# A Mathematical Model of Contact Tracing during the 2014–2016 West African Ebola Outbreak

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{7}

^{8}

^{9}

^{10}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model

## 3. Reproductive Number

## 4. Parameter Estimation

- Fit the data with a low J value;
- In each class, we wanted reasonable dynamics, meaning approximately the correct magnitude in the size of each compartment.

## 5. Importance of Contact Tracing

## 6. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Data

## Appendix B. Initial Fitting Results

**Figure A1.**First attempt match to the data of cumulative cases and cumulative deaths with all parameters constant. The value of J is $0.1963$.

## References

- Frieden, T.R.; Damon, I.; Bell, B.P.; Kenyon, T.; Nichol, S. Ebola 2014–New Challenges, New Global Response and Responsibility. N. Engl. J. Med.
**2014**, 371, 1177–1180. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Aylward, B.; Barboza, P.; Bawo, L.; Bertherat, E.; Bilivogui, P.; Blake, I.; Brennan, R.; Briand, S.; Chakauya, J.M.; Chitala, K.; et al. Ebola virus disease in West Africa-The first 9 months of the epidemic and forward projections. N. Engl. J. Med.
**2014**, 371, 1481–1495. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Bogoch, I.I.; Creatore, M.I.; Cetron, M.S.; Brownstein, J.S.; Pesik, N.; Miniota, J.; Tam, T.; Hu, W.; Nicolucci, A.; Ahmed, S.; et al. Assessment of the potential for international dissemination of Ebola virus via commercial air travel during the 2014 west African outbreak. Lancet
**2015**, 385, 29–35. [Google Scholar] [CrossRef] [Green Version] - Brainard, J.; Hooper, L.; Pond, K.; Edmunds, K.; Hunter, P.R. Risk factors for transmission of Ebola or Marburg virus disease: A systematic review and meta-analysis. Int. J. Epidemiol.
**2016**, 45, 102–116. [Google Scholar] [CrossRef] [Green Version] - Dietz, P.M.; Jambai, A.; Paweska, J.T.; Yoti, Z.; Ksaizek, T.G. Epidemiology and risk factors for ebola virus disease in Sierra Leone-23 May 2014 to 31 January 2015. Clin. Infect. Dis.
**2015**, 61, 1648–1654. [Google Scholar] [CrossRef] [Green Version] - Drake, J.M.; Bakach, I.; Just, M.R.; O’Regan, S.M.; Gambhir, M.; Chun-Hai, I.F. Transmission models of historical ebola outbreaks. Emerg. Infect. Dis.
**2015**, 21, 1447–1450. [Google Scholar] [CrossRef] [Green Version] - Gomes, M.F.; Piontti, A.P.Y.; Rossi, L.; Chao, D.; Longini, I.; Halloran, M.E.; Vespignani, A. Assessing the international spreading risk associated with the 2014 West African Ebola outbreak. PLoS Curr.
**2014**, 6. [Google Scholar] [CrossRef] [PubMed] - Skrip, L.A.; Fallah, M.P.; Gaffney, S.G.; Yaari, R.; Yamin, D.; Huppert, A.; Bawo, L.; Nyenswah, T.; Galvani, A.P. Characterizing risk of Ebola transmission based on frequency and type of case–contact exposures. Philos. Trans. R. Soc. B Biol. Sci.
