# Controlling COVID-19 Outbreaks with Financial Incentives

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model and Numerical Solution Algorithm

## 3. Computational Experiments

#### 3.1. Estimation without Financial Incentives

#### 3.2. Estimation with Financial Incentives

**Step****1.**- Before the implementation of the incentive policy, we first estimate $\beta ,\phantom{\rule{3.33333pt}{0ex}}\gamma ,\phantom{\rule{3.33333pt}{0ex}}{S}^{n+1},$ and ${U}^{n+1}$ using the SUC model with the number of confirmed cases up to now. Here, $p=7$ is fixed.
**Step****2.**- Assuming that financial incentives are provided, we compute $S\left(t\right),\phantom{\rule{3.33333pt}{0ex}}U\left(t\right),\phantom{\rule{3.33333pt}{0ex}}C\left(t\right)$, and T using the SUC model and the iterative method until $U\left(t\right)<tol$. Here, we use the same parameters as those used and obtained in
**Step 1**except for $\gamma $. Instead of $\gamma $, we use ${\gamma}_{I}$ that is greater than 0.

**Step 1**, we obtain $\beta \approx 0.0891$ and $\gamma \approx 0.1665$ at $t=6$. We reset $t=6$ to $t=0$. When ${\gamma}_{I}$ is taken to be the same as $\gamma $, the epidemic may be ended at $t=65$. Assuming that ${\gamma}_{I}=1.1\gamma \approx 0.1832$ due to an incentive policy, we estimate that the epidemic may end at $t=54$. Figure 4 shows the numerical results. The increase in ${\gamma}_{I}$ shortens the average time it takes to identify unconfirmed infected people, which can rapidly isolate the unidentified infected cases. Thus, it prevents the spread of the infectious disease so that it can be seen that the time T is reduced.

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Data

**Table A1.**Confirmed cases of COVID-19 domestic outbreak in South Korea from 25 March to 17 July 2020 [4].

No | Date | Cases | No | Date | Cases | No | Date | Cases | No | Date | Cases |
---|---|---|---|---|---|---|---|---|---|---|---|

1 | 25-March | 8910 | 30 | 23-April | 9677 | 59 | 22-May | 9941 | 88 | 20-June | 10,945 |

2 | 26-March | 8957 | 31 | 24-April | 9681 | 60 | 23-May | 9960 | 89 | 21-June | 10,985 |

3 | 27-March | 9023 | 32 | 25-April | 9687 | 61 | 24-May | 9977 | 90 | 22-June | 10,996 |

4 | 28-March | 9115 | 33 | 26-April | 9688 | 62 | 25-May | 9990 | 91 | 23-June | 11,012 |

5 | 29-March | 9171 | 34 | 27-April | 9691 | 63 | 26-May | 10,006 | 92 | 24-June | 11,043 |

6 | 30-March | 9185 | 35 | 28-April | 9693 | 64 | 27-May | 10,043 | 93 | 25-June | 11,066 |

7 | 31-March | 9268 | 36 | 29-April | 9697 | 65 | 28-May | 10,111 | 94 | 26-June | 11,093 |

8 | 01-April | 9327 | 37 | 30-April | 9697 | 66 | 29-May | 10,166 | 95 | 27-June | 11,124 |

9 | 02-April | 9375 | 38 | 01-May | 9698 | 67 | 30-May | 10,193 | 96 | 28-June | 11,164 |

10 | 03-April | 9415 | 39 | 02-May | 9698 | 68 | 31-May | 10,208 | 97 | 29-June | 11,194 |

11 | 04-April | 9392 | 40 | 03-May | 9701 | 69 | 01-June | 10,238 | 98 | 30-June | 11,217 |

12 | 05-April | 9433 | 41 | 04-May | 9701 | 70 | 02-June | 10,274 | 99 | 01-July | 11,252 |

13 | 06-April | 9464 | 42 | 05-May | 9701 | 71 | 03-June | 10,320 | 100 | 02-July | 11,296 |

