233
Views
11
CrossRef citations to date
0
Altmetric
Articles

Nonstandard finite difference method revisited and application to the Ebola virus disease transmission dynamics

, , , , , & show all
Pages 818-854 | Received 27 Mar 2020, Accepted 23 May 2020, Published online: 20 Jul 2020

References

  • F.B. Agusto, M.I. Teboh-Ewungkem, and A.B. Gumel, Mathematical assessment of the effect of traditional beliefs and customs on the transmission dynamics of the 2014 Ebola outbreaks, BMC Med. 13(96) (2015), pp. 1–18.
  • H. Al-Kahby, F. Dannan, and S. Elaydi, Nonstandard discretization methods for some biological models, in Applications of Nonstandard Finite Difference Schemes, R.E. Mickens, ed., World Scientific, Singapore, 2000, pp. 155–180.
  • R. Anguelov and J.M.-S. Lubuma, Contributions to the mathematics of the nonstandard finite difference method and applications, Numer. Methods Partial Differ. Equ. 17 (2001), pp. 518–543. doi: 10.1002/num.1025
  • R. Anguelov and J.M.-S. Lubuma, Nonstandard finite difference method by nonlocal approximation, Math. Comput. Simul. 61 (2003), pp. 465–475. doi: 10.1016/S0378-4754(02)00106-4
  • R. Anguelov, P. Kama, and J.M.-S. Lubuma, On non-standard finite difference models of reaction-diffusion equations, J. Comput. Appl. Math. 175 (2005), pp. 11–29. doi: 10.1016/j.cam.2004.06.002
  • R. Anguelov, J.K. Djoko, P. Kama, and J.M.-S. Lubuma, On elementary stable and dissipative nonstandard finite difference schemes for dynamical systems, Proceedings of the International Conference of Computational Methods in Science and Engineering (Crete, Greece, 27 October–1 November 2006), Lecture Series on Computer and Computational Sciences Vol. 7A, VSP International Science Publishers, Utrecht, 2006, pp. 24–27
  • R. Anguelov, J.M.-S. Lubuma, and M. Shillor, Topological dynamic consistency of nonstandard finite difference schemes for dynamical systems, J. Differ. Equ. Appl. 17 (2011), pp. 1769–1791. doi: 10.1080/10236198.2010.488226
  • R. Anguelov, Y. Dumont, J.M.-S. Lubuma, and M. Shillor, Dynamically consistent nonstandard finite difference schemes for epidemiological models, J. Comput. Appl. Math. 255 (2014), pp. 161–182. doi: 10.1016/j.cam.2013.04.042
  • A.R. Appadu, J.M.-S Lubuma, and N. Mphephu, Computational study of three numerical methods for some linear and nonlinear advection-diffusion-reaction problems, Prog. Comput. Fluid Dyn. 17 (2017), pp. 114–129. doi: 10.1504/PCFD.2017.082520
  • T. Berge, J.M.S. Lubuma, G.M. Moremedi, N. Morris, and R. Kondera-Shava, A simple mathematical model for Ebola in Africa, J. Biol. Dyn. 11(1) (2017), pp. 42–74. doi: 10.1080/17513758.2016.1229817
  • T. Berge, M. Chapwanya, J.M.S. Lubuma, and Y.A. Terefe, A mathematical model for Ebola epidemic with self-protection measures, J. Biol. Syst. 26(1) (2018), pp. 107–132. doi: 10.1142/S0218339018500067
  • T. Berge, A.J.O.Tassé H.M.Tenkam, and J.M.S Lubuma, Mathematical modeling of contact tracing as a control strategy of Ebola virus disease, Int. J. Biomath. 11(7) (2018), 1850093 (36 pages). doi: 10.1142/S1793524518500936
  • T. Berge, S. Bowong, J.M.S. Lubuma, and L.M. Manyombe, Modeling Ebola virus disease transmissions with reservoir in a complex virus life ecology, Math. Biosci. Eng. 15(1) (2018), pp. 21–56. doi: 10.3934/mbe.2018002
  • Y. Boum, The DRC is on the road to being Ebola free: How it got there? The Conversation, 5 March 2020. Available at https://theconversation.com/the-drc-is-on-the-road-to-being-ebola-free-how-it-got-here-132992.
  • G. Chowell, N.W. Hengartner, C. Castillo-Chavez, P.W. Fenimore, and J.M. Hyman, The basic reproductive number of Ebola and the effects of public health measures: The cases of Congo and Uganda, J. Theor. Biol. 229 (2004), pp. 119–126. doi: 10.1016/j.jtbi.2004.03.006
  • D.T. Dimitrov and H.V. Kojouharov, Positive and elementary stable nonstandard numerical methods with applications to predator-prey models, J. Comput. Appl. Math. 189 (2006), pp. 98–108. doi: 10.1016/j.cam.2005.04.003
  • D.T. Dimitrov, H.V. Kojouharov, and B.M. Chen-Charpentier, Reliable finite difference schemes with applications in mathematical biology, in Advances in the Applications of Nonstandard Finite Difference Schemes, R.E. Mickens, ed., World Scientific, Singapore, 2005, pp. 249–285.
