# An Actuarial Approach for Modeling Pandemic Risk

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Overview of Previous Research

## 3. A Deterministic Epidemic Model

## 4. Actuarial Valuation of an Insurance Plan

**Proposition**

**1.**

**Proof.**

**Corollary**

**1.**

**Proposition**

**2.**

**Proof.**

## 5. Empirical Illustration

## 6. Time-Changed Extension

**Proposition**

**3.**

**Proof.**

## 7. Reinsurance in the Time-Changed Model

**Proposition**

**4.**

**Proof.**

## 8. Estimation and Illustration

## 9. A Jump Diffusion Model

**Proposition**

**5.**

**Proof.**

**Proposition**

**6.**

**Corollary**

**2.**

**Proposition**

**7.**

**Proof.**

## 10. Reinsurance in the Jump Diffusion Model

**Proposition**

**8.**

**Proof.**

## 11. Estimation of the Jump Diffusion Model

## 12. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Anderson, Roy M., and Robert M. May. 2008. Population biology of infectious diseases. Nature 280: 361–67. [Google Scholar] [CrossRef] [PubMed]
- Brauer, Fred. 2017. Mathematical epidemiology: Past, present, and future. Infectious Disease Modelling 2: 113–27. [Google Scholar] [CrossRef] [PubMed]
- Caraballo, Tomas, and Renato Colucci. 2017. A comparison between random and stochastic modeling for a SIR model. Communications on Pure and Applied Analysis 16: 151–62. [Google Scholar] [CrossRef] [Green Version]
- Caraballo, Tomas, and Sami Keraani. 2018. Analysis of a stochastic SIR model with fractional Brownian motion. Stochastic Analysis and Applications 36: 895–908. [Google Scholar] [CrossRef]
- Chen, Hua, and Samuel H. Cox. 2009. An option-based operational risk management model for pandemics. North American Actuarial Journal 13: 54–76. [Google Scholar] [CrossRef]
- Clara-Rahola, Joaquim. 2020. An empirical model for the spread and reduction of the COVID19 pandemic. Estudios de Economia Aplicada 38. [Google Scholar] [CrossRef]
- Daley, David J., and J. Gani. 1999. Epidemic Models: An Introduction. Cambridge Studies in Mathematical Biology 15. Cambridge: Cambridge University Press. [Google Scholar]
- Diekmann, O., J. A. P. Heesterbeek, and J. A. J. Metz. 1995. The legacy of Kermack and McKendrick. In Epidemic Models: Their Structure and Relation to Data. Edited by D. Mollison. Cambridge: Cambridge University Press, pp. 95–115. [Google Scholar]
- Diekmann, Odo, J. A. P. Heesterbeek, and Johan A. J. Metz. 1990. On the definition and the computation of the basic reproductive ratio R0 in models for infectious diseases in heterogeneous populations. Journal of Mathematical Biology 28: 365–82. [Google Scholar] [CrossRef] [Green Version]
- Feng, Runhuan, and Jose Garrido. 2011. Actuarial applications of epidemiological models. North American Actuarial Journal 15: 112–27. [Google Scholar] [CrossRef]
- Gathy, Maude, and Claude Lefèvre. 2009. From damage models to SIR epidemics and cascading failures. Advances in Applied Probability 41: 247–69. [Google Scholar] [CrossRef] [Green Version]
- Hainaut, Donatien, and Franck Moraux. 2019. A switching self-exciting jump diffusion process for stock prices. Annals of Finance Volume 15: 267–306. [Google Scholar] [CrossRef]
- Hethcote, Herbert W. 2000. The Mathematics of Infectious Diseases. SIAM Review 42: 599–653. [Google Scholar] [CrossRef] [Green Version]
- Jia, Na, and Lawrence Tsui. 2005. Epidemic Modelling Using SARS as a Case Study. North American Actuarial Journal 9: 28–42. [Google Scholar] [CrossRef]
- Kermack, William Ogilvy, and A. G. McKendrick. 1927. Contributions to the mathematical theory of epidemics—Part I. Proceedings of the Royal Society of London. Series A 115: 700–21. [Google Scholar]
- Lefèvre, Claude, and Sergey Utev. 1999. Branching Approximation for the Collective Epidemic Model. Methodology and Computing in Applied Probability Volume 1: 211–28. [Google Scholar] [CrossRef]
- Lefèvre, Claude, Picard Philippe, and Matthieu Simon. 2017. Epidemic risk and insurance coverage. Journal of Applied Probabilit 54: 286–303. [Google Scholar] [CrossRef] [Green Version]
- Rhodes, Tim, Kari Lancaster, Shelley Lees, and Melissa Parker. 2020. Modelling the pandemic: Attuning models to their contexts. BMJ Global Health 5: e002914. [Google Scholar] [CrossRef]
- Smith, Dominic. 2017. Pandemic Risk Modelling. In The Palgrave Handbook of Unconventional Risk Transfer. Edited by M. Pompella and N. A. Scordis. Cham: Palgrave Macmillan, pp. 463–495. [Google Scholar]
- Tchuenche, Jean M., Alexander Nwagwo, and Richard Levins. 2007. Global behaviour of an SIR epidemic model with time delay. Mathematical Methods in the Applied Sciences 30: 733–49. [Google Scholar] [CrossRef]
- Van den Driessche, P., and James Watmough. 2020. Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission. Mathematical Biosciences 180: 29–48. [Google Scholar] [CrossRef]
- Walters, Caroline E., Margaux M.I. Meslé, and Ian M. Hall. 2018. Modelling the global spread of diseases: A review of current practice and capability. Epidemics 25: 1–8. [Google Scholar] [CrossRef]
- Watson, H. W., and Francis Galton. 1874. On the probability of the extinction of families. The Journal of the Anthropological Institute of Great Britain and Ireland 4: 138–44. [Google Scholar] [CrossRef]
- Zhang, Xianghua, and Ke Wang. 2013. Stochastic SIR model with jumps. Applied Mathematics Letters 26: 867–74. [Google Scholar] [CrossRef]