14 | 07-April | 9494 | 43 | 06-May | 9701 | 72 | 04-June | 10,353 | 101 | 03-July | 11,348 |

15 | 08-April | 9523 | 44 | 07-May | 9702 | 73 | 05-June | 10,387 | 102 | 04-July | 11,384 |

16 | 09-April | 9539 | 45 | 08-May | 9703 | 74 | 06-June | 10,430 | 103 | 05-July | 11,427 |

17 | 10-April | 9561 | 46 | 09-May | 9720 | 75 | 07-June | 10,483 | 104 | 06-July | 11,449 |

18 | 11-April | 9579 | 47 | 10-May | 9746 | 76 | 08-June | 10,516 | 105 | 07-July | 11,469 |

19 | 12-April | 9587 | 48 | 11-May | 9775 | 77 | 09-June | 10,551 | 106 | 08-July | 11,499 |

20 | 13-April | 9596 | 49 | 12-May | 9797 | 78 | 10-June | 10,594 | 107 | 09-July | 11,526 |

21 | 14-April | 9611 | 50 | 13-May | 9819 | 79 | 11-June | 10,634 | 108 | 10-July | 11,548 |

22 | 15-April | 9627 | 51 | 14-May | 9845 | 80 | 12-June | 10,677 | 109 | 11-July | 11,568 |

23 | 16-April | 9638 | 52 | 15-May | 9867 | 81 | 13-June | 10,720 | 110 | 12-July | 11,589 |

24 | 17-April | 9646 | 53 | 16-May | 9876 | 82 | 14-June | 10,751 | 111 | 13-July | 11,608 |

25 | 18-April | 9655 | 54 | 17-May | 9882 | 83 | 15-June | 10,774 | 112 | 14-July | 11,622 |

26 | 19-April | 9658 | 55 | 18-May | 9887 | 84 | 16-June | 10,795 | 113 | 15-July | 11,633 |

27 | 20-April | 9664 | 56 | 19-May | 9896 | 85 | 17-June | 10,826 | 114 | 16-July | 11,647 |

28 | 21-April | 9668 | 57 | 20-May | 9920 | 86 | 18-June | 10,877 | 115 | 17-July | 11,668 |