**2017**, 372, 20160301. [Google Scholar] [CrossRef] [Green Version] - Li, S.L.; Bjørnstad, O.N.; Ferrari, M.J.; Mummah, R.; Runge, M.C.; Fonnesbeck, C.J.; Tildesley, M.J.; Probert, W.J.; Shea, K. Essential information: Uncertainty and optimal control of Ebola outbreaks. Proc. Natl. Acad. Sci. USA
**2017**, 114, 5659–5664. [Google Scholar] [CrossRef] [Green Version] - Cook, B.W.M.; Cutts, T.A.; Nikiforuk, A.M.; Poliquin, P.G.; Court, D.A.; Strong, J.E.; Theriault, S.S. Evaluating environmental persistence and disinfection of the Ebola virus Makona variant. Viruses
**2015**, 7, 1975–1986. [Google Scholar] [CrossRef] - Senga, M.; Pringle, K.; Ramsay, A.; Brett-Major, D.M.; Fowler, R.A.; French, I.; Vandi, M.; Sellu, J.; Pratt, C.; Saidu, J.; et al. Factors underlying Ebola virus infection among health workers, Kenema, Sierra Leone, 2014–2015. Clin. Infect. Dis.
**2016**, 63, 454–459. [Google Scholar] [CrossRef] [Green Version] - Garske, T.; Cori, A.; Ariyarajah, A.; Blake, I.M.; Dorigatti, I.; Eckmanns, T.; Fraser, C.; Hinsley, W.; Jombart, T.; Mills, H.L.; et al. Heterogeneities in the case fatality ratio in the west African Ebola outbreak 2013–2016. Philos. Trans. R. Soc. B Biol. Sci.
**2017**, 372, 20160308. [Google Scholar] [CrossRef] [Green Version] - De Arazoza, H.; Lounes, R. A non-linear model for a sexually transmitted disease with contact tracing. IMA J. Math. Appl. Med. Biol.
**2002**, 19, 221–234. [Google Scholar] [CrossRef] [PubMed] - Hsieh, Y.H.; Wang, Y.S.; de Arazoza, H.; Lounes, R. Modeling secondary level of HIV contact tracing: Its impact on HIV intervention in Cuba. BMC Infect. Dis.
**2010**, 10, 1–9. [Google Scholar] [CrossRef] [Green Version] - Hyman, J.M.; Li, J.; Stanley, E.A. Modeling the impact of random screening and contact tracing in reducing the spread of HIV. Math. Biosci.
**2003**, 181, 17–54. [Google Scholar] [CrossRef] - CDC. Increases in Heroin Overdose Deaths—28 States, 2010 to 2012. MMWR Morb. Mortal. Wkly. Rep.
**2014**, 63, 849–854. [Google Scholar] - Browne, C.; Gulbudak, H.; Webb, G. Modeling contact tracing in outbreaks with application to Ebola. J. Theor. Biol.
**2015**, 384, 33–49. [Google Scholar] [CrossRef] [PubMed] [Green Version] - 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. PLoS Curr.
**2015**, 7, 1–8. [Google Scholar] [CrossRef] - Rivers, C.M.; Lofgren, E.T.; Marathe, M.; Eubank, S.; Lewis, B.L. Modeling the impact of interventions on an epidemic of Ebola in Sierra Leone and Liberia. PLoS Curr.
**2014**, 6, 1–12. [Google Scholar] [CrossRef] - Stehling-Ariza, T.; Rosewell, A.; Moiba, S.A.; Yorpie, B.B.; Ndomaina, K.D.; Jimissa, K.S.; Leidman, E.; Rijken, D.J.; Basler, C.; Wood, J.; et al. The impact of active surveillance and health education on an Ebola virus disease cluster - Kono District, Sierra Leone, 2014–2015. BMC Infect. Dis.
**2016**, 16, 1–7. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Olu, O.O.; Lamunu, M.; Nanyunja, M.; Dafae, F.; Samba, T.; Sempiira, N.; Kuti-George, F.; Abebe, F.Z.; Sensasi, B.; Chimbaru, A.; et al. Contact Tracing during an Outbreak of Ebola Virus Disease in the Western Area Districts of Sierra Leone: Lessons for Future Ebola Outbreak Response. Front. Public Health
**2016**, 4, 1–9. [Google Scholar] [CrossRef] [Green Version] - Washington, M.; (Centers for Disease Control and Prevention, Atlanta, GA, USA). Personal communication, 2017.
- Wolfe, C.M.; Hamblion, E.L.; Schulte, J.; Williams, P.; Koryon, A.; Enders, J.; Sanor, V.; Wapoe, Y.; Kwayon, D.; Blackley, D.J.; et al. Ebola virus disease contact tracing activities, lessons learned and best practices during the Duport Road outbreak in Monrovia, Liberia, November 2015. PLoS Negl. Trop. Dis.