1 | See, https://github.com/RamiKrispin/coronavirus, package developed by Rami Krispin. |

2 | In practice, the reinsurer would use a set of parameters more conservative than the estimated one in order to include a safety margin. |

**Figure 1.**Graphs of the number of COVID-19 reported cases and deaths from March to June, for Belgium, Germany, Italy and Spain.

**Figure 3.**Comparison of $\mathbb{E}\left({I}_{t}|{\mathcal{F}}_{0}\right)$ with the observed values and those computed with the deterministic model of Section 6.

**Figure 4.**Simulation of three sample paths for ${I}_{t}$ and comparison with the simulated average of 1000 sample paths for Italy.

**Figure 5.**Histograms of the cumulated number of deaths at time $t=0.1$ and $t=0.25$ for the Italian population. 2000 simulations.

**Figure 6.**Plot of ${Z}_{k}$ time series for Belgium, Germany, Italy and Spain. The threshold is calculated with $q=90\%$.

**Figure 7.**Simulations of 1000 sample paths of ${I}_{t}$, with parameter estimates in Table 9 and constant J. The thick dotted blue line is the expectation of ${I}_{t}$, whereas the thick black line is the observed number of infected cases.

Country | Starting Date | Number of Days | Number of Infected Individuals | Number of Deaths | Population Size, ${\mathit{S}}_{0}$ |
---|---|---|---|---|---|

Belgium | 7/3/2020 | 101 | 59,991 | 9661 | 11,589,623 |

Germany | 8/3/2020 | 100 | 186,883 | 8807 | 83,770,952 |

Italy | 27/2/2020 | 110 | 236,837 | 34,359 | 60,461,826 |

Spain | 7/3/2020 | 101 | 243,709 | 27,131 | 46,934,632 |

$\widehat{\mathit{\alpha}}$ | $\widehat{\mathit{\gamma}}$ | $\widehat{\mathit{\beta}}$ | $\widehat{\mathit{\mu}}$ | ${\mathit{t}}_{\mathit{max}}$(days) | ${\mathit{I}}_{\mathit{max}}$ | SSE | |
---|---|---|---|---|---|---|---|

Belgium | 40.718 | 4.74 | 6.606 | 4.457 | 38 | 17,829 | 535,988 |

Germany | 40.633 | 3.124 | 3.693 | 1.239 | 27 | 65,665 | 12,513,688 |

Italy | 30.878 | 3.382 | 3.709 | 3.931 | 35 | 65,103 | 3,093,842 |

Spain | 46.631 | 3.937 | 6.979 | 2.966 | 29 | 89,479 | 24,203,834 |

Adjusted ${\widehat{\mathit{S}}}_{0}$ | ${\widehat{\mathit{\alpha}}}_{\mathit{SIR}}$ | ${\widehat{\mathit{\beta}}}_{\mathit{SIR}}$ | $\widehat{\mathit{\mu}}$ | SSE | |
---|---|---|---|---|---|