29 | 22-April | 9673 | 58 | 21-May | 9930 | 87 | 19-June | 10,909 |

## Appendix B. MATLAB Code

## References

- Lai, C.C.; Shih, T.P.; Ko, W.C.; Tang, H.J.; Hsueh, P.R. Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) and corona virus disease-2019 (COVID-19): The epidemic and the challenges. Int. J. Antimicrob. Agents
**2020**, 55, 105924. [Google Scholar] [CrossRef] [PubMed] - Younes, A.B.; Hasan, Z. COVID-19: Modeling, prediction, and control. Appl. Sci.
**2020**, 10, 3666. [Google Scholar] [CrossRef] - World Health Organization (WHO). Coronavirus Disease 2019 (COVID-19) Situation Report 179, accesed 17 July 2020. Available online: https://www.who.int/docs/default-source/coronaviruse/situation-reports/20200717-covid-19-sitrep-179.pdf?sfvrsn=2f1599fa_28 (accessed on 6 January 2021).
- Novel Coronavirus (COVID-19) Situation Reports Published by the Korea Disease Control and Prevention Agency (KDCA). Available online: http://www.kdca.go.kr/board/board.es?mid=a20501000000&bid=0015 (accessed on 6 January 2021).
- Khalifa, S.A.; Mohamed, B.S.; Elashal, M.H.; Du, M.; Guo, Z.; Zhao, C.; Musharraf, S.G.; Boskabady, M.H.; El-Seedi, H.H.R.; Efferth, T.; et al. Comprehensive overview on multiple strategies fighting COVID-19. Int. J. Environ. Res. Public Health
**2020**, 17, 5813. [Google Scholar] [CrossRef] [PubMed] - Ha, B.T.T.; La Quang, N.; Mirzoev, T.; Tai, N.T.; Thai, P.Q.; Dinh, P.C. Combating the COVID-19 Epidemic: Experiences from Vietnam. Int. J. Environ. Res. Public Health
**2020**, 17, 3125. [Google Scholar] [CrossRef] [PubMed] - Pung, R.; Chiew, C.J.; Young, B.E.; Chin, S.; Chen, M.I.; Clapham, H.E.; Cook, A.R.; Maurer-Stroh, S.; Toh, M.P.H.S.; Poh, C.; et al. Investigation of three clusters of COVID-19 in Singapore: Implications for surveillance and response measures. Lancet
**2020**, 395, 1039–1046. [Google Scholar] [CrossRef] - Chu, D.K.; Akl, E.A.; Duda, S.; Solo, K.; Yaacoub, S.; Schünemann, H.J. Physical distancing, face masks, and eye protection to prevent person-to-person transmission of SARS-CoV-2 and COVID-19: A systematic review and meta-analysis. Lancet
**2020**, 395, 1973–1987. [Google Scholar] [CrossRef] - Guzzetta, G.; Poletti, P.; Ajelli, M.; Trentini, F.; Marziano, V.; Cereda, D.; Tirani, M.; Diurno, G.; Bodina, A.; Barone, A.; et al. Potential short-term outcome of an uncontrolled COVID-19 epidemic in Lombardy, Italy, February to March 2020. Eurosurveillance
**2020**, 25, 2000293. [Google Scholar] [CrossRef] - Xie, K.; Liang, B.; Dulebenets, M.A.; Mei, Y. The impact of risk perception on social distancing during the COVID-19 pandemic in China. Int. J. Environ. Res. Public Health
**2020**, 17, 6256. [Google Scholar] - Ghinai, I.; McPherson, T.D.; Hunter, J.C.; Kirking, H.L.; Christiansen, D.; Joshi, K.; Rubin, R.; Morales-Estrada, S.; Black, S.R.; Pacilli, M.; et al. First known person-to-person transmission of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) in the USA. Lancet
**2020**, 395, 1137–1144. [Google Scholar] [CrossRef] - Peeri, N.C.; Shrestha, N.; Rahman, M.S.; Zaki, R.; Tan, Z.; Bibi, S.; Baghbanzadeh, M.; Aghamohammadi, N.; Zhang, W.; Haque, U. The SARS, MERS and novel coronavirus (COVID-19) epidemics, the newest and biggest global health threats: What lessons have we learned? Int. J. Epidemiol.
**2020**, 49, 717–726. [Google Scholar] [CrossRef][Green Version] - Day, M. Covid-19: Identifying and isolating asymptomatic people helped eliminate virus in Italian village. BMJ
**2020**, 368. [Google Scholar] [CrossRef] [PubMed][Green Version] - Sjödin, H.; Wilder-Smith, A.; Osman, S.; Farooq, Z.; Rocklöv, J. Only strict quarantine measures can curb the coronavirus disease (COVID-19) outbreak in Italy, 2020. Eurosurveillance
**2020**, 25, 2000280. [Google Scholar] [CrossRef] [PubMed] - Han, D.; Shao, Q.; Li, D.; Sun, M. How the individuals’ risk aversion affect the epidemic spreading. Appl. Math. Comput.
**2020**, 369, 124894. [Google Scholar] [CrossRef] - Reuters News. Available online: https://www.reuters.com/article/us-china-health-rewards/china-city-offers-1400-reward-for-virus-patients-who-report-to-authorities-idUSKCN20L0GE (accessed on 6 January 2021).
- Chamie, G.; Ndyabakira, A.; Marson, K.G.; Emperador, D.M.; Kamya, M.R.; Havlir, D.V.; Kwarisiima, D.; Thirumurthy, H. A pilot randomized trial of incentive strategies to promote HIV retesting in rural Uganda. PLoS ONE
**2020**, 15, e0233600. [Google Scholar] [CrossRef] [PubMed] - Adamson, B.; El-Sadr, W.; Dimitrov, D.; Gamble, T.; Beauchamp, G.; Carlson, J.J.; Garrison, L.; Donnell, D. The cost-effectiveness of financial incentives for viral suppression: HPTN 065 study. Value Health
**2019**, 22, 194–202. [Google Scholar] [CrossRef] [PubMed][Green Version] - Rat-Aspert, O.; Fourichon, C. Modelling collective effectiveness of voluntary vaccination with and without incentives. Prev. Vet. Med.
**2010**, 93, 265–275. [Google Scholar] [CrossRef] - Lee, C.; Li, Y.; Kim, J. The susceptible-unidentified infected-confirmed (SUC) epidemic model for estimating unidentified infected population for COVID-19. Chaos Soliton. Fract.
**2020**, 139, 110090. [Google Scholar] [CrossRef] - Capasso, V.; Serio, G. A generalization of the Kermack–McKendrick deterministic epidemic model. Math. Biosci.
**1978**, 42, 43–61. [Google Scholar] [CrossRef] - Gray, A.; Greenhalgh, D.; Hu, L.; Mao, X.; Pan, J. A stochastic differential equation SIS epidemic model. SIAM J. Appl. Math.
**2011**, 71, 876–902. [Google Scholar] [CrossRef][Green Version] - Ma, Y.; Liu, J.B.; Li, H. Global dynamics of an SIQR model with vaccination and elimination hybrid strategies. Mathematics
**2018**, 6, 328. [Google Scholar] [CrossRef][Green Version] - Khan, M.A.; Khan, Y.; Islam, S. Complex dynamics of an SEIR epidemic model with saturated incidence rate and treatment. Physica A
**2018**, 493, 210–227. [Google Scholar] [CrossRef] - Wu, M.; Dai, W.; Lu, Z.; Zhao, Y.; Wang, M. The method for risk evaluation in assembly process based on the discrete-time SIRS epidemic model and information entropy. Entropy
**2019**, 21, 1029. [Google Scholar] [CrossRef][Green Version] - Nistal, R.; De la Sen, M.; Alonso-Quesada, S.; Ibeas, A. On a new discrete SEIADR model with mixed controls: Study of its properties. Mathematics
**2019**, 7, 18. [Google Scholar] [CrossRef][Green Version]