**2017**, 11, 1–16. [Google Scholar] [CrossRef] [Green Version] - Swanson, K.C.; Altare, C.; Wesseh, C.S.; Nyenswah, T.; Ahmed, T.; Eyal, N.; Hamblion, E.L.; Lessler, J.; Peters, D.H.; Altmann, M. Contact tracing performance during the Ebola epidemic in Liberia, 2014–2015. PLoS Negl. Trop. Dis.
**2018**, 12, 2014–2015. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Chowell, G.; Nishiura, H. Transmission dynamics and control of Ebola virus disease (EVD): A review. BMC Med.
**2014**, 12, 1–17. [Google Scholar] [CrossRef] [Green Version] - Ebola Situation Report, Ministry of Health and Sanitation, Sierra Leone. 2015. Available online: https://web.archive.org/web/20150314233800/http:/health.gov.sl/?page_id=583 (accessed on 28 February 2020).
- Ebola Situation Report, Ministry of Health and Sanitation, Sierra Leone. 2016. Available online: https://web.archive.org/web/20160509014636/http:/health.gov.sl/?page_id=583 (accessed on 28 February 2020).
- Senga, M.; Koi, A.; Moses, L.; Wauquier, N.; Barboza, P.; Fernandez-Garcia, M.D.; Engedashet, E.; Kuti-George, F.; Mitiku, A.D.; Vandi, M.; et al. Contact tracing performance during the ebola virus disease outbreak in kenema district, Sierra Leone. Philos. Trans. R. Soc. B Biol. Sci.
**2017**, 372, 20160300. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Lokuge, K.; Caleo, G.; Greig, J.; Duncombe, J.; McWilliam, N.; Squire, J.; Lamin, M.; Veltus, E.; Wolz, A.; Kobinger, G.; et al. Successful Control of Ebola Virus Disease: Analysis of Service Based Data from Rural Sierra Leone. PLoS Negl. Trop. Dis.
**2016**, 10, 1–12. [Google Scholar] [CrossRef] [Green Version] - Diekmann, O. Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis, and Interpretation; Wiley series in mathematical and computational biology; John Wiley: Chichester, UK, 2000. [Google Scholar]
- Diekmann, O.; Heesterbeek, H.; Britton, T. Mathematical Tools for Understanding Infectious Disease Dynamics; Princeton University Press: Princeton, NJ, USA, 2012; Volume 7. [Google Scholar]
- Van den Driessche, P.; Watmough, J. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci.
**2002**, 180, 29–48. [Google Scholar] [CrossRef] - Van den Driessche, P.; Watmough, J. Further notes on the basic reproduction number. Lect. Notes Math.
**2008**, 1945, 159–178. [Google Scholar] [CrossRef] [Green Version] - 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.
**2004**, 229, 119–126. [Google Scholar] [CrossRef] [Green Version] - Levy, B.; Edholm, C.; Gaoue, O.; Kaondera-Shava, R.; Kgosimore, M.; Lenhart, S.; Lephodisa, B.; Lungu, E.; Marijani, T.; Nyabadza, F. Modeling the role of public health education in Ebola virus disease outbreaks in Sudan. Infect. Dis. Model.
**2017**, 2, 323–340. [Google Scholar] [CrossRef] - Halloran, M.E.; Vespignani, A.; Bharti, N.; Feldstein, L.R.; Alexander, K.; Ferrari, M.; Shaman, J.; Drake, J.M.; Porco, T.; Eisenberg, J.N.; et al. Ebola: Mobility data. Science
**2014**, 346, 433. [Google Scholar] [CrossRef] [PubMed] [Green Version]

