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On First-Passage Times and Sojourn Times in Finite QBD Processes and Their Applications in Epidemics

Author

Listed:
  • Antonio Gómez-Corral

    (Department of Statistics and Operations Research, School of Mathematical Sciences, Plaza de Ciencias 3, Complutense University of Madrid, 28040 Madrid, Spain)

  • Martín López-García

    (Department of Applied Mathematics, School of Mathematics, University of Leeds, Leeds LS2 9JT, UK)

  • Maria Jesus Lopez-Herrero

    (Department of Statistics and Data Science, School of Statistical Studies, Avda. Puerta de Hierro s/n, Complutense University of Madrid, 28040 Madrid, Spain)

  • Diana Taipe

    (Department of Statistics and Operations Research, School of Mathematical Sciences, Plaza de Ciencias 3, Complutense University of Madrid, 28040 Madrid, Spain)

Abstract

In this paper, we revisit level-dependent quasi-birth-death processes with finitely many possible values of the level and phase variables by complementing the work of Gaver, Jacobs, and Latouche (Adv. Appl. Probab. 1984), where the emphasis is upon obtaining numerical methods for evaluating stationary probabilities and moments of first-passage times to higher and lower levels. We provide a matrix-analytic scheme for numerically computing hitting probabilities, the number of upcrossings, sojourn time analysis, and the random area under the level trajectory. Our algorithmic solution is inspired from Gaussian elimination, which is applicable in all our descriptors since the underlying rate matrices have a block-structured form. Using the results obtained, numerical examples are given in the context of varicella-zoster virus infections.

Suggested Citation

  • Antonio Gómez-Corral & Martín López-García & Maria Jesus Lopez-Herrero & Diana Taipe, 2020. "On First-Passage Times and Sojourn Times in Finite QBD Processes and Their Applications in Epidemics," Mathematics, MDPI, vol. 8(10), pages 1-25, October.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:10:p:1718-:d:424434
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    References listed on IDEAS

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    1. Yang Woo Shin, 2009. "Fundamental Matrix Of Transient Qbd Generator With Finite States And Level Dependent Transitions," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 26(05), pages 697-714.
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    3. Artalejo, J.R. & Economou, A. & Lopez-Herrero, M.J., 2015. "The stochastic SEIR model before extinction: Computational approaches," Applied Mathematics and Computation, Elsevier, vol. 265(C), pages 1026-1043.
    4. Claude Lefèvre & Matthieu Simon, 2020. "SIR-Type Epidemic Models as Block-Structured Markov Processes," Methodology and Computing in Applied Probability, Springer, vol. 22(2), pages 433-453, June.
    5. Moghaddass, Ramin & Zuo, Ming J. & Wang, Wenbin, 2011. "Availability of a general k-out-of-n:G system with non-identical components considering shut-off rules using quasi-birth–death process," Reliability Engineering and System Safety, Elsevier, vol. 96(4), pages 489-496.
    6. Economou, A. & Gómez-Corral, A. & López-García, M., 2015. "A stochastic SIS epidemic model with heterogeneous contacts," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 421(C), pages 78-97.
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    Cited by:

    1. Gómez-Corral, A. & Lopez-Herrero, M.J. & Taipe, D., 2023. "A Markovian epidemic model in a resource-limited environment," Applied Mathematics and Computation, Elsevier, vol. 458(C).

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