A Phase-Type Distribution for the Sum of Two Concatenated Markov Processes Application to the Analysis Survival in Bladder Cancer

[EN] Stochastic processes are useful and important for modeling the evolution of processes that take different states over time, a situation frequently found in fields such as medical research and engineering. In a previous paper and within this framework, we developed the sum of two independent phase-type (PH)-distributed variables, each of them being associated with a Markovian process of one absorbing state. In that analysis, we computed the distribution function, and its associated survival function, of the sum of both variables, also PH-distributed. In this work, in one more step, we have developed a first approximation of that distribution function in order to avoid the calculation of an inverse matrix for the possibility of a bad conditioning of the matrix, involved in the expression of the distribution function in the previous paper. Next, in a second step, we improve this result, giving a second, more accurate approximation. Two numerical applications, one with simulated data and the other one with bladder cancer data, are used to illustrate the two proposed approaches to the distribution function. We compare and argue the accuracy and precision of each one of them by means of their error bound and the application to real data of bladder cancer.This paper has been supported by the Generalitat Valenciana grant AICO/2020/114.García Mora, MB.; Santamaria Navarro, C.; Rubio Navarro, G. (2020). A Phase-Type Distribution for the Sum of Two Concatenated Markov Processes Application to the Analysis Survival in Bladder Cancer. Mathematics. 8(12):1-15. https://doi.org/10.3390/math8122099S115812Rodríguez, J., Lillo, R. E., & Ramírez-Cobo, P. (2015). Failure modeling of an electrical N-component framework by the non-stationary Markovian arrival process. Reliability Engineering & System Safety, 134, 126-133. doi:10.1016/j.ress.2014.10.020García‐Mora, B., Santamaría, C., & Rubio, G. (2020). Markovian modeling for dependent interrecurrence times in bladder cancer. Mathematical Methods in the Applied Sciences, 43(14), 8302-8310. doi:10.1002/mma.6593Montoro-Cazorla, D., & Pérez-Ocón, R. (2014). Matrix stochastic analysis of the maintainability of a machine under shocks. Reliability Engineering & System Safety, 121, 11-17. doi:10.1016/j.ress.2013.07.002Fackrell, M. (2008). Modelling healthcare systems with phase-type distributions. Health Care Management Science, 12(1), 11-26. doi:10.1007/s10729-008-9070-yGarg, L., McClean, S., Meenan, B. J., & Millard, P. (2011). Phase-Type Survival Trees and Mixed Distribution Survival Trees for Clustering Patients’ Hospital Length of Stay. Informatica, 22(1), 57-72. doi:10.15388/informatica.2011.314Marshall, A. H., & McClean, S. I. (2003). Conditional phase-type distributions for modelling patient length of stay in hospital. International Transactions in Operational Research, 10(6), 565-576. doi:10.1111/1475-3995.00428Marshall, A. H., & McClean, S. I. (2004). Using Coxian Phase-Type Distributions to Identify Patient Characteristics for Duration of Stay in Hospital. Health Care Management Science, 7(4), 285-289. doi:10.1007/s10729-004-7537-zFackrell, M. (2012). A semi-infinite programming approach to identifying matrix-exponential distributions. International Journal of Systems Science, 43(9), 1623-1631. doi:10.1080/00207721.2010.549582García-Mora, B., Santamaría, C., Rubio, G., & Pontones, J. L. (2013). Computing survival functions of the sum of two independent Markov processes: an application to bladder carcinoma treatment. International Journal of Computer Mathematics, 91(2), 209-220. doi:10.1080/00207160.2013.765560Kenney, C., & Laub, A. J. (1989). Condition Estimates for Matrix Functions. SIAM Journal on Matrix Analysis and Applications, 10(2), 191-209. doi:10.1137/0610014Jackson, C. H. (2011). Multi-State Models for Panel Data: ThemsmPackage forR. Journal of Statistical Software, 38(8). doi:10.18637/jss.v038.i08Mullin, L., & Raynolds, J. (2014). Scalable, Portable, Verifiable Kronecker Products on Multi-scale Computers. Studies in Computational Intelligence, 111-129. doi:10.1007/978-3-319-04280-0_1