Belgium | 38,203 | 12.407 | 107.76 | 4.457 | 16,109 |

Germany | 128,250 | 16.816 | 131.649 | 1.239 | 1,372,666 |

Italy | 118,250 | 10.616 | 113.425 | 3.931 | 1,175,335 |

Spain | 171,750 | 15.914 | 146.516 | 2.966 | 2,743,222 |

b | c | Fair p | $\mathit{\beta}+1\%$ | $\mathit{\beta}-1\%$ | $\mathit{\alpha}+1\%$ | $\mathit{\alpha}-1\%$ | $\mathit{\gamma}+1\%$ | $\mathit{\gamma}-1\%$ | |
---|---|---|---|---|---|---|---|---|---|

Be | 365,000 | 0 | 138.54 | 145.23 | 132.09 | 131.58 | 145.93 | 136.83 | 140.32 |

0 | 200,000 | 338.35 | 354.7 | 322.59 | 321.36 | 356.41 | 334.19 | 342.71 | |

Ge | 365,000 | 0 | 62.38 | 64.35 | 60.45 | 59.95 | 64.94 | 60.21 | 64.65 |

0 | 200,000 | 42.35 | 43.69 | 41.04 | 40.70 | 44.09 | 40.88 | 43.89 | |

It | 365,000 | 0 | 107.02 | 110.68 | 103.44 | 102.95 | 111.28 | 103.89 | 110.27 |

0 | 200,000 | 230.52 | 238.42 | 222.81 | 221.77 | 239.7 | 223.78 | 237.53 | |

Sp | 365,000 | 0 | 143.1 | 148.82 | 137.54 | 136.64 | 149.94 | 140.5 | 145.8 |

0 | 200,000 | 232.57 | 241.87 | 223.54 | 222.07 | 243.68 | 228.35 | 236.95 |

b | c | Fair p, SIR | ${\mathit{\beta}}_{\mathit{SIR}}+1\%$ | ${\mathit{\beta}}_{\mathit{SIR}}-1\%$ | ${\mathit{\alpha}}_{\mathit{SIR}}+1\%$ | ${\mathit{\alpha}}_{\mathit{SIR}}-1\%$ | |
---|---|---|---|---|---|---|---|

Be | 365,000 | 0 | 142.43 | 142.45 | 142.4 | 141.38 | 143.49 |

0 | 200,000 | 347.84 | 347.89 | 347.79 | 345.29 | 350.43 | |

Ge | 365,000 | 0 | 61.86 | 61.86 | 61.85 | 61.28 | 62.44 |

0 | 200,000 | 42.00 | 42.00 | 41.99 | 41.61 | 42.39 | |

It | 365,000 | 0 | 97.95 | 97.95 | 97.94 | 97.24 | 98.66 |

0 | 200,000 | 210.98 | 211.00 | 210.97 | 209.47 | 212.52 | |

Sp | 365,000 | 0 | 141.51 | 141.52 | 141.51 | 140.33 | 142.72 |

0 | 200,000 | 229.99 | 230 | 229.98 | 228.07 | 231.95 |

$\widehat{\mathit{\alpha}}$ | $\widehat{\mathit{\gamma}}$ | $\widehat{\mathit{\beta}}$ | $\widehat{\mathit{\mu}}$ | $\widehat{\mathit{\lambda}}$ | SSE | |
---|---|---|---|---|---|---|

Belgium | 44.806 | 105.021 | 1.113 | 0.149 | 27.776 | 1,005,094 |

Germany | 17.208 | 29.866 | 1.295 | 0.041 | 29.353 | 18,592,728 |

Italy | 12.407 | 25.088 | 1.09 | 0.149 | 25.307 | 5,906,960 |

Spain | 67.311 | 81.602 | 2.133 | 0.112 | 23.427 | 40,570,084 |

b | c | Fair p | Model 1, Fair p | $\mathit{\lambda}+1\%$ | $\mathit{\lambda}-1\%$ | |
---|---|---|---|---|---|---|

Be | 365,000 | 0 | 148.76 | 138.54 | 147.29 | 150.26 |

0 | 200,000 | 336.76 | 338.35 | 336.77 | 336.76 | |

Ge | 365,000 | 0 | 63.98 | 62.38 | 63.34 | 64.62 |

0 | 200,000 | 41.9 | 42.35 | 41.9 | 41.9 | |

It | 365 000 | 0 | 110.99 | 107.02 | 109.9 | 112.11 |

0 | 200 000 | 228.32 | 230.52 | 228.32 | 228.31 | |

Sp | 365,000 | 0 | 161.6 | 143.1 | 160 | 163.23 |

0 | 200,000 | 227.73 | 232.57 | 227.73 | 227.73 |

**Table 8.**Expectations, standard deviations, 5% and 95% percentiles of simulated ${D}_{t}$. 2000 simulations.

$\mathbb{E}\left({\mathit{D}}_{0.25}\right)$ | $\sqrt{\mathbb{V}\left({\mathit{D}}_{0.25}\right)}$ | $\mathbb{E}\left({\mathit{D}}_{0.10}\right)$ | $\sqrt{\mathbb{V}\left({\mathit{D}}_{0.10}\right)}$ | ${\mathit{D}}_{0.10}$: 5% | ${\mathit{D}}_{0.10}$: 95% | |
---|---|---|---|---|---|---|

