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

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## 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

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**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 |

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**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