Deterministic solvency thresholds and RL-based premium calibration for life insurance under age-structured epidemics

i

This article has been updated. A Correction to this article was published on .

i

The Original Article was published on

i

The Original Article was published on

i

This article has been updated

i

This article was retracted on

i

This article has been updated

Abstract

We develop a deterministic framework that links epidemic propagation, mortality risk, and life-insurance solvency by modeling the population through an age-structured SEIRD system in which disease-induced deaths drive an insurer’s surplus process governed by an ordinary differential equation with a continuous premium inflow and death-benefit outflow. We characterize the disease-free equilibrium and show that its local stability depends on the basic reproduction number R₀, and for any finite horizon [0,T], we derive an explicit solvency threshold in the form of a critical premiump_crit(T) that guarantees a nonnegative surplus and depends on the age-weighted infection burden. For large horizons, we obtain an analytically tractable approximation of this threshold using multi-group final-size relations and show that it monotonically increases with epidemic severity. To illustrate practical implications, we formulate an age-specific premium selection as a one-shot continuous-action reinforcement-learning problem in which a Proximal Policy Optimization (PPO) agent is trained on simulated age-structured epidemic scenarios to choose premiums that avoid ruin while remaining near actuarially fair levels; out-of-sample tests confirm that the learned premiums satisfy the analytical solvency constraints, deliver low ruin probabilities, and achieve tight fairness calibration. The combined analytical and numerical results provide a transparent basis for epidemic-sensitive premium design and stress testing of life-insurance portfolios.

Keywords

epidemic risk modelling / life insurance solvency / age-structured SEIRD model / reinforcement learning / premium calibration / numerical stress testing / insurance mathematics

CHANGE HISTORY

A Correction to this paper has been published:
https://doi.org

The Retraction Note was updated to reflect reception of statements of disagreement from all authors.

This article has been retracted.Please see the Retraction Notice for more detail:
https://doi.org

References

[1] D. I. J. Samaranayake, D. L. M. Samaranayake, Insurance contract for epidemiological diseases spread, Sri Lanka J. Econ. Res., 5 (2018), 103-124. http://doi.org/10.4038/sljer.v5i2.51

[2] C. Lefévre, P. Picard, M. Simon, Epidemic risk and insurance coverage, J. Appl. Prob., 54 (2017), 286-303. http://doi.org/10.1017/jpr.2016.100

[3] S. Kadyrov, G. Kayumova, A. Yallaboyev, B. Sagidolla, A compartmental epidemic model with age stratification for insurance premium calculation, Math. Biosci. Eng., 22 (2025), 3088-3106. http://doi.org/10.3934/mbe.2025114

[4] D. Hainaut, An actuarial approach for modeling pandemic risk, Risks, 9 (2021), 3. http://doi.org/10.3390/risks9010003

[5] C. Zhai, P. Chen, Z. Jin, T. K. Siu, Epidemic modelling and actuarial applications for pan- demic insurance: a case study of Victoria, Australia, Ann. Actuarial Sci., 18 (2024), 242-269. http://doi.org/10.1017/S1748499523000246

[6] A. Zinihi, M. Ehrhardt, M. R. S. Ammi, Actuarial analysis of an infectious dis- ease insurance based on an SEIARD epidemiological model, preprint, arXiv:2508.06580. https://doi.org/10.48550/arXiv.2508.06580

[7] J. Hoseana, F. Kusnadi, G. Stephanie, L. Loanardo, C. Wijaya, Design and financial analy- sis of a health insurance based on an SIH-type epidemic model, preprint, arXiv:2408.05397. https://doi.org/10.48550/arXiv.2408.05397

[8] M. S. Jameaba, Digitalization, emerging technologies, and financial stability: Chal- lenges and opportunities for the Indonesian banking industry and beyond, Qeios, 2022. https://doi.org/10.32388/CSTTYQ.2

[9] J. Achiam, D. Held, A. Tamar, P. Abbeel, Constrained policy optimization, in Proceedings of the 34th International Conference on Machine Learning, PMLR, 70 (2017), 22-31.

