Special Issue on Fractal AI-Based Analyses and Applications to Complex Systems IV
Guest Editors: Yeliz Karaca, Dumitru Baleanu, Majaz Moonis, Yu-Dong Zhang and Osvaldo Gervasi

Abstract

In this paper, a mathematical model of Ebola virus with contact tracing as a control strategy was developed and analyzed. We considered the model without contact tracing and with perfect contact tracing. The two sub-models have been explicitly analyzed. In the first sub-model, it has been found that the disease-free equilibrium (DFE) is both locally and globally asymptotically stable whenever the associated control reproduction number is less than one. The endemic equilibrium point (EEP) is globally asymptotically stable whenever the associated control reproduction number is greater than one. In the second sub-model, it has been found that the DFE is both locally and globally asymptotically stable whenever the control reproduction number is less than one. The EEP is globally asymptotically stable whenever the control reproduction number is greater than one. The full model has also been analyzed, which shows that the DFE is both locally and globally asymptotically stable whenever the associated control reproduction number is less than one. The EEP is globally asymptotically stable whenever the control reproduction number is greater than one. In sensitivity analysis part, effective contact rate for humans was very sensitive in increasing the basic reproduction number and personal hygiene was very sensitive in decreasing the basic reproduction number also, numerical simulation shows that the Ebola virus can be wiped out in society if contact tracing and personal hygiene can be implemented perfectly in the human population.

Keywords:

References

1. D. Burton et al., A mathematical model of contact tracing during the 2014–2016 West African Ebola outbreak, Mathematics 9 (2021) 608, https://doi.org/10.3390/math9060608 Crossref, Web of Science, Google Scholar
2. B. Aylward et al., Ebola virus disease in West Africa — The first 9 months of the epidemic and forward projections, N. Engl. J. Med. 371 (2014) 1481–1495. Crossref, Web of Science, Google Scholar
3. 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) 42–74, https://doi.org/10.1080/17513758.2016.1229817 Crossref, Web of Science, Google Scholar
4. UN News, Latest deadly Ebola virus outbreak in DR Congo declared over [A plastic sheet separates a mother from her son at an Ebola treatment centre in Beni, North Kivu province, Democratic Republic of the Congo], 2021, https://news.un.org/en/story/2021/05/1091162. Google Scholar
5. 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 13(96) (2015) 1–18. Google Scholar
6. T. Berge, S. Bowong, J. M.-S. Lubuma and M. L. M. Manyombe , Modelling Ebola virus disease transmissions with reservoir in a complex virus life ecology, Math. Biosci. 15(1) (2018) 21–56. Crossref, Web of Science, Google Scholar
7. T. Berge, J. S.-M. Lubuma, A. J. O. Tassé and H. M. Tenkam , Dynamics of host reservoir transmission of Ebola with spillover potential to humans, Electron. J. Qual. Theory Differ. Equ. 14 (2018) 1–32. Google Scholar
8. H. Gumsova, A. Shittu, M. Sanbul and H. Leblebicioglu , Ebola virus disease and the veterinary perspective, BMC 14(30) (2015) 1–5. Google Scholar
9. F. B. Agusto , Mathematical model of Ebola transmission dynamics with relapse and reinfection, Math. Biosci. 283 (2017) 48–59. Crossref, Web of Science, Google Scholar
10. Ebola “threat to world security”-UN Security Council, BBC (2014), www.bbc.com/news/world-africa-29262968. Google Scholar
11. D. Gao, Transmission dynamics of some epidemiological patch model, Ph.D. thesis, University of Miami (2012). Google Scholar
12. T. Berge, A. J. Ouemba Tassé and H. M. Tenkam , Mathematical modeling of contact tracing as a control strategy of Ebola virus disease, Int. J. Biomath. 11(7) (2018) 1850093, https://doi.org/10.1142/S1793524518500936 Link, Web of Science, Google Scholar
13. H. De Arazoza and R. Lounes , A non-linear model for a sexually transmitted disease with contact tracing, Math. Med. Biol. 19 (2002) 221–234. Crossref, Web of Science, Google Scholar
14. Y. H. Hsieh, Y. S. Wang, H. de Arazoza and R. Lounes , Modeling secondary level of HIV contact tracing: Its impact on HIV intervention in Cuba, BMC Infect. Dis. 10 (2010) 1–9. Crossref, Web of Science, Google Scholar
15. J. M. Hyman, J. Li and E. A. Stanley , Modeling the impact of random screening and contact tracing in reducing the spread of HIV, Math. Biosci. 181 (2003) 17–54. Crossref, Web of Science, Google Scholar
16. CDC, Increases in Heroin Overdose Deaths — 28 States, 2010 to 2012, MMWR Morb. Mortal. Wkly. Rep. 63 (2014) 849–854. Web of Science, Google Scholar
17. C. Browne, H. Gulbudak and G. Webb , Modeling contact tracing in outbreaks with application to Ebola, J. Theor. Biol. 384 (2015) 33–49. Crossref, Web of Science, Google Scholar
18. T. Attenborough , Modelling the Ebola Outbreak in West Africa and Community Responses (Complex University College, London, 2015), pp. 1–21. Google Scholar
19. D. Butler , How disease detectives are fighting Ebola’s spread, Nature 10 (2014) 1–2, www.nature.com/news/how-disease-detectives-are-ghting-ebola-s-spread-1.16069. Google Scholar
20. G. A. Ngwa and M. I. Teboh-Ewungkem , A mathematical model with quarantine states for the dynamics of Ebola virus disease in human populations, Comput. Math. Methods Med. 2016 (2016) 9352725, https://doi.org/10.1155/2016/9352725 Crossref, Web of Science, Google Scholar
21. J. Zhang , Infectious disease model with the latent period and quarantine and comparison of drug distribution scheme based on Ebola virus, in 1st International Symposium on Social Science (Atlantis Press, 2015), pp. 362–366. Crossref, Google Scholar
22. D. Salem and R. Smith , A mathematical model of Ebola virus disease: Using sensitivity analysis to determine effective intervention targets, in SummerSim-SCS, Montreal, QC, Canada, 2016, pp. 16–23. Google Scholar
23. C. M. Rivers, E. T. Lofgren, M. Marathe, S. Eubank and B. L. Lewis , Modeling the impact of interventions on an epidemic of Ebola in Sierra Leone and Liberia, PLoS Curr. 6 (2014) 1–9, https://doi.org/10.1371/currents.outbreaks.fd38dd85078565450b0be3fcd78f5ccf Google Scholar
24. W. O. Kermack and A. G. Mckendrick , Contributions to the mathematical theory of epidemics, Proc. R. Soc. Lond. A 115 (1927) 700–721. Crossref, Google Scholar
25. C. Browne, X. Huo, P. Magal, M. Seydi, O. Seydi and G. Webb , A model of the 2014 Ebola epidemic in West Africa with contact tracing, PLoS Curr. 7 (2015) 1–20, https://doi.org/10.1371/currents.outreaks.846b2a31ef37018b7d1126a9c8adf22a Google Scholar
26. R. M. Anderson, R. M. May and S. Gupta , Non-linear phenomena in Host–Parasite interactions, Parasitology 99(S1) (1989) S59–S79. Crossref, Web of Science, Google Scholar
27. I. Area, F. Ndaïrou, J. J. Nieto, C. J. Silva and D. F. M. Torres , Ebola model and optimal controls with vaccination constraints, J. Ind. Manag. Optim. 278 (2017) 1–21. Google Scholar
28. World Health Organisation (WHO), Ebola virus disease, Fact Sheet 103, WHO Press (2015), http://www.who.int/mediacentre/factsheets/fs103/en/. Google Scholar
29. Center for Disease Control and Prevention, Ebola virus disease: Signs and symptoms, CDC (2014), http://www.cdc.gov/vhf/ebola/symptoms/. Google Scholar
