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Ruin problems for epidemic insurance

Part of: Mathematical economics Markov processes

Published online by Cambridge University Press:  01 July 2021

Claude Lefèvre  and
Matthieu Simon Open the ORCID record for Matthieu Simon [Opens in a new window]
Claude Lefèvre*
Affiliation:
Université Libre de Bruxelles
Matthieu Simon*
Affiliation:
University of Melbourne
*
*Postal address: Département de Mathématique, Campus de la Plaine, CP 210, B-1050 Bruxelles, Belgium. Email: [email protected]
**Postal address: School of Mathematics and Statistics, Parkville, VIC 3010, Australia. Email: [email protected]
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Abstract

The paper discusses the risk of ruin in insurance coverage of an epidemic in a closed population. The model studied is an extended susceptible–infective–removed (SIR) epidemic model built by Lefèvre and Simon (Methodology Comput. Appl. Prob.22, 2020) as a block-structured Markov process. A fluid component is then introduced to describe the premium amounts received and the care costs reimbursed by the insurance. Our interest is in the risk of collapse of the corresponding reserves of the company. The use of matrix-analytic methods allows us to determine the distribution of ruin time, the probability of ruin, and the final amount of reserves. The case where the reserves are subjected to a Brownian noise is also studied. Finally, some of the results obtained are illustrated for two particular standard SIR epidemic models.

Keywords

SIR-type epidemicsinsurance reservesrisk of ruinblock-structured Markov processesMarkov-modulated fluid flowsmatrix-analytic methodsBrownian noise

MSC classification

Primary: 91B30: Risk theory, insurance 92D30: Epidemiology
Secondary: 60J28: Applications of continuous-time Markov processes on discrete state spaces
Type
Original Article
Information
Advances in Applied Probability , Volume 53 , Issue 2 , June 2021 , pp. 484 - 509
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Andersson, H. and Britton, T. (2000). Stochastic Epidemic Models and Their Statistical Analysis. Springer, New York.CrossRefGoogle Scholar
Asmussen, S. (1995). Stationary distributions for fluid flow models with or without Brownian noise. Commun. Statist. Stoch. Models 11, 21–49.CrossRefGoogle Scholar
Asmussen, S. and Albrecher, H. (2010). Ruin Probabilities. World Scientific, Singapore.CrossRefGoogle Scholar
Badescu, A. L. and Landriault, D. (2009). Applications of fluid flow matrix analytic methods in ruin theory—a review. Rev. Real Acad. Cienc. A 103, 353–372.Google Scholar
Bean, N., O’Reilly, M. and Taylor, P. G. (2005). Hitting probabilities and hitting times for stochastic fluid flows. Stoch. Process. Appl. 115, 1530–1556.CrossRefGoogle Scholar
Bini, D., Iannazzo, B., Latouche, G. and Meini, B. (2006). On the solution of algebraic Riccati equations arising in fluid queues. Linear Algebra Appl. 413, 474–494.CrossRefGoogle Scholar
Daley, D. J. and Gani, J. (1999). Epidemic Modelling: an Introduction. Cambridge University Press.Google Scholar
Diekmann, O. and Heesterbeek, H. (2000). Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation. Wiley, New York.Google Scholar
Diekmann, O., Heesterbeek, H. and Britton, T. (2013). Mathematical Tools for Understanding Infectious Disease Dynamics. Princeton University Press.Google Scholar
Feng, R. and Garrido, J. (2011). Actuarial applications of epidemiological models. N. Amer. Actuarial J. 15, 112–136.CrossRefGoogle Scholar
Gardiner, J. D., Laub, A. J., Amato, J. J. and Moler, C. B. (1992). Solution of the Sylvester matrix equation ${AXB}^{{T}}+{CXD}^{T}={E}$. ACM Trans. Math. Software 18, 232–238.CrossRefGoogle Scholar
Guo, C. (2006). Efficient methods for solving a nonsymmetric algebraic Riccati equation arising in stochastic fluid models. J. Comput. Appl. Math. 192, 353–373.CrossRefGoogle Scholar
He, Q.-M. (2014). Fundamentals of Matrix-Analytic Methods. Springer, New York.CrossRefGoogle Scholar
Kressner, D., Luce, R. and Statti, F. (2017). Incremental computation of block triangular matrix exponentials with application to option pricing. Electron. Trans. Numer. Anal. 47, 57–72.Google Scholar
Latouche, G. and Nguyen, G. (2015). The morphing of fluid queues into Markov-modulated Brownian motion. Stoch. Systems 5, 62–86.CrossRefGoogle Scholar
Latouche, G. and Nguyen, G. (2018). Analysis of fluid flow models. Queueing Models Service Manag. 1, 1–29.Google Scholar
Latouche, G. and Ramaswami, V. (1999). Introduction to Matrix Analytic Methods in Stochastic Modeling. Society for Industrial and Applied Mathematics, Philadelphia.CrossRefGoogle Scholar
Lefèvre, C., Picard, P. and Simon, M. (2017). Epidemic risk and insurance coverage. J. Appl. Prob. 54, 286–303.CrossRefGoogle Scholar
Lefèvre, C. and Simon, M. (2020). SIR-type epidemic models as block-structured Markov processes. Methodology Comput. Appl. Prob. 22, 433–453.CrossRefGoogle Scholar
Picard, P. and Lefèvre, C. (1993). Distribution of the final state and severity of epidemics with fatal risk. Stoch. Process. Appl. 48, 277–294.CrossRefGoogle Scholar
Ramaswami, V. (1999). Matrix-analytic methods for stochastic fluid flows. In Teletraffic Engineering in a Competitive World (Proc. 16th Internat. Teletraffic Congress), eds Smith, D. and Hey, P., Elsevier, Amsterdam, pp. 1019–1030.Google Scholar
Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.Google Scholar