Stochastic descriptors in an SIR epidemic model for heterogeneous individuals in small networks

Abstract

We continue here the work initiated in [13], and analyse an SIR epidemic model for the spread of an epidemic among the members of a small population of N individuals, defined in terms of a continuous-time Markov chain X . We propose a structure by levels and sub-levels of the state space of the process X , and present two different orders, Orders A and B, for states within each sub-level, which are related to a matrix and a scalar formalism, respectively, when developing our analysis. Stochastic descriptors regarding the length and size of an outbreak, the maximum number of individuals simultaneously infected during an outbreak, the fate of a particular individual within the population, and the number of secondary cases caused by a certain individual until he recovers, are deeply analysed. Our approach is illustrated by carrying out a set of numerical results regarding the spread of the nosocomial pathogen Methicillin-resistant Staphylococcus Aureus among the patients within an intensive care unit. In this application, our interest is in analysing the effectiveness of control strategies (the isolation of the patient initiating the outbreak and the proper room configuration of the intensive care unit) that intrinsically introduce heterogeneities among the members of the population.Keywords Continuous-time Markov chain · Maximum number of infected individuals · Length and size of an outbreak · Individual fate · Nosocomial infections

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

“…Therefore, the use of Algorithms , and is expected to be more convenient when studying more complex epidemic models, such as those incorporating population heterogeneities at the individual level (considering different individual susceptibilities, infectivities, or recovery periods), leading to LD‐QBD processes defined on networks, these models being specially useful for analyzing epidemic processes in highly heterogeneous environments such as hospital units . The type of local sensitivity analysis carried out here is specially interesting in this type of epidemic processes on networks because the number of parameters in these models grows combinatorially with the number N of individuals in the network.…”

“…The type of local sensitivity analysis carried out here is specially interesting in this type of epidemic processes on networks because the number of parameters in these models grows combinatorially with the number N of individuals in the network. For instance, in the case of an S I R epidemic model on a directed network (see the work by López‐García ) with external sources of infection and N individuals, the number of parameters in the model amounts to , corresponding to infectious contact rates, N external infection rates, and N recovery rates; this means that, for a population of N =10 heterogeneous individuals, the number of parameters may be as large as s =110.…”

“…individual susceptibilities, infectivities, or recovery periods), leading to LD-QBD processes defined on networks, 18,40 these models being specially useful for analyzing epidemic processes in highly heterogeneous environments such as hospital units. 40,41 The type of local sensitivity analysis carried out here is specially interesting in this type of epidemic processes on networks because the number of parameters in these models grows combinatorially with the number N of individuals in the network. ) infectious contact rates, N external infection rates, and N recovery rates; this means that, for a population of N = 10 heterogeneous individuals, the number of parameters may be as large as s = 110.…”

Exact analysis of summary statistics for continuous-time discrete-state Markov processes on networks using graph-automorphism lumping

“…In contrast to the use of approximate mean-field theories or simulation studies (Stoll et al 2012), recent approaches in mathematical epidemiology have focused on the exact analysis of infection spread dynamics occurring on small networks, for example to quantify the importance of nodes, in terms of outbreak size, vaccination and early infection in SIR epidemics (Holme 2017), and to compute SIS extinction times using computational algebra for all sufficiently small graphs (Holme and Tupikina 2018). By focusing on small networks, it is possible to include heterogeneous rates of infection and recovery in the context of particular applications, such as the spread of hospital-acquired infections in intensive care units (López-García 2016), and to analyse these systems in terms of a number of performance measures. Thus, the aim is usually to compute summary statistics related to the dynamical process (Economou et al 2015), instead of focusing on analysing the complete transient dynamics of the process, which are usually more complex to study (Keeling and Ross 2009).…”

On SIR epidemic models with generally distributed infectious periods: Number of secondary cases and probability of infection

“…In these situations, preventive strategies that do not require for detection of the disease should prevail; see, for example, the paper by López-García [12] where the efficacy of preventive (room configuration design of the unit) and responsive (isolation of patients) strategies is analyzed by means of a SIR-model on an heterogeneous population for the spread of antibiotic resistant bacteria in an intensive care unit. We refer the reader to the paper [4], where the efficacy of responsive strategies (vaccination and isolation of individuals after the first removal occurs) is analyzed for a SEIR-model in a population partitioned into households; this analysis was extended by Ball et al [5] by including imperfect vaccination, latent individuals being also vaccine-sensitive and both 9 constant and exponential infectious and latent periods.…”