30. F. B. Agusto et al., The impact of bed-net use on malaria prevalence, J. Theor. Biol. 320 (2013) 50–65. Crossref, Web of Science, Google Scholar
31. C. Castillo-Chavez, S. Blower, P. Van den Driesche, D. Kirschner and A. A. Yakubu , Mathematical Approaches for Emergence and Reemergence Infectious Diseases (Springer-Verlag, New York, 2002). Google Scholar
32. X. S. Wang and L. Zhong , Ebola outbreak in West Africa: Real-time estimation and multiple wave prediction, Math. Biosci. Eng. 12(5) (2015) 1055–1063, https://doi.org/10.3934/mbe.2015.12.1055 Crossref, Web of Science, Google Scholar
33. S. Towers, O. Patterson-Lomba and C. Castillo-Chavez , Temporal variations in the effective reproduction number of the 2014 West Africa Ebola outbreak, PLoS Curr. 6 (2014). Google Scholar
34. A. Rachah and D. F. M. Torres , Mathematical modelling, simulation, and optimal control of the 2014 Ebola outbreak in West Africa, Discrete Dyn. Nat. Soc. 2015 (2015) 842792, https://doi.org/10.1155/2015/842792 Crossref, Web of Science, Google Scholar
35. 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) 55–68. Crossref, Google Scholar
36. P. E. Lekone and B. F. Finkenstädt , Statistical inference in a stochastic epidemic SEIR model with control intervention: Ebola as a case study, Biometrics 62 (2006) 1170–1177. Crossref, Web of Science, Google Scholar
37. C. Althaus , Estimating the reproduction number of Ebola (EBOV) during outbreak in West Africa, PLoS Curr. 1 (2014). Google Scholar
38. G. Chowell and H. Nishiura , Transmission dynamics and control of Ebola virus disease (EVD): A review, BMC Med. 12 (2014) 196. Crossref, Web of Science, Google Scholar
39. F. O. Fasina, A. Shittu, D. Lazarus, O. Tomori, L. Simonsen, C. Viboud and G. Chowell , Transmission dynamics and control of Ebola virus disease outbreak in Nigeria, July to September 2014, Eurosurveillance 19(40) (2014) pii=20920, http://www.eurosurveillance.org/ViewArticle.aspx?ArticleId=20920. Crossref, Web of Science, Google Scholar
40. B. Ivorra, D. Ngom and A. M. Ramos , Be-CoDiS: A mathematical model to predict the risk of human diseases spread between countries-validation and application to the 2014–2015 Ebola virus disease epidemic, Bull. Math. Biol. 77 (2015) 1668–1704, https://doi.org/10.1007/s11538-015-0100-x Crossref, Web of Science, Google Scholar
41. J. Legrand, R. F. Grais, P. Y. Boelle, A. J. Valleron and A. Flahault , Understanding the dynamics of Ebola epidemics, Epidemiol. Infect. 135 (2007) 610–621. Crossref, Web of Science, Google Scholar
42. C. Althaus , Estimating the reproduction number of Ebola (EBOV) during outbreak in West Africa, PLoS Curr. 6 (2014) 1–5, https://doi.org/10.1371/currents.outbreaks.91afb5e0f279e7f29e7056095255b288 Google Scholar
43. C. L. Althaus, N. Low, E. O. Musa, F. Schuaib and S. Gsteiger , Ebola virus disease outbreak in Nigeria: Transmission dynamics and rapid control, Epidemics 11 (2015) 80–84. Crossref, Web of Science, Google Scholar
44. A. Denes and A. B. Gumel , Modelling the impact of quarantine during an outbreak of Ebola virus disease, Infect. Dis. Model. 4 (2019) 12–27. Google Scholar
45. T. Berge, S. Bowong and J. S. M. Lubuma , Global stability of a two-patch cholera model with fast and slow transmissions, Math. Comput. Simulation 133 (2017) 142–164. Crossref, Web of Science, Google Scholar
46. V. Lakshmikantham, S. Leela and A. A. Martynyuk , Stability Analysis of Nonlinear Systems (Marcel Dekker, New York, Basel, 1989). Google Scholar
47. H. W. Hethcot , The mathematics of infectious diseases, SIAM Rev. 42(4) (2000) 599–653. Crossref, Web of Science, Google Scholar
48. O. Diekmann, J. A. Heesterbeek and J. A. Metz , On the definition and the computation of the basic reproduction ratio $R_{0}$ $R_{0}$ in models for infectious diseases in heterogeneous populations, J. Math. Biol. 28(4) (1990) 365–382. Crossref, Web of Science, Google Scholar
49. P. Van De Driessche and J. Watmough , Reproduction number and sub threshold Endemic Equilibrium for compartmental models of disease transmission, Math. Biosci. 180 (2002) 29–48. Crossref, Web of Science, Google Scholar
50. C. V. De-León , Constructions of Lyapunov functions for classics SIS, SIR and SIRS epidemic model with variable population size, Foro-Red-Mat: Rev. Electrón. Conten. Mat. 26(5) (2009) 1. Google Scholar
51. J. K. K. Asamoah, F. T. Oduro, E. Bonyah and B. Seidu , Modelling of rabies transmission dynamics using optimal control analysis, Hindawi, J. Appl. Math. 2017 (2017) 2451237, https://doi.org/10.1155/2017/2451237 Crossref, Google Scholar
52. J. P. Lasalle , The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics (SIAM, Philadelphia, 1976). Crossref, Google Scholar
53. U. T. Mustapha, S. Qureshi, A. Yusuf and E. Hincal , Fractional modeling for the spread of Hookworm infection under Caputo operator, Chaos Solitons Fractals 137 (2020) 109878. Crossref, Web of Science, Google Scholar
54. S. Usaini, U. T. Mustapha and S. M. Sabiu , Modelling scholastic underachievement as a contagious disease, Math. Methods Appl. Sci. 41(18) (2018) 8603–8612. Crossref, Web of Science, Google Scholar
55. M. A. Safi, A. B. Gumel , Mathematical analysis of a disease transmission model with quarantine, isolation and an imperfect vaccine, Comput. Math. Appl. 61 (2011) 3044–3070. Crossref, Web of Science, Google Scholar
56. T. Berge, J. M. Lubuma, G. M. Moremedi, N. Moris and G. M. Kondera-Shava , A simple mathematical model for Ebola in Africa, J. Biol. Dynam. 11(1) (2018) 42–74. Crossref, Web of Science, Google Scholar
57. N. Abdulrahman, S. Abdulrahman and A. Abulrazaq , A mathematical model for controlling the spread of Ebola virus disease in Nigeria, Int. J. Humanities and Management Sciences (IJHMS) 3(3) (2015) 2320–4044. Google Scholar

