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

Antonio Gómez-Corral (), Martín López-García (), Maria Jesus Lopez-Herrero () and Diana Taipe ()
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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

Mathematics, 2020, vol. 8, issue 10, 1-25

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.

Keywords: epidemic modeling; first-passage times; hitting probabilities; quasi-birth-death processes; sojourn times (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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