Percentile | Percentile | |||||

Belgium | 9574.027 | 1005.182 | 5138.069 | 4504.648 | 0 | 9718.298 |

Germany | 8708.296 | 432.437 | 6288.87 | 3403.712 | 0 | 8741.932 |

Italy | 33,735.473 | 3497.944 | 19071.41 | 14,798.839 | 0 | 34,382.855 |

Spain | 26,533.64 | 1116.695 | 19,959.171 | 10,804.449 | 0 | 26,602.104 |

$\widehat{\mathit{\alpha}}$ | $\widehat{\mathit{\gamma}}$ | $\widehat{\mathit{\beta}}$ | $\widehat{\mathit{\mu}}$ | $\widehat{\mathit{\sigma}}$ | $\widehat{\mathit{\lambda}}$ | $\widehat{{\mathit{\mu}}_{\mathit{J}}}$ | $\widehat{{\mathit{\sigma}}_{\mathit{J}}}$ | SSE | |
---|---|---|---|---|---|---|---|---|---|

Belgium | 41.944 | 4.740 | 6.606 | 4.457 | 0.589 | 20.278 | 0.06 | 0.007 | 535,988 |

Germany | 43.758 | 3.125 | 3.693 | 1.239 | 0.727 | 30.845 | 0.101 | 0.022 | 12,513,688 |

Italy | 31.88 | 3.382 | 3.709 | 3.931 | 0.375 | 13.519 | 0.074 | 0.024 | 3,093,842 |

Spain | 51.465 | 3.937 | 6.979 | 2.966 | 1.512 | 25.347 | 0.191 | 0.026 | 24,203,834 |

Maximum | 90% Percentile | 95% Percentile | Ratio 95% | |
---|---|---|---|---|

Number of | Max. Number | Max. Number | Percentile on | |

Infected | of Infected | of Infected | Max. Cases | |

Belgium | 18,225 | 23,702 | 25,260 | 138.6% |

Germany | 71,219 | 93,506 | 100,764 | 141.49% |

Italy | 70,233 | 79,827 | 85,141 | 121.23% |

Spain | 97,400 | 164707 | 197,224 | 202.49% |

**Table 11.**Expectation, standard deviation and 5%–95% percentiles of simulated cumulated number of deaths, ${D}_{t}$. 1000 simulations.

${\mathit{D}}_{0.25}$ | Expected | Standard | 5% | 95% |
---|---|---|---|---|

Deviation | Percentile | Percentile | ||

Belgium | 9379.404 | 1850.385 | 6664.908 | 12,717.609 |

Germany | 8576.788 | 2079.589 | 5671.554 | 12,163.033 |

Italy | 33,166.37 | 4704.36 | 26,483.31 | 41,795.193 |

Spain | 24,948.895 | 12,595.321 | 10,641.107 | 46,904.273 |

K | ${\mathit{e}}^{-\mathit{r}\phantom{\rule{0.166667em}{0ex}}0.10}\mathbb{E}\left({\left({\mathit{I}}_{0.10}-\mathit{K}\right)}_{+}\right)$ | K | ${\mathit{e}}^{-\mathit{r}\phantom{\rule{0.166667em}{0ex}}0.25}\mathbb{E}\left({\left({\mathit{D}}_{0.25}-\mathit{K}\right)}_{+}\right)$ | |
---|---|---|---|---|

ine Belgium | 14,998 | 3154.536 | 9661 | 610.114 |

Germany | 46,721 | 11,411.364 | 8807 | 705.235 |

Italy | 59,209 | 7088.726 | 34,359 | 1362.358 |

Spain | 60,927 | 25,778.164 | 27,131 | 3739.249 |

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**MDPI and ACS Style**

Hainaut, D.
An Actuarial Approach for Modeling Pandemic Risk. *Risks* **2021**, *9*, 3.
https://doi.org/10.3390/risks9010003

**AMA Style**

Hainaut D.
An Actuarial Approach for Modeling Pandemic Risk. *Risks*. 2021; 9(1):3.
https://doi.org/10.3390/risks9010003

**Chicago/Turabian Style**

Hainaut, Donatien.
2021. "An Actuarial Approach for Modeling Pandemic Risk" *Risks* 9, no. 1: 3.
https://doi.org/10.3390/risks9010003