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

A novel queue-based stochastic epidemic model with adaptive stabilising control

The main objective of this paper is to propose a novel SEIR stochastic epidemic model. A distinguishing feature of this new model is that it allows us to consider a setup under general latency and infectious period distributions. To some extent, queuing systems with infinitely many servers and a Markov chain with time-varying transition rate comprise the very technical underpinning of the paper. Although more general, the Markov chain is as tractable as previous models for exponentially distributed latency and infection periods. It is also significantly more straightforward and tractable than semi-Markov models with a similar level of generality. Based on stochastic stability, we derive a sufficient condition for a shrinking epidemic regarding the queuing system's occupation rate that drives the dynamics. Relying on this condition, we propose a class of ad-hoc stabilising mitigation strategies that seek to keep a balanced occupation rate after a prescribed mitigation-free period. We validate the approach in the light of the COVID-19 epidemic in England and in the state of Amazonas, Brazil, and assess the effect of different stabilising strategies in the latter setting. Results suggest that the proposed approach can curb the epidemic with various occupation rate levels if the mitigation is timely.

A geometric analysis of the SIRS epidemiological model on a homogeneous network

We study a fast–slow version of an SIRS epidemiological model on homogeneous graphs, obtained through the application of the moment closure method. We use GSPT to study the model, taking into account that the infection period is much shorter than the average duration of immunity. We show that the dynamics occurs through a sequence of fast and slow flows, that can be described through 2-dimensional maps that, under some assumptions, can be approximated as 1-dimensional maps. Using this method, together with numerical bifurcation tools, we show that the model can give rise to periodic solutions, differently from the corresponding model based on homogeneous mixing.

Dimension-reduction of dynamics on real-world networks with symmetry

We derive explicit formulae to quantify the Markov chain state-space compression, or lumping, that can be achieved in a broad range of dynamical processes on real-world networks, including models of epidemics and voting behaviour, by exploiting redundancies due to symmetries. These formulae are applied in a large-scale study of such symmetry-induced lumping in real-world networks, from which we identify specific networks for which lumping enables exact analysis that could not have been done on the full state-space. For most networks, lumping gives a state-space compression ratio of up to 10 7 , but the largest compression ratio identified is nearly 10 12 . Many of the highest compression ratios occur in animal social networks. We also present examples of types of symmetry found in real-world networks that have not been previously reported.

Assessing intervention strategies for non-homogeneous populations using a closed form formula for R0

A general stochastic model for susceptible → infective → recovered (SIR) epidemics in non-homogeneous populations is considered. The heterogeneity is a very important aspect here since it allows more realistic but also more complex models. The basic reproduction number R₀, an indication of the probability of an outbreak for homogeneous populations does not indicate the probability of an outbreak for non-homogeneous models anymore, because it changes with the initially infected case. Therefore, we use "individual R₀" that is the expected number of secondary cases for a unique given initially infected individual. Thus, the effectiveness of intervention strategies can be assessed by their capability to reduce individual R₀ values. Also a vaccination plan based on individual R₀ values for fully heterogeneous populations is proposed. It is based on the recursive calculation of individual R₀ values.

Measuring Infection Transmission in a Stochastic SIV Model with Infection Reintroduction and Imperfect Vaccine

An additional compartment of vaccinated individuals is considered in a SIS stochastic epidemic model with infection reintroduction. The quantification of the spread of the disease is modeled by a continuous time Markov chain. A well-known measure of the initial transmission potential is the basic reproduction number [Formula: see text], which determines the herd immunity threshold or the critical proportion of immune individuals required to stop the spread of a disease when a vaccine offers a complete protection. Due to repeated contacts between the typical infective and previously infected individuals, [Formula: see text] overestimates the average number of secondary infections and leads to, perhaps unnecessary, high immunization coverage. Assuming that the vaccine is imperfect, alternative measures to [Formula: see text] are defined in order to study the influence of the initial coverage and vaccine efficacy on the transmission of the epidemic.

