Perturbation analysis in finite LD-QBD processes and applications to epidemic models

In this paper, we adapt arguments from the paper by Caswell [11] to level-dependent quasi-birth-and-death (LD-QBD) processes, which constitute a wide class of structured Markov chains. A LD-QBD process has the special feature that its space of states can be structured by levels (groups of states), so that a tridiagonal-by-blocks structure is obtained for its infinitesimal generator. For these processes, a number of algorithmic procedures exist in the literature in order to compute several performance measures while exploiting the underlying matrix structure; among others, these measures are related to first-passage times to a certain level L(0) and hitting probabilities at this level, the maximum level visited by the process before reaching states of level L(0), and the stationary distribution. For the case of a finite number of states, our aim here is to develop analogous algorithms to the ones analyzing these measures, for their perturbation analysis. This approach uses matrix calculus and exploits the specific structure of the infinitesimal generator, which allows us to obtain additional information during the perturbation analysis of the LD-QBD process by dealing with specific matrices carrying probabilistic insights of the dynamics of the process. We illustrate the approach by means of applying multi-type versions of SI and SIS epidemic models to the spread of antibiotic-sensitive and antibiotic-resistant bacterial strains in a hospital ward.