# Transition Intensities for Critical Illness: A Study on Canadian Health Data

## Authors

• Beng Eik Neoh School of Mathematical Sciences, Sunway University, 47500 Petaling Jaya, Selangor, Malaysia
• Kai An Sim School of Mathematical Sciences, Sunway University, 47500 Petaling Jaya, Selangor, Malaysia
• Lay Guat Chan School of Mathematical Sciences, Sunway University, 47500 Petaling Jaya, Selangor, Malaysia
• Chee Kit Ho School of Mathematical Sciences, Sunway University, 47500 Petaling Jaya, Selangor, Malaysia

## Keywords:

Multiple state models, Transition intensities, Gompertz-Makeham, Prevalence rates, Critical illness insurance

## Abstract

Multiple state model is a mathematical model which is characterised by two important elements, transition intensity and transition probability. Critical illness has increased rapidly which is alarmed by the healthcare experts, and becoming an important concern in society. In this paper, by using the Canadian health data, we provide an estimation of transition intensities from the healthy state to the critical illness state with the application of prevalence rate. We provide a discrete calculation of transition intensities with some mathematical formula discussed by some previous studies. Next, we assume that the transition intensities of critical illnesses and death due to other causes are modelled by Gompertz and Makeham mortality models. We also compare and estimate the transition intensities of critical illnesses and dead due to other causes between these two models using a model selection method. We observe the sensitivity of the Gompertz and Makeham models with the different values of extra mortality . Lastly, we obtain and present the numerical results of the transition intensities of healthy lives to critical illness with the Canadian health data.

## References

Baione, F., & Levantesi, S. (2014). A health insurance pricing model based on prevalence rates: Application to critical illness insurance. Insurance: Mathematics and Economics, 58, 174-184.

Baione, F., & Levantesi, S. (2018). Pricing critical illness insurance from prevalence rates: Gompertz versus Weibull. North American Actuarial Journal, 22(2), 270-288.

Brink, A. (2010). Practical example of split benefit accelerated critical illness insurance product. In Transitions of the 29th International Congress of Actuaries, 677-703.

Castellares, F., Patrício, S., & Lemonte, A. J. (2022). On the gompertz–makeham law: A useful mortality model to deal with human mortality. Brazilian Journal of Probability and Statistics, 36(3), 613-639.

Canadian Chronic Disease Surveillance System (CCDSS). (n.d.). Public Health Infobase.

Christiansen, M. C. (2012). Multistate models in health insurance. AStA Advances in Statistical Analysis, 96(2), 155-186.

Dash, A. C., & Grimshaw, D. L. (1993). Dread Disease Cover–An Actuarial Perspective. Journal of the Staple Inn Actuarial Society, 33(1), 149-193.

Dickson, D. C., Hardy, M. R., & Waters, H. R. (2019). Actuarial mathematics for life contingent risks. Cambridge University Press.

Forfar, D. O., McCutcheon, J. J., & Wilkie, A. D. (1988). On graduation by mathematical formula. Journal of the Institute of Actuaries, 115(1), 1-149.

Forfar, D. O. (2006). Mortality laws. Encyclopedia of Actuarial Science, 2.

Golubev, A. (2009). How could the gompertz–makeham law evolve. Journal of Theoretical Biology, 258(1), 1-17.

Golubev, A. (2019). A 2d analysis of correlations between the parameters of the gompertz–makeham model (or law?) of relationships between aging, mortality, and longevity. Biogerontology, 20(6), 799-821.

Gompertz, B. (1825). XXIV. On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies. In a letter to Francis Baily, Esq. FRS &c. Philosophical Transactions of the Royal Society of London, (115), 513-583.

Haberman, S. (1984). Decrement tables and the measurement of morbidity: II. Journal of the Institute of Actuaries, 111(1), 73-86.

Haberman, S., & Pitacco, E. (2018). Actuarial models for disability insurance. Routledge.

Jones, B. L. (1994). Actuarial calculations using a Markov model. Transactions of the Society of Actuaries, 46, 227-250.

Li, J. (2019). A multi-state model for pricing critical illness insurance products.

Makeham, W. M. (1867). On the Law of Mortality. Journal of the Institute of Actuaries, 13(6), 325-358.

Melnikov, A., & Romaniuk, Y. (2006). Evaluating the performance of Gompertz, Makeham and Lee-Carter mortality models for risk management with unit-linked contracts. Insurance: Mathematics and Economics, 39(3), 310-329.

Michael, O. G. (2021). On empirical analysis of gompertzian mortality dynamics. Journal of Science, 2, 1-18.

Olivieri, A. (1996). Sulle basi tecniche per coperture “Long Term Care”. Giornale dell’Istituto Italiano degli Attuari, 49, 87-116.

Pasaribu, U. S., Husniah, H., Sari, R. K., & Yanti, A. R. (2019). Pricing Critical Illness Insurance Premiums Using Multiple State Continous Markov Chain Model. In Journal of Physics: Conference Series, 1366(1), 012112. IOP Publishing.

Pitacco, E., & Tabakova, D. (2021). Actuarial mathematics. time-continuous models for life insurance. Time-Continuous Models for Life Insurance (October 9, 2021).

Salhi, Y., Thérond, P. E., & Tomas, J. (2016). A credibility approach of the Makeham mortality law. European Actuarial Journal, 6(1), 61-96.

Shklovskii, B. (2005). A simple derivation of the gompertz law for human mortality. Theory in Biosciences, 123(4), 431-433.

Table 13-10-0751-01 Number of prevalent cases and prevalence proportions of primary cancer, by prevalence duration, cancer type, attained age group and sex. (2019). Statistics Canada.

Table 17–10-0005-01 Population estimates on July 1st, by age and sex. (2021). Statistics Canada.

Table 13–10-0392-01 Deaths and age-specific mortality rates, by selected grouped causes. (2022). Statistics Canada.

Waters, H. R. (1984). An approach to the study of multiple state models. Journal of the Institute of Actuaries, 111(2), 363-374.