Vol. 31, No. 10

Metrics

Downloaded 131 times

History

Received 21 June 2022

Accepted 28 April 2023

Published: October 14, 2023

Information

This is an Open Access article in the “Special Issue on Fractal AI-Based Analyses and Applications to Complex Systems: Part IV”, edited by Yeliz Karaca https://orcid.org/0000-0001-8725-6719 (University of Massachusetts Chan Medical School, USA), Dumitru Baleanu https://orcid.org/0000-0002-0286-7244 (Cankaya University, Turkey & Institute of Space Sciences, Romania), Majaz Moonis https://orcid.org/0000-0002-9747-0003 (University of Massachusetts Chan Medical School, USA), Yu-Dong Zhang https://orcid.org/0000-0002-4870-1493 (University of Leicester, Leicester, UK) & Osvaldo Gervasi https://orcid.org/0000-0003-4327-520X (Perugia University, Perugia, Italy) published by World Scientific Publishing Company. It is distributed under the terms of the Creative Commons Attribution 4.0 (CC BY) License which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

Keywords

PDF download

System Upgrade on Tue, May 28th, 2024 at 2am (EDT)

UNRAVELING THE DYNAMICS OF EBOLA VIRUS WITH CONTACT TRACING AS CONTROL STRATEGY

Abstract

Recommended