Number of infections suffered by a focal individual in a two‐strain SIS model with partial cross‐immunity

Abstract

We study a stochastic model for the spread of two pathogen strains—termed type 1 and type 2—among a homogeneously mixing community consisting of a finite number of individuals. In the model, we assume partial cross‐immunity, exogenous streams of infection, and that the degree of severity of a newly infective individual depends on who this infective individual was infected by. The aim is to characterize the joint probability distribution of the numbers M1 and M2 of type‐1 and type‐2 infections suffered by a focal individual during an outbreak of the disease. We present iterative procedures for computing the probability mass function of (M1,M2) under the assumption that the initial state of the focal individual is known, and a numerical study of the model is performed to investigate the influence of certain key parameters on the spread of resistant bacteria in hospitals.

Epidemiological Consequences of Viral Interference: A Mathematical Modeling Study of Two Interacting Viruses

“…Epidemiological models have also been developed in an attempt to understand how viruses interact at the population level. Many of these models assume interaction between the viruses based on some form of cross-immunity (Ackleh and Allen, 2005 ; Allen and Kirupaharan, 2005 ; Nuno et al, 2005 ; Saunders, 1981 ; Andreason, 2018 ; Garba et al, 2013 ; Ackleh and Salceanu, 2014 ; Bhattacharyya et al, 2015 ; Baguelin and Eggo, 2018 ; Almaraz and Gomez-Corral, 2019 ; Amador et al, 2019 ; Gutierrez-Jara et al, 2019 ), finding competitive exclusion or at least a diminished presence of one virus depending on the strength and mathematical formulation of the cross-immunity. Other interactions between viruses have also been considered.…”

A structured Markov chain model to investigate the effects of pre-exposure vaccines in tuberculosis control

“…Extreme value properties (Section 3.1), the cumulative number V of vaccinations (Section 3.2) and a random version R exact,0 of the basic reproduction number (Section 3.3) have been analytically derived by using absorption probabilities and first-passage times in suitably defined quasi-birth-death processes, and Algorithms 1-5 have been inspired from Gaussian elimination (Latouche and Ramaswami [35,Chapter 10]), which is applicable in all descriptors of Sections 3.1-3.3 since underlying rate matrices are band matrices. The analytical approach is inherently linked to the use of blockstructured Markov processes (Baumann and Sandmann [10]; Lefèvre and Simon [36]) and phase-type random variables (Neuts [46]) in epidemic modelling (Almaraz and Gómez-Corral [4]; Amador et al [5]; Gómez-Corral and López-García [28]).…”

“…Despite the significant literature in this field, the above-mentioned mod-els evaluate TB dynamics and vaccination strategies in terms of systems of differential equations, instead of using stochastic modelling. Both deterministic and stochastic perspectives are important for practical use, but stochastic models are usually preferable (Artalejo and López-Herrero [8]; Britton [12]) in the setting of small communities sharing confined spaces, such as intensive care units, households, schools, work places, prisons and elderly residences (Almaraz and Gómez-Corral [4]; Amador et al [5]; Economou et al [21]; Gómez-Corral and López-García [28]). In the setting of stochastic vaccination models, recent work (Chen and Kang [17]; Kribs-Zaleta and Martchena [33]; Liu and Jiang [39]; Shim [48]; Witbooi et al [53]; Zhang et al [57]) incorporates stochastic fluctuations into deterministic models by means of a suitably defined system of stochastic differential equations.…”

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

“…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 ).…”