**Figure 3.**Estimation of the unidentified confirmed patients U from 17 July 2020 to the end of the epidemic with different p values.

**Figure 5.**Computational results of (

**a**) $U\left(t\right)$, (

**b**) T, (

**c**) $\Delta {C}_{i}$, and (

**d**) $\Delta C$ with respect to ${\gamma}_{I}$.

**Figure 7.**Plot of the fitting function for a power cost function, ${\gamma}_{I}\left(x\right)/{\gamma}_{\mathrm{ref}}=1+exp\left(A\right){x}^{b}$.

**Table 1.**Computational results for the best fitting confirmed (C) and the unidentified infected (U) of COVID-19 with respect to p. The confirmed case data were $11,668$ on 17 July 2020.

p | 7 | 14 | 21 |
---|---|---|---|

C | 11,664 | 11,664 | 11,675 |

U | 75 | 42 | 55 |

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**MDPI and ACS Style**

Lee, C.; Kwak, S.; Kim, J.
Controlling COVID-19 Outbreaks with Financial Incentives. *Int. J. Environ. Res. Public Health* **2021**, *18*, 724.
https://doi.org/10.3390/ijerph18020724

**AMA Style**

Lee C, Kwak S, Kim J.
Controlling COVID-19 Outbreaks with Financial Incentives. *International Journal of Environmental Research and Public Health*. 2021; 18(2):724.
https://doi.org/10.3390/ijerph18020724

**Chicago/Turabian Style**

Lee, Chaeyoung, Soobin Kwak, and Junseok Kim.
2021. "Controlling COVID-19 Outbreaks with Financial Incentives" *International Journal of Environmental Research and Public Health* 18, no. 2: 724.
https://doi.org/10.3390/ijerph18020724