**Figure 1.**Flow diagram with Susceptible–Exposed–Infectious–Recovered ($SEIR$) standard disease compartments, $F,{E}_{F}$ compartments due to contact tracing, and $H,D$ for Hospitalized and Dead bodies as appropriate for Ebola. The coefficient f represents transitions due to contact tracing. The parameters and compartments are defined in Table 1.

**Figure 5.**Effect of varying the number of contact tracers available from 0 to 2000, with 1200 as the corresponding number in our model.

**Figure 6.**Effect of varying contact tracing parameters ${\kappa}_{1}$ and ${\kappa}_{2}$ on the total number of deaths by day 504 of the epidemic.

Symbol | Interpretation | Units |
---|---|---|

${\beta}_{1}$ | transmission from interactions between I and S | per person per time |

${\beta}_{2}$ | transmission from interactions between D and S | per person per time |

$1/\theta $ | number of days a person is traced | time |

$1/\alpha $ | length of the exposed period | time |

r | rate of hospitalization for traced individuals | per time |

$\gamma $ | rate of hospitalization for untraced individuals | per time |

${\varphi}_{1}$ | recovery rate for untreated | per time |

${\varphi}_{2}$ | recovery rate for treated | per time |

$\nu $ | death rate for untreated | per time |

$\mu $ | death rate for treated | per time |

$\omega $ | rate at which dead bodies become non-infectious | per time |

${\kappa}_{1}$ | contacts recruited from hospitalization of one person | unitless |

${\kappa}_{2}$ | contacts recruited from funeral of one person | unitless |

q | scaling factor for exposed contacts | unitless |

S | susceptibles | individuals |

F | susceptibles being traced | individuals |

E | exposed | individuals |

${E}_{F}$ | exposed being traced | individuals |

I | infectious | individuals |

H | hospitalized | individuals |

D | dead bodies | individuals |

R | recovered | individuals |

**Table 2.**Values for parameters, with five parameters having early and late values. Parameters with * were taken from the data or the literature. Others were estimated.

Parameter | Value | Parameter | Value |
---|---|---|---|

${\beta}_{1}$ early | $1.00\times {10}^{-9}$ | r | $0.056$ |

${\beta}_{1}$ late | $1.00\times {10}^{-9}$ | p | $0.90$ |

${\beta}_{2}$ early | $1.00\times {10}^{-6}$ | $\nu $ | $0.024$ |

${\beta}_{2}$ late | $1.00\times {10}^{-7}$ | $\mu $ | $0.010$ |

$\gamma $ early | $0.41$ | ${\varphi}_{1}$ | $0.020$ |

$\gamma $ late | $0.062$ | ${\varphi}_{2}$ | $0.028$ |

${\kappa}_{1}$ early | $29.74$ | $F\left(0\right)$ | $2451.10$ |

${\kappa}_{1}$ late | $44.93$ | $E\left(0\right)$ | $32.04$ |

${\kappa}_{2}$ early | $44.62$ | ${E}_{F}\left(0\right)$ | $124.88$ |

${\kappa}_{2}$ late | $16.61$ | $I\left(0\right)$ | $71.76$ |

$D\left(0\right)$ | $6.09$ | ||

${\alpha}^{\ast}$ | $0.1$ | $1/{\omega}^{\ast}$ | $4.5$ |

$H{\left(0\right)}^{\ast}$ | 94 | $S{\left(0\right)}^{\ast}$ | $6,348,350$ |

$R{\left(0\right)}^{\ast}$ | 0 | $1/{\theta}^{\ast}$ | 21 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Burton, D.; Lenhart, S.; Edholm, C.J.; Levy, B.; Washington, M.L.; Greening, B.R., Jr.; White, K.A.J.; Lungu, E.; Chimbola, O.; Kgosimore, M.;
et al. A Mathematical Model of Contact Tracing during the 2014–2016 West African Ebola Outbreak. *Mathematics* **2021**, *9*, 608.
https://doi.org/10.3390/math9060608

**AMA Style**

Burton D, Lenhart S, Edholm CJ, Levy B, Washington ML, Greening BR Jr., White KAJ, Lungu E, Chimbola O, Kgosimore M,
et al. A Mathematical Model of Contact Tracing during the 2014–2016 West African Ebola Outbreak. *Mathematics*. 2021; 9(6):608.
https://doi.org/10.3390/math9060608

**Chicago/Turabian Style**

Burton, Danielle, Suzanne Lenhart, Christina J. Edholm, Benjamin Levy, Michael L. Washington, Bradford R. Greening, Jr., K. A. Jane White, Edward Lungu, Obias Chimbola, Moatlhodi Kgosimore,
and et al. 2021. "A Mathematical Model of Contact Tracing during the 2014–2016 West African Ebola Outbreak" *Mathematics* 9, no. 6: 608.
https://doi.org/10.3390/math9060608