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

In this paper, the interest is in a structured Markov chain model to describe the transmission dynamics of tuberculosis (TB) in the setting of small communities of hosts sharing confined spaces, and to explore the potential impact of new pre-exposure vaccines on reducing the number of new TB cases during an outbreak of the disease. The model under consideration incorporates endogenous reactivation of latent tubercle bacilli, exogenous reinfection of latently infected TB hosts, loss of effectiveness of the vaccine protection, and death of hosts due to tubercle bacilli and from causes beyond TB. Various probabilistic measures are defined and analytically studied to describe extreme values and the number of vaccinations during an outbreak, and a random version of the basic reproduction number is used to measure the transmission potential during the initial phase of the epidemic. Our numerical experiments allow us to compare different pre-exposure vaccines versus the level of coverage in terms of these probabilistic measures.

A unified stochastic modelling framework for the spread of nosocomial infections

Over the last years, a number of stochastic models have been proposed for analysing the spread of nosocomial infections in hospital settings. These models often account for a number of factors governing the spread dynamics: spontaneous patient colonization, patient–staff contamination/colonization, environmental contamination, patient cohorting or healthcare workers (HCWs) hand-washing compliance levels. For each model, tailor-designed methods are implemented in order to analyse the dynamics of the nosocomial outbreak, usually by means of studying quantities of interest such as the reproduction number of each agent in the hospital ward, which is usually computed by means of stochastic simulations or deterministic approximations. In this work, we propose a highly versatile stochastic modelling framework that can account for all these factors simultaneously, and which allows one to exactly analyse the reproduction number of each agent at the hospital ward during a nosocomial outbreak. By means of five representative case studies, we show how this unified modelling framework comprehends, as particular cases, many of the existing models in the literature. We implement various numerical studies via which we (i) highlight the importance of maintaining high hand-hygiene compliance levels by HCWs, (ii) support infection control strategies including to improve environmental cleaning during an outbreak and (iii) show the potential of some HCWs to act as super-spreaders during nosocomial outbreaks.

Population-level mathematical modeling of antimicrobial resistance: a systematic review

BACKGROUND

Mathematical transmission models are increasingly used to guide public health interventions for infectious diseases, particularly in the context of emerging pathogens; however, the contribution of modeling to the growing issue of antimicrobial resistance (AMR) remains unclear. Here, we systematically evaluate publications on population-level transmission models of AMR over a recent period (2006-2016) to gauge the state of research and identify gaps warranting further work.

METHODS

We performed a systematic literature search of relevant databases to identify transmission studies of AMR in viral, bacterial, and parasitic disease systems. We analyzed the temporal, geographic, and subject matter trends, described the predominant medical and behavioral interventions studied, and identified central findings relating to key pathogens.

RESULTS

We identified 273 modeling studies; the majority of which (> 70%) focused on 5 infectious diseases (human immunodeficiency virus (HIV), influenza virus, Plasmodium falciparum (malaria), Mycobacterium tuberculosis (TB), and methicillin-resistant Staphylococcus aureus (MRSA)). AMR studies of influenza and nosocomial pathogens were mainly set in industrialized nations, while HIV, TB, and malaria studies were heavily skewed towards developing countries. The majority of articles focused on AMR exclusively in humans (89%), either in community (58%) or healthcare (27%) settings. Model systems were largely compartmental (76%) and deterministic (66%). Only 43% of models were calibrated against epidemiological data, and few were validated against out-of-sample datasets (14%). The interventions considered were primarily the impact of different drug regimens, hygiene and infection control measures, screening, and diagnostics, while few studies addressed de novo resistance, vaccination strategies, economic, or behavioral changes to reduce antibiotic use in humans and animals.

CONCLUSIONS

The AMR modeling literature concentrates on disease systems where resistance has been long-established, while few studies pro-actively address recent rise in resistance in new pathogens or explore upstream strategies to reduce overall antibiotic consumption. Notable gaps include research on emerging resistance in Enterobacteriaceae and Neisseria gonorrhoeae; AMR transmission at the animal-human interface, particularly in agricultural and veterinary settings; transmission between hospitals and the community; the role of environmental factors in AMR transmission; and the potential of vaccines to combat AMR.

Reducing Spreading Processes on Networks to Markov Population Models

Stochastic processes on complex networks, where each node is in one of several compartments, and neighboring nodes interact with each other, can be used to describe a variety of real-world spreading phenomena. However, computational analysis of such processes is hindered by the enormous size of their underlying state space.

