Abstract
A major Ebola outbreak occurs in West Africa since March 2014, being the deadliest epidemic in history. As an infectious disease epidemiology, Ebola is the most lethal and is moving faster than in previous outbreaks. On 8 August 2014, the World Health Organization (WHO) declared the outbreak a public health emergency of international concern. Last update on 7 July 2015 by WHO reports 27,609 cases of Ebola with a total of 11,261 deaths. In this work, we present a mathematical description of the spread of Ebola virus based on the SEIR (Susceptible–Exposed–Infective–Recovered) model and optimal strategies for Ebola control. In order to control the propagation of the virus and to predict the impact of vaccine programmes, we investigate several strategies of optimal control of the spread of Ebola: control infection by vaccination of susceptible; minimize exposed and infected; reduce Ebola infection by vaccination and education.
Similar content being viewed by others
References
Althaus, C.L.: Estimating the reproduction number of Ebola virus (EBOV) during the 2014 outbreak in West Africa. PLOS Curr. Outbreaks (2014).doi:10.1371/currents.outbreaks.91afb5e0f279e7f29e7056095255b288
Area, I., Batarfi, H., Losada, J., Nieto, J.J., Shammakh, W., Torres, A.: On a fractional order Ebola epidemic model, Adv. Differ. Equ. 2015, 278 (2015), 12 pp
Area, I., Losada, J., Ndaïrou, F., Nieto, J.J., Tcheutia, D.D.: Mathematical modeling of 2014 Ebola outbreak. Math. Methods Appl. Sci. doi:10.1002/mma.3794 (in press)
Ariens, D., Houska, B., Ferreau, H.J.: ACADO toolkit user’s manual. Toolkit for Automatic Control and Dynamic Optimization (2010). http://www.acadotoolkit.org
Astacio, J., Briere, D., Guilléon, M., Martinez, J., Rodriguez, F., Valenzuela-Campos, N.: Mathematical models to study the outbreaks of Ebola. Report BU-1365-M, Mathematical and Theoretical Biology Institute (MTBI), Cornell University (1996)
Banks, H.T., Castillo-Chavez, C.: Bioterrorism. Frontiers in Applied Mathematics, vol. 28. SIAM, Philadelphia (2003)
Barry, M., Traoré, F.A., Sako, F.B., Kpamy, D.O., Bah, E.I., Poncin, M., Keita, S., Cisse, M., Touré, A.: Ebola outbreak in Conakry, Guinea: epidemiological, clinical, and outcome features. Médecine et Maladies Infectieuses 44(11–12), 491–494 (2014)
Berry, C.M.: Vaccine control of avian influenza H5N1 in poultry: need for a positive marker. J. Vaccin. Vaccin. 4(1), Art. ID 1000168 (2013), 4 pp
Bock, H.G., Pitt, K.J.: A multiple shooting algorithm for direct solution of optimal control problems. Proc. 9th IFAC World Congress, Pergamon Press, Budapest, pp. 243–247 (1984)
Borio, L., et al. [Working Group on Civilian Biodefense; Corporate Author].: Hemorrhagic fever viruses as biological weapons: medical and public health management. J. Am. Med. Assoc. 287(18), 2391–2405 (2002)
Chowell, G., Hayman, J.M., Bettencourt, L.M.A., Castillo-Chavez, C.: Mathematical and Statistical Estimation Approaches in Epidemiology. Springer, Dordrecht (2009)
Chowell, G., Hengartner, N.W., Castillo-Chavez, C., Fenimore, P.W., Hyman, J.M.: The basic reproductive number of Ebola and the effects of public health measures: the cases of Congo and Uganda. J. Theor. Biol. 229(1), 119–126 (2004)
Dowell, S.F., Mukunu, R., Ksiazek, T.G., Khan, A.S., Rollin, P.E., Peters, C.J.: Transmission of Ebola hemorrhagic fever: a study of risk factors in family members, Kikwit, Democratic Republic of the Congo, 1995. Commission de Lutte contre les Epidémies à Kikwit. J. Infect. Dis. 179(Suppl. 1), S87–S91 (1999)
Gao, S., Teng, Z., Nieto, J.J., Torres, A.: Analysis of an SIR epidemic model with pulse vaccination and distributed time delay. J. Biomed. Biotechnol. 2007, Art. ID 64870 (2007), 10 pp
Houska, B., Ferreau, H.J., Diehl, M.: ACADO toolkit—an open-source framework for automatic control and dynamic optimization. Optim. Control Appl. Methods 32(3), 298–312 (2011)
