Transmission dynamics of epidemic spread and outbreak of Ebola in West Africa: fuzzy modeling and simulation

  • Renu VermaEmail author
  • S. P. Tiwari
  • Ranjit Kumar Upadhyay
Original Research


In this paper, an attempt is made to understand the transmission dynamics of Ebola virus disease (EVD) incorporating fuzziness in all biological parameters due to its natural variability. To characterize the transmission trajectories of Ebola outbreak, we propose and analyze two SEIR and SEIRHD type transmission models. Using triangular fuzzy numbers for the imprecise parameters, we first study the existence of the equilibria and their stability. Both of the model have two equilibria, namely the disease-free and endemic. Stability of the disease-free and endemic equilibria is related with basic reproduction number that has been calculated from next generation matrix. Stability analysis of the system shows that the disease free equilibrium is locally as well as globally asymptotically stable when the basic reproduction number is less than unity. Under some additional conditions, the model system becomes locally asymptotically stable at unique endemic equilibrium when basic reproduction number is greater than unity. Finally, we perform some numerical experiments to justify the theoretical estimate.


Ebola virus epidemic model Fuzzy number Basic reproduction number Global stability 

Mathematics Subject Classifications

97Mxx 37C75 03B52 



The first author acknowledge with thanks the support received through a research grant, provided by the Council of Scientific and Industrial Research (CSIR) (Grant No. 09/085(0113)/2015-EMR-1), New Delhi, under which this work has been carried out.