  • Y. Dumont and J.M.-S. Lubuma, Non-Standard finite-difference methods for vibro-impact problems, Proc. R. Soc. Lond. Ser. A: Math. Phys. Eng. Sci. 461 (2005), pp. 1927–1950. doi: 10.1098/rspa.2004.1425
  • L.C. Evans, Partial Differential Equations, American Mathematical Society, Providence, RI, 1998.
  • A.B. Gumel (ed.), Journal of Difference Equations and Application, Vol. 9, 2003, Special Issue no 11–12 dedicated to R.E. Mickens on the occasion of his 60th birthday
  • A.B. Gumel (ed.), Mathematics of Discrete and Continuous Dynamical Systems, Vol. 618, 2014, Contemporary Mathematics, American Mathematical Society, (Volume dedicated to R.E. Mickens on the occasion of his 70th birthday)
  • B. Ivorra, D. Ngom, and A.M. Ramos, A mathematical model to predict the risk of human disease spread between countries-validation and application to 2014–2015 Ebola virus disease epidemic, Bull. Math. Biol. 77 (2015), pp. 1668–1704. doi: 10.1007/s11538-015-0100-x
  • K. Kupferschmidt, Ebola veteran promises an end to Congo's epidemic, Africa, Health, Aug. 6, 2019. Available at https://doi.org/10.1126/science.aaz0268.
  • J.D. Lambert, Numerical Methods for Ordinary Differential Systems, John Wiley & Sons, New York, 1991.
  • J.M.-S. Lubuma and K.C. Patidar, Contributions to the theory of non-standard finite difference methods and applications to singular perturbation problems, in Advances in the Applications of Nonstandard Finite Difference Schemes, R.E. Mickens, ed., World Scientific, Singapore, 2005, pp. 513–560.
  • J.M.-S. Lubuma and A. Roux, An improved theta method for systems of ordinary differential equations, J. Differ. Equ. Appl. 9 (2003), pp. 1023–1035. doi: 10.1080/1023619031000146904
  • A. Maxmen, Two Ebola drugs show promise amid ongoing outbreak, Nature News, 12 August 2019. Available at https://dx.doi.org/10.1038/d41586-019-02442-6.
  • R.E. Mickens, Nonstandard Finite Difference Models of Differential Equations, World Scientific, Singapore, 1994.
  • R.E. Mickens (ed.), Applications of Nonstandard Finite Difference Schemes, World Scientific, Singapore, 2000.
  • R.E. Mickens, Discrete models of differential equations: The roles of dynamic consistency and positivity, in Difference Equations and Discrete Dynamical Systems, (Proceedings of the 9th International Conference, Los Angeles, USA, 2–7 August 2004), L.J.S. Allen, B. Aulbach, S. Elaydi, and R. Sacker, eds., World Scientific, Singapore, 2005, pp. 51–70
  • R.E. Mickens (ed.), Nonstandard finite difference methods, in Advances in the Applications of Nonstandard Finite Difference Schemes, World Scientific, Singapore, 2005
  • R.E. Mickens, Calculation of denominator functions for nonstandard finite difference schemes for differential equations satisfying a positivity condition, Numer Methods Partial Differ. Equ. 23 (2007), pp. 672–691. doi: 10.1002/num.20198
  • D. Ndanguza, J.M. Tchuenche, and H. Haario, Statistical data analysis of the 1995 Ebola outbreak in the Democratic Republic of Congo, Afrika Mat. 24 (2013), pp. 55–68. doi: 10.1007/s13370-011-0039-5
  • K.C. Patidar, Nonstandard finite difference methods: Recent trends and further developments, J. Differ. Equ. Appl. 22 (2016), pp. 817–849. doi: 10.1080/10236198.2016.1144748
  • L.-I.W. Roeger, Exact finite difference schemes, in Mathematics of Discrete and Continuous Dynamical Systems, Contemporary Mathematics, A.B. Gumel, ed., Vol. 618, American Mathematical Society, Providence, 2014, pp. 147–161.
  • A.M. Stuart and A.R. Humphries, Dynamical Systems and Numerical Analysis, Cambridge University Press, New York, 1998.
  • World Health Organization (WHO), Ebola virus disease, Democratic Republic of Congo, External Situation Report 78/2019, (4 February 2020).
  • World Health Organization (WHO), Ebola Virus Disease, Democratic Republic of Congo, External Situation Reports 82/2019, (4 February 2020) and 81/2019, (25 February 2020).
  • P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartment models of disease transmission, Math. Biosci. 180 (2002), pp. 29–48. doi: 10.1016/S0025-5564(02)00108-6

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.