Jung, E., Iwami, S., Takeuchi, Y., Jo, T.-C.: Optimal control strategy for prevention of avian influenza pandemic. J. Theor. Biol. 260(2), 220–229 (2009)
Kaurov, V.: Modeling a pandemic like Ebola with the Wolfram language. Tech. Commun. Strategy (2014). http://blog.wolfram.com/2014/11/04/modeling-a-pandemic-like-ebola-with-the-wolfram-language
Legrand, J., Grais, R.F., Boelle, P.Y., Valleron, A.J., Flahault, A.: Understanding the dynamics of Ebola epidemics. Epidemiol. Infect. 135(4), 610–621 (2007)
Lewnard, J.A.: Ndeffo Mbah, M.L., Alfaro-Murillo, J.A., Altice, F.L., Bawo, L., Nyenswah, T.G., Galvani, A.P.: Dynamics and control of Ebola virus transmission in Montserrado, Liberia: a mathematical modelling analysis. Lancet Infect. Dis. 14(12), 1189–1195 (2014)
Peters, C.J., LeDuc, J.W.: An introduction to Ebola: the virus and the disease. J. Infect. Dis. 179(Suppl. 1), ix–xvi (1999)
Rachah, A., Torres, D.F.M.: Mathematical modelling, simulation, and optimal control of the 2014 Ebola outbreak in West Africa. Discrete Dyn. Nat. Soc. 2015, Art. ID 842792 (2015), 9 pp
Rachah, A., Torres, D.F.M.: Modelling and numerical simulation of the recent outbreak of Ebola. In: Proceedings of the 2nd International Conference on Numerical and Symbolic Computation: Developments and Applications (SYMCOMP 2015), Universidade do Algarve, Faro, March 26–27, 2015. Edited by APMTAC (Editors: A. Loja, J. I. Barbosa and J. A. Rodrigues), pp. 179–190. ISBN 978-989-96264-7-8
Rachah, A., Torres, D.F.M.: Predicting and controlling the Ebola infection. Math. Methods Appl. Sci. doi:10.1002/mma.3841 (in press)
Report of an International Commission: Ebola haemorrhagic fever in Zaire, 1976. Bull. World Health Organ. 56(2), 271–293 (1978)
Rodrigues, H.S., Monteiro, M.T.T., Torres, D.F.M.: Dynamics of dengue epidemics when using optimal control. Math. Comput. Model. 52(9–10), 1667–1673 (2010)
Rodrigues, H.S., Monteiro, M.T.T., Torres, D.F.M.: Vaccination models and optimal control strategies to dengue. Math. Biosci. 247(1), 1–12 (2014)
Rodrigues, H.S., Monteiro, M.T.T., Torres, D.F.M.: Optimal control and numerical software: an overview. In: Miranda, F. (ed.) Systems Theory: Perspectives, Applications and Developments, pp. 93–110. Nova Science Publishers, New York (2014)
Rodrigues, P., Silva, C.J., Torres, D.F.M.: Cost-effectiveness analysis of optimal control measures for tuberculosis. Bull. Math. Biol. 76(10), 2627–2645 (2014)
Silva, C.J., Torres, D.F.M.: Optimal control for a tuberculosis model with reinfection and post-exposure interventions. Math. Biosci. 244(2), 154–164 (2013)
Tchuenche, J.M., Khamis, S.A., Agusto, F.B., Mpeshe, S.C.: Optimal control and sensitivity analysis of an influenza model with treatment and vaccination. Acta Biotheoretica 59(1), 1–28 (2011)
Uganda Ministry of Health.: An outbreak of Ebola in Uganda. Trop. Med. Int. Health. 7(12), 1068–1075 (2002)
Valleron, A.J., Schwartz, D., Goldberg, M., Salamon, R.: Collectif Lépidémiologie humaine, Conditions de son développement en France, et rôle des mathématiques, vol. 462. Institut de France Académie des Sciences, Paris (2006)
WHO.: Report of an International Study Team. Ebola haemorrhagic fever in Sudan 1976. Bull. World Health Organ. 56(2), 247–270 (1978)
WHO.: Ebola data and statistics. http://apps.who.int/gho/data/node.ebola-sitrep
Zeng, D., Chen, H., Castillo-Chavez, C., Lober, W.B., Thurmond, M.: Infectious disease Informatics and Biosurveillance. Integrated Series in Information Systems, vol. 27. Springer, New York (2011)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Rachah, A., Torres, D.F.M. Dynamics and Optimal Control of Ebola Transmission. Math.Comput.Sci. 10, 331–342 (2016). https://doi.org/10.1007/s11786-016-0268-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11786-016-0268-y