  1. 1.
    Agusto, F.B.: Mathematical model of Ebola transmission dynamics with relapse and reinfection. Math. Biosci. 283, 48–59 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Althaus, C.L., Low, N., Musa, E.O., Shuaib, F., Gsteiger, S.: Ebola virus disease outbreak in Nigeria: transmission dynamics and rapid control. Epidemics 11, 80–84 (2015)CrossRefGoogle Scholar
  3. 3.
    Barro, S., Marin, R.: Fuzzy Logic in Medicine. Physica, heidelberg (2002)CrossRefzbMATHGoogle Scholar
  4. 4.
    Barros, L.C., Bassanezi, R.C., Leite, M.B.F.: The \(SI\) epidemiological models with a fuzzy transmission parameter. Comput. Math. Appl. 45, 1619–1628 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Barros, L.C., Tonelli, P.: About fuzzy dynamical systems: theory and applications, Ph.D thesis of the Institute of Mathematics, Statistics and Computer Science of University of Campinas, SaĂo Paulo, Brazil (1997)Google Scholar
  6. 6.
    Bassanezi, R.C., Barros, L.C.: A simple model of life expectancy with subjective parameters. Kybernets 24, 91–98 (1995)Google Scholar
  7. 7.
    Castillo-Chavez, C., Song, B.: Dynamical models of tuberculosis and their applications. Math. Biosci. Eng. 1, 361–404 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Centers for Disease Control and Prevention. Questions and answers: estimating the future number of cases in the Ebola epidemic -Liberia and Sierra Leone, 2014–2015. (2014). Accessed 10 Jan 2016
  9. 9.
    Centers for Disease Control and Prevention. Ebola virus disease. (2015). Accessed 28 Dec 2015
  10. 10.
    Chowell, G., Nishiura, H.: Transmission dynamics and control of Ebola virus disease: a review. BMC Med. 12, 1–16 (2014)CrossRefGoogle Scholar
  11. 11.
    Coltart, C.E., Lindsey, B., Ghinai, I., Johnson, A.M., Heymann, D.L.: The Ebola outbreak, 2013–2016: old lessons for new epidemics. Philos. Trans. R. Soc. B 372, 20160297 (2017)CrossRefGoogle Scholar
  12. 12.
    Das, A., Pal, M.: A mathematical study of an imprecise \(SIR\) epidemic model with treatment control. J. Appl. Math. Comput. 56, 477–500 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Datta, D.P.: The golden mean, scale free extension of real number system, fuzzy sets and \(\frac{1}{f}\) spectrum in physics and biology. Chaos Solitons Fractals 17, 781–788 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Dia, P., Constantine, P., Kalmbach, K., Jones, E., Pankavich, S.: A modified \(SEIR\) model for the spread of Ebola in Western Africa and metrics for resource allocation. Appl. Math. Comput. 324, 141–155 (2018)MathSciNetGoogle Scholar
  15. 15.
    Dixon, M.G., Schafer, I.J., et al.: Ebola viral disease outbreak in West Africa. MMWR Morb. Mortal. Wkly. Rep. 63, 548–551 (2014)Google Scholar
  16. 16.
    D’Silva, J.P., Eisenberg, M.C.: Modeling spatial invasion of Ebola in West Africa. J. Theor. Biol. 428, 65–75 (2017)CrossRefzbMATHGoogle Scholar
  17. 17.
    Du Toit, A.: Ebola virus in West Africa. Nat. Rev. Microbiol. 12, 312 (2014)CrossRefGoogle Scholar
  18. 18.
    EI Naschie, M.S.: On a fuzzy Kähler manifold which is consistent with the two slit experiment. Int. J. Nonlinear Sci. Numer. Simul. 7, 95–98 (2005)Google Scholar
  19. 19.
    EI Naschie, M.S.: From experimental quantum optics to quantum gravity via a fuzzy Kähler manifold. Chaos Solitons Fractals 25, 969–977 (2005)CrossRefzbMATHGoogle Scholar
  20. 20.
    Farahi, M.H., Barati, S.: Fuzzy time-delay dynamical systems. J. Math. Comput. Sci. 2, 44–53 (2011)CrossRefGoogle Scholar
  21. 21.
    Hale, J.K.: Theory of Functional Differential Equations. Springer, New York (1977)CrossRefzbMATHGoogle Scholar
  22. 22.
    Hanss, M.: Applied Fuzzy Arithmetic: An Introduction with Engineering Applications. Springer, Berlin (2005)zbMATHGoogle Scholar
  23. 23.
    Klir, G., Yuan, B.: Fuzzy Sets and Fuzzy Logic. Prentice Hall, New Jersey (1995)zbMATHGoogle Scholar
  24. 24.
    Li, L.: Transmission dynamics of Ebola virus disease with human mobility in Sierra Leone. Chaos, Solitons and Fractals 104, 575–579 (2017)CrossRefzbMATHGoogle Scholar
  25. 25.
    Martin, R.H.: Logarithmic norms and projections applied to linear differential systems. J. Math. Anal. Appl. 45, 432–454 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Massad, E., Ortega, N.R.S., Barros, L.C., Struchiner, C.J.: Fuzzy Logic in Action: Applications in Epidemiology and Beyond, Studied in Fuzziness and Soft Computing. Springer, Berlin (2008)CrossRefzbMATHGoogle Scholar
  27. 27.
    Mishra, B.K., Pandey, S.K.: Fuzzy epidemic model for the transmission of worms in computer network. Nonlinear Anal. Real World Appl. 11, 4335–4341 (2010)CrossRefzbMATHGoogle Scholar
  28. 28.
    Njankou, S.D.D., Nyabadza, F.: An optimal control model for Ebola virus disease. J. Biol. Syst. 24, 1–21 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Ortega, N.R.S., Sallum, P.C., Massad, E.: Fuzzy dynamical systems in epidemic modeling. Kybernetes 29, 201–218 (2000)CrossRefzbMATHGoogle Scholar
  30. 30.
    Pal, D., Mahaptra, G.S., Samanta, G.P.: Optimal harvesting of prey-predator system with interval biological parameters: a bioeconomic model. Math. Biosci. 241, 181–187 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Panja, P., Mondal, S.K., Chattopadhyay, J.: Dynamical study in fuzzy threshold dynamics of a Cholera epidemic model. Fuzzy Inf. Eng. 9, 381–401 (2017)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Puri, M., Ralescu, D.: Differentials of fuzzy functions. J. Math. Anal. Appl. 91, 552–558 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Rachab, A., Torres, D.F.M.: Mathematical modeling, simulation and optimal control of the \(2014\) Ebola outbreak in West Africa, Discrete Dynamics in Nature and Society,
  34. 34.
    Richards, P., Amara, J., Ferme, M., Mokuwa, E., Koroma, P., Sheriff, I., Suluku, R., Voors, M.: Social pathways for Ebola virus disease in rural Sierra Leone and some implications for containment. PLoS Negl. Trop. Dis. (2015).
  35. 35.
    Rivers, C., Lofgren, E., Marathe, M., Eubank, S., Lewis, B.: Modeling the impact of interventions on an epidemic of Ebola in Sierra Leone and Liberia. PLoS Curr. 6, 1–24 (2014)Google Scholar
  36. 36.
    Roy, P., Upadhyay, R.K.: Spatiotemporal transmission dynamics of recent Ebola outbreak in Sierra Leone, West Africa: impact of control measures. J. Biol. Syst. 25, 1–29 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Sadhukhan, D., Sahoo, L.N., Mondal, B., Maitri, M.: Food chain model with optimal harvesting in fuzzy environment. J. Appl. Math. Comput. 34, 1–18 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Seikkala, S.: On the fuzzy initial value problem. Fuzzy Sets Syst. 24, 309–330 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Sugeno, M.: Theory of fuzzy integrals and its applications, Doctoral Thesis, Tokyo Institute of Technology (1974)Google Scholar
  40. 40.
    Upadhyay, R.K., Roy, P.: Deciphering dynamics of recent epidemic spread and outbreak in West Africa: the case of Ebola virus. Int. J. Bifurc. Chaos 26, 1–25 (2016)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Van den Driessche, P., Watmough, J.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180, 29–48 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Verma, R., Tiwari, S.P., Upadhyay, R.K.: Fuzzy modeling for the spread of influemza virus and its possible control. Comput. Ecol. Softw. 8, 32–45 (2018)Google Scholar
  43. 43.
    Webb, G., Browne, C., Huo, X., Seydi, O., Seydi, M., Magal, P.: A model of the \(2014\) Ebola epidemic in West Africa with contact tracing. In: PLOS Currents Outbreaks, 1st edn. (2015).
  44. 44.
    Weitzand, J., Dushoff, J.: Modeling post-death transmission of Ebola: challenges for inference and opportunities for control. Sci. Rep. 5, 8751 (2015)CrossRefGoogle Scholar
  45. 45.
    World Health Organization. Ebola situation report-29 April 2015. (2015). Accessed 09 Jan 2016

Copyright information

© Korean Society for Computational and Applied Mathematics 2019

Authors and Affiliations

  • Renu Verma
    • 1
    Email author
  • S. P. Tiwari
    • 1
  • Ranjit Kumar Upadhyay
    • 1
  1. 1.Department of Applied MathematicsIndian Institute of Technology (ISM)DhanbadIndia

Personalised recommendations