://www.msf.org.uk/article/guinea-ebola-epidemic-declared. Retrieved 9 October, 2017.
[22]	How Ebola sped out of control. Washington Post. http://www.washingtonpost.com/sf/national/2014/10/04/how-ebola-sped-out-of-control/. Retrieved 9 October, 2017.
[23]	EVD affected districts in Guinea, Liberia and Sierra Leone. (https://www.google.com). Retrieved 9 October, 2017.
[24]	Test positive for Ebola in Liberia (2 of 5), Liberian observer, 31 March 2014. http://www.liberianobserver.com. Retrieved 9 October, 2017.
[25]	Grady, D., Fink, S.: Tracing Ebola's breakout to an African 2-year-old. The New York Times. http://www.nytimes.com/2014/08/10/world/africa/tracing-ebolas-breakout-to-an-african-2-year-old.html (2014). Retrieved 9 October, 2017.
[26]	Nassos, S.: How world's worst Ebola outbreak began with one boy's death. BBC News. http://www.bbc.co.uk/news/world-africa-30199004 (2014). Retrieved 9 October, 2017.
[27]	WHO Ebola Response Team: Ebola Virus Disease in West Africa-the first 9 months of the epidemic and forward projections (2014). The New England J. of Med. 371 (16), 1481-1495.
[28]	Mystery hemorrhagic fever kills 23 in Guinea. Reuters. 19 March 2014. http://www.reuters.com/article/2014/03/19/us-guinea-fever-idusbrea2i0qm20140319. Retrieved 9 October, 2017.
[29]	Genetics of the 2014 Ebola outbreak. http://www.nih.gov/researchmatters/september2014/09152014ebola.htm. Retrieved 9 October, 2017.
[30]	Ebola outbreak blamed on infected bushmeat; first family to catch virus hunted bats. International Business http://au.ibtimes.com/articles/570107/20141020/bushmeat-ebola-outbreak-fruit-bats.htm#.VEaivSihFnU.Retrieved 9 October, 2017.Times.
[31]	Johnston, W.M.: Statistics on the 2014 - 2015 West Africa Ebola Outbreak. http://www.johnstonsarchive.net>policy>westafrica(2015). Retrieved 18 September, 2017.
[32]	Anderson, H., Britton, T.: Stochastic epidemics in dynamic populations: Quasi- stationarity and extinction. J. Math. Bio. 41, 559-580 (2000).
[33]	Darroch, J.N., Seneta, E.: On Quasi-stationary distribution in absorbing continuous-time finite Markov Chains. J. of App. Prob. 4, 192-196 (1967).