[10] L. Mercuri, A. Perchiazzo, E. Rroji, A Hawkes model with CARMA (p,q) intensity, Insur. Math. Econ., 116 (2024), 1-26. https://doi.org/10.1016/j.insmatheco.2024.01.007

[11] A. Swishchuk, R. Zagst, G. Zeller, Hawkes processes in insurance: risk model, applica- tion to empirical data and optimal investment, Insur. Math. Econ., 101 (2021), 107-124. https://doi.org/10.1016/j.insmatheco.2020.12.005

[12] F. E. Benth, C. Klüppelberg, G. Müller, L. Vos, Futures pricing in electricity mar- kets based on stable CARMA spot models, Energy Econ., 44 (2014), 392-406. https://doi.org/10.1016/j.eneco.2014.03.020

[13] European Parliament and Council , Directive 2009/138/EC of 25 November 2009 on the taking-up and pursuit of the business of Insurance and Reinsurance (Solvency II) , Official Journal of the European Union, 2009. Available at: https://eur-lex.europa.eu/eli/dir/2009/138/oj.

[14] C. Baumgart, J. Krebs, R. Lempertseder, O. Pfaffel, Quantifying life insurance risk using least- squares Monte Carlo, preprint, arXiv:1910.03951. https://doi.org/10.48550/arXiv.1910.03951

[15] L. Dede, N. Parolini, A. Quarteroni, G. Villani, G. Ziarelli, SEIHRDV: a multi-age multi- group epidemiological model and its validation on the COVID-19 epidemics in Italy, preprint, arXiv:2501.04148. https://doi.org/10.48550/arXiv.2501.04148

[16] I. Rodiah, P. Vanella, A. Kuhlmann, V. K. Jaeger, M. Harries, G. Krause, et al., Age-specific contribution of contacts to transmission of SARS-CoV-2 in Japan, Eur. J. Epidemiol., 38 (2023), 39-58. https://doi.org/10.1007/s10654-022-00938-6

[17] Z. Vizi, E. K. Korir, N. Bogya, C. Rosztoczi, G. Makay, P. Boldog, Age group sensitivity analysis of epidemic models: investigating the impact of contact matrix structure, preprint, arXiv:2502.19206. https://doi.org/10.48550/arXiv.2502.19206

[18] M. De la Sen, A. Ibeas, S. Alonso-Quesada, R. Nistal, On a new epidemic model with asymptomatic and dead-infective subpopulations with feedback controls useful for Ebola disease, Discrete Dyn. Nat. Soc., 2017 (2017), 4232971. https://doi.org/10.1155/2017/4232971

[19] G. Stamov, I. Stamova, N. Simeonova, K. Gabrovska, S. Simeonov, Stability of sets for Ebola virus disease models through impulsive conformable approach, Mathematics, 14 (2026), 1108. https://doi.org/10.3390/math14071108

[20] N. Wang, L. Zhang, Y. Luo, Z. Teng, Global stability for a cumulative release Ebola epidemic model from the corpses and infected individuals, Infect. Dis. Modell., 10 (2025), 1291-1306. https://doi.org/10.1016/j.idm.2025.07.002

[21] T. Rolski, H. Schmidli, V. Schmidt, J. L. Teugels, Stochastic Processes for Insurance and Finance , John Wiley & Sons, 2009. https://doi.org/10.1002/9780470317044

[22] L. Palmborg, F. Lindskog, Premium control with reinforcement learning, ASTIN Bull., 53 (2023), 233-257. https://doi.org/10.1017/asb.2023.13

[23] S. Singireddy, Reinforcement learning approaches for pricing condo insurance policies, Am. J. Anal. Artif. Intell., 2023.

[24] C. Xuan, F. Zhang, F. Yin, H. K. Lam, Constrained proximal policy optimization, preprint, arXiv:2305.14216. https://doi.org/10.48550/arXiv.2305.14216

[25] J. Schulman, F. Wolski, P. Dhariwal, A. Radford, O. Klimov, Proximal policy optimization algo- rithms, preprint, arXiv:1707.06347. https://doi.org/10.48550/arXiv.1707.06347