In this work, we demonstrate that lumping can be used to reduce any epidemic model to a Markov Population Model (MPM). Therefore, we propose a novel lumping scheme based on a partitioning of the nodes. By imposing different types of counting abstractions, we obtain coarse-grained Markov models with a natural MPM representation that approximate the original systems. This makes it possible to transfer the rich pool of approximation techniques developed for MPMs to the computational analysis of complex networks’ dynamics.

We present numerical examples to investigate the relationship between the accuracy of the MPMs, the size of the lumped state space, and the type of counting abstraction.

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.

Role of genetic heterogeneity in determining the epidemiological severity of H1N1 influenza

Genetic differences contribute to variations in the immune response mounted by different individuals to a pathogen. Such differential response can influence the spread of infectious disease, indicating why such diseases impact some populations more than others. Here, we study the impact of population-level genetic heterogeneity on the epidemic spread of different strains of H1N1 influenza. For a population with known HLA class-I allele frequency and for a given H1N1 viral strain, we classify individuals into sub-populations according to their level of susceptibility to infection. Our core hypothesis is that the susceptibility of a given individual to a disease such as H1N1 influenza is inversely proportional to the number of high affinity viral epitopes the individual can present. This number can be extracted from the HLA genetic profile of the individual. We use ethnicity-specific HLA class-I allele frequency data, together with genome sequences of various H1N1 viral strains, to obtain susceptibility sub-populations for 61 ethnicities and 81 viral strains isolated in 2009, as well as 85 strains isolated in other years. We incorporate these data into a multi-compartment SIR model to analyse the epidemic dynamics for these (ethnicity, viral strain) epidemic pairs. Our results show that HLA allele profiles which lead to a large spread in individual susceptibility values can act as a protective barrier against the spread of influenza. We predict that populations skewed such that a small number of highly susceptible individuals coexist with a large number of less susceptible ones, should exhibit smaller outbreaks than populations with the same average susceptibility but distributed more uniformly across individuals. Our model tracks some well-known qualitative trends of influenza spread worldwide, suggesting that HLA genetic diversity plays a crucial role in determining the spreading potential of different influenza viral strains across populations.

Levels of immunity to strains of H1N1 influenza can vary, depending on the individual. This strongly influences how the disease spreads in a population. Accounting for such variations is a major challenge for the epidemiology of infectious diseases. We study the effect of population-level genetic heterogeneity on the epidemic spread of different strains of H1N1 influenza. We model the immune response of specific ethnicities to a number of H1N1 viral strains, using this information to study disease spread for these (ethnicity, viral strain) epidemic pairs. Our results show that larger genetic diversity at the level of immune response, leading to the presence of susceptibility sub-populations with a broad distribution of susceptibilities, protects against the spread of influenza in a population. We also show that populations with a small number of highly susceptible individuals, but with a large number of less susceptible ones, should exhibit smaller outbreaks than populations with the same average susceptibility but where it is more uniformly distributed. Our work captures some qualitative trends of influenza spread worldwide, providing a first attempt at understanding how susceptibility heterogeneities arising from variations in immune response determine disease spread in populations.

Mathematical models of infection transmission in healthcare settings: recent advances from the use of network structured data

PURPOSE OF REVIEW

Mathematical modeling approaches have brought important contributions to the study of pathogen spread in healthcare settings over the last 20 years. Here, we conduct a comprehensive systematic review of mathematical models of disease transmission in healthcare settings and assess the application of contact and patient transfer network data over time and their impact on our understanding of transmission dynamics of infections.

RECENT FINDINGS

Recently, with the increasing availability of data on the structure of interindividual and interinstitution networks, models incorporating this type of information have been proposed, with the aim of providing more realistic predictions of disease transmission in healthcare settings. Models incorporating realistic data on individual or facility networks often remain limited to a few settings and a few pathogens (mostly methicillin-resistant Staphylococcus aureus).

SUMMARY

To respond to the objectives of creating improved infection prevention and control measures and better understanding of healthcare-associated infections transmission dynamics, further innovations in data collection and parameter estimation in modeling is required.

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

Recently, Clancy [SIR epidemic models with general infectious period distribution, Statist. Prob. Lett. 85 (2014) 1–5] has shown how SIR epidemics in which individuals’ infection periods are not necessarily exponentially distributed may be modeled in terms of a piecewise-deterministic Markov process (PDMP). In this paper, we present a more detailed description of the underlying PDMP, from which we analyze the population transmission number and the infection probability of a certain susceptible individual.

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

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.