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

Mathematics

Martín López-García¹  MariaJesus Lopez-Herrero²  Antonio Gómez-Corral³  Diana Taipe³ 

[1] Department of Applied Mathematics, School of Mathematics, University of Leeds, Leeds LS2 9JT, UK;Department of Statistics and Data Science, School of Statistical Studies, Avda. Puerta de Hierro s/n, Complutense University of Madrid, 28040 Madrid, Spain;Department of Statistics and Operations Research, School of Mathematical Sciences, Plaza de Ciencias 3, Complutense University of Madrid, 28040 Madrid, Spain;

关键词: epidemic modeling;    first-passage times;    hitting probabilities;    quasi-birth-death processes;    sojourn times;   

DOI  :  10.3390/math8101718

来源: DOAJ

【 摘 要 】

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.

【 授权许可】

Unknown