Model for disease dynamics of a waterborne pathogen on a random network

Modelling techniques in cholera epidemiology: A systematic and critical review

Diverse modelling techniques in cholera epidemiology have been developed and used to (1) study its transmission dynamics, (2) predict and manage cholera outbreaks, and (3) assess the impact of various control and mitigation measures. In this study, we carry out a critical and systematic review of various approaches used for modelling the dynamics of cholera. Also, we discuss the strengths and weaknesses of each modelling approach. A systematic search of articles was conducted in Google Scholar, PubMed, Science Direct, and Taylor & Francis. Eligible studies were those concerned with the dynamics of cholera excluding studies focused on models for cholera transmission in animals, socio-economic factors, and genetic & molecular related studies. A total of 476 peer-reviewed articles met the inclusion criteria, with about 40% (32%) of the studies carried out in Asia (Africa). About 52%, 21%, and 9%, of the studies, were based on compartmental (e.g., SIRB), statistical (time series and regression), and spatial (spatiotemporal clustering) models, respectively, while the rest of the analysed studies used other modelling approaches such as network, machine learning and artificial intelligence, Bayesian, and agent-based approaches. Cholera modelling studies that incorporate vector/housefly transmission of the pathogen are scarce and a small portion of researchers (3.99%) considers the estimation of key epidemiological parameters. Vaccination only platform was utilized as a control measure in more than half (58%) of the studies. Research productivity in cholera epidemiological modelling studies have increased in recent years, but authors used diverse range of models. Future models should consider incorporating vector/housefly transmission of the pathogen and on the estimation of key epidemiological parameters for the transmission of cholera dynamics.

Edge-based compartmental modeling for the spread of cholera on random networks: A case study in Somalia

Cholera remains a major public health problem that threatens human health worldwide and its severity is continuing. In this paper, an edge-based model for cholera transmission on random networks is proposed and investigated. The model assumes that two communities share a common water source and includes three transmission routes, namely intra- and inter-community human-to-human transmission as well as water-to-human transmission. Intra-community human-to-human contacts are modeled through a random contact network, while both inter-community and water-to-human transmission are modeled through external nodes that reach each individual in the network to the same extent. The basic reproduction number and the equations of the final epidemic size are obtained. In addition, our study considers the cholera situation in Banadir, which is one of the most severely infected regions in Somalia, during the period (2019-2021). According to the geographical location, two adjacent districts are selected and our model fits well with the real data on the monthly cumulative cholera cases of these two districts during the above-mentioned period. From the perspective of network topology, cutting off high-risk contacts by supervising, isolating, quarantining and closing places with high-degree cholera-infected individuals to reduce degree heterogeneity is an effective measure to control cholera transmission. Our findings might offer some useful insights on cholera control.

The Basic Reproduction Number as a Loop Gain Matrix

The COVID-19 pandemic and the disordered reactions of most governments made the importance of mathematical modelling and model-based predictions evident, even outside the scientific community. The basic reproduction number [Formula: see text] quickly entered the common jargon, as a concise but effective tool to communicate the spreading power of a disease and estimate, at least roughly, the possible outcomes of the epidemic. However, while [Formula: see text] is easily defined for simple models, its proper definition is more subtle for larger, state-of-the-art models. Here we show that it is nothing else than the spectral radius of the gain matrix of a linear system, and that this matrix generalizes [Formula: see text] in the computation of the vector-valued final epidemic size and epidemic threshold, in a large class of finite-dimensional SIR-like models.

Analysis of a mathematical model for the transmission dynamics of human melioidosis

A deterministic model for the transmission dynamics of melioidosis disease in human population is designed and analyzed. The model is shown to exhibit the phenomenon of backward bifurcation, where a stable disease-free equilibrium co-exists with a stable endemic equilibrium when the basic reproduction number [Formula: see text] is less than one. It is further shown that the backward bifurcation dynamics is caused by the reinfection of individuals who recovered from the disease and relapse. The existence of backward bifurcation implies that bringing down [Formula: see text] to less than unity is not enough for disease eradication. In the absence of backward bifurcation, the global asymptotic stability of the disease-free equilibrium is shown whenever [Formula: see text]. For [Formula: see text], the existence of at least one locally asymptotically stable endemic equilibrium is shown. Sensitivity analysis of the model, using the parameters relevant to the transmission dynamics of the melioidosis disease, is discussed. Numerical experiments are presented to support the theoretical analysis of the model. In the numerical experimentations, it has been observed that screening and treating individuals in the exposed class has a significant impact on the disease dynamics.

Direct transmission via households informs models of disease and intervention dynamics in cholera

While several basic properties of cholera outbreaks are common to most settings-the pathophysiology of the disease, the waterborne nature of transmission, and others-recent findings suggest that transmission within households may play a larger role in cholera outbreaks than previously appreciated. Important features of cholera outbreaks have long been effectively modeled with mathematical and computational approaches, but little is known about how variation in direct transmission via households may influence epidemic dynamics. In this study, we construct a mathematical model of cholera that incorporates transmission within and between households. We observe that variation in the magnitude of household transmission changes multiple features of disease dynamics, including the severity and duration of outbreaks. Strikingly, we observe that household transmission influences the effectiveness of possible public health interventions (e.g. water treatment, antibiotics, vaccines). We find that vaccine interventions are more effective than water treatment or antibiotic administration when direct household transmission is present. Summarizing, we position these results within the landscape of existing models of cholera, and speculate on its implications for epidemiology and public health.

Hepatitis C virus modelled as an indirectly transmitted infection highlights the centrality of injection drug equipment in disease dynamics

The hepatitis C virus (HCV) epidemic often occurs through the persistence of injection drug use. Mathematical models have been useful in understanding various aspects of the HCV epidemic, and especially, the importance of new treatment measures. Until now, however, few models have attempted to understand HCV in terms of an interaction between the various actors in an HCV outbreak-hosts, viruses and the needle injection equipment. In this study, we apply perspectives from the ecology of infectious diseases to model the transmission of HCV among a population of injection drug users. The products of our model suggest that modelling HCV as an indirectly transmitted infection-where the injection equipment serves as an environmental reservoir for infection-facilitates a more nuanced understanding of disease dynamics, by animating the underappreciated actors and interactions that frame disease. This lens may allow us to understand how certain public health interventions (e.g. needle exchange programmes) influence HCV epidemics. Lastly, we argue that this model is of particular importance in the light of the modern opioid epidemic, which has already been associated with outbreaks of viral diseases.

The large graph limit of a stochastic epidemic model on a dynamic multilayer network

We consider a Markovian SIR-type (Susceptible → Infected → Recovered) stochastic epidemic process with multiple modes of transmission on a contact network. The network is given by a random graph following a multilayer configuration model where edges in different layers correspond to potentially infectious contacts of different types. We assume that the graph structure evolves in response to the epidemic via activation or deactivation of edges of infectious nodes. We derive a large graph limit theorem that gives a system of ordinary differential equations (ODEs) describing the evolution of quantities of interest, such as the proportions of infected and susceptible vertices, as the number of nodes tends to infinity. Analysis of the limiting system elucidates how the coupling of edge activation and deactivation to infection status affects disease dynamics, as illustrated by a two-layer network example with edge types corresponding to community and healthcare contacts. Our theorem extends some earlier results describing the deterministic limit of stochastic SIR processes on static, single-layer configuration model graphs. We also describe precisely the conditions for equivalence between our limiting ODEs and the systems obtained via pair approximation, which are widely used in the epidemiological and ecological literature to approximate disease dynamics on networks. The flexible modeling framework and asymptotic results have potential application to many disease settings including Ebola dynamics in West Africa, which was the original motivation for this study.

Edge-based epidemic spreading in degree-correlated complex networks

Networks that grow through the addition of new nodes or edges may acquire degree-degree correlations. When one considers a short epidemic on a slowly growing network, such as the spread of a strain of influenza in a population for one season, it is reasonable to assume that the degree-correlated network is static during the course of an epidemic. In this case using only information about the network degree distribution is not enough to capture the exponential growth phase, the epidemic peak or the final epidemic size. Hence, in this paper we formulate an edge-based SIR epidemic model on degree-correlated networks, which includes the Miller model on configuration networks as a special case. The model is relatively low-dimensional; in particular, considering the fact that it captures degree correlations. Moreover, we derive rate equations to compute two node degree correlations in a growing network. Predictions of our model agree well with the corresponding stochastic SIR process on degree-correlated networks, such as the exponential growth phase, the epidemic peak and the final epidemic size. The basic reproduction number R₀ and the final epidemic size are theoretically derived, which are equivalent to those based on the percolation theory. However, our model has the advantage that it can trace the dynamic spread of an epidemic on degree-correlated networks. This provides us with more accurate information to predict and control the spread of diseases in growing populations with biased-mixing. Finally, our model is tested on degree-correlated networks with clustering, and it is shown that our model is robust to degree-correlated networks with small clustering.

An edge-based SIR model for sexually transmitted diseases on the contact network

Sexually transmitted diseases, which are infections through sexual contact, pose severe public health threat nowadays. In this paper, we develop a novel model for such diseases on a bipartite random contact network. Our model is precise with arbitrary initial conditions, which makes it suitable to study preventative vaccination strategies. We derive the reproduction number and show that R₀=1 is the disease threshold. An implicit formula for the final epidemic size is also derived, and we show that the formula gives a unique positive final epidemic size when the reproduction number is larger than unity. We find that the final size in either sex is heavily influenced by the degree distribution of the opposite sex.

Lessons from a decade of individual-based models for infectious disease transmission: a systematic review (2006-2015)

Background

Individual-based models (IBMs) are useful to simulate events subject to stochasticity and/or heterogeneity, and have become well established to model the potential (re)emergence of pathogens (e.g., pandemic influenza, bioterrorism). Individual heterogeneity at the host and pathogen level is increasingly documented to influence transmission of endemic diseases and it is well understood that the final stages of elimination strategies for vaccine-preventable childhood diseases (e.g., polio, measles) are subject to stochasticity. Even so it appears IBMs for both these phenomena are not well established. We review a decade of IBM publications aiming to obtain insights in their advantages, pitfalls and rationale for use and to make recommendations facilitating knowledge transfer within and across disciplines.

Methods

We systematically identified publications in Web of Science and PubMed from 2006-2015 based on title/abstract/keywords screening (and full-text if necessary) to retrieve topics, modeling purposes and general specifications. We extracted detailed modeling features from papers on established vaccine-preventable childhood diseases based on full-text screening.

Results

We identified 698 papers, which applied an IBM for infectious disease transmission, and listed these in a reference database, describing their general characteristics. The diversity of disease-topics and overall publication frequency have increased over time (38 to 115 annual publications from 2006 to 2015). The inclusion of intervention strategies (8 to 52) and economