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

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.

Measures to assess a warning vaccination level in a stochastic SIV model with imperfect vaccine

“…45 Most of the theoretical work for QBD concerns stationary distributions and moments of first passage times, but it can be applied to derive other interesting characteristics such as absorption and hitting probabilities. 46,47 In this sense, we point out that the distribution of the random variable 𝑆 𝑤 can be derived from hitting probabilities of reaching states showing 𝑤 vaccinated for an auxiliary absorbing process that takes values in Ŵ, while the random variable 𝑅 𝑤 can be seen as a first passage time to states in the set {(𝑣, 𝑖) ∶ 0 ≤ 𝑣 ≤ 𝑤, 0 ≤ 𝑖 ≤ 𝑁 − 𝑣 0 + 𝑤 − 𝑣}. Hence results presented in this paper can be derived by the matrix-analytics methodology.…”

“…These processes are extensively used in modeling stochastic systems and their analysis can be addressed by the matrix‐analytics methodology 45 . Most of the theoretical work for QBD concerns stationary distributions and moments of first passage times, but it can be applied to derive other interesting characteristics such as absorption and hitting probabilities 46,47 . In this sense, we point out that the distribution of the random variable $$S_w$$ can be derived from hitting probabilities of reaching states showing $$w$$ vaccinated for an auxiliary absorbing process that takes values in $$ \widehat{W}$$, while the random variable $$R_w$$ can be seen as a first passage time to states in the set $$\lbrace (v,i): 0\le v \le w, 0 \le i \le N-v_0+w-v \rbrace$$.…”

A Markov chain model to investigate the spread of antibiotic‐resistant bacteria in hospitals

“…15 as first-passage times, hitting probabilities, extreme values, and stationary regime, respectively, in the underlying LD-QBD process. See, for example, the papers by De Nitto Personè and Grassi, 19 Gaver et al, 20 and Gómez-Corral et al 21 for a detailed discussion on LD-QBD processes and their applications in the context of varicella-zoster virus infections. The work to be presented here is part of our ongoing study on the use of Markov chains, including LD-QBD processes, and related matrix-analytic methods in a variety of stochastic epidemic models, such as SIS and SIR models with two strains and cross-immunity (Almaraz and Gómez-Corral, 22 Amador et al 23 ), discrete and continuous versions of SIS models (Chalub and Sousa, 24 Gómez-Corral et al 25 ), quarantine of hosts (Amador and Gómez-Corral 26 ), limited resources in epidemics (Amador and López-Herrero 27 ), and vaccination strategies (Fernández-Peralta and Gómez-Corral, 28 Gamboa and López-Herrero 29 ).…”

“…15 as first‐passage times, hitting probabilities, extreme values, and stationary regime, respectively, in the underlying LD‐QBD process. See, for example, the papers by De Nitto Personè and Grassi, 19 Gaver et al., 20 and Gómez‐Corral et al 21 . for a detailed discussion on LD‐QBD processes and their applications in the context of varicella‐zoster virus infections.…”

A Stochastic SVIR Model with Imperfect Vaccine and External Source of Infection

“…We do this by representing the epidemic process in terms of a multidimensional continuous-time Markov chain (CTMC), and studying a time to absorption in this process. We show how a particular organization of states in this CTMC leads to the study of a level-dependent quasi birth-and-death process (LD-QBD) [28], and propose an efficient scheme to analyse the summary statistic of interest. Our methodology is based on the analysis of Laplace-Stieltjes transforms and the implementation of first-step arguments, adapting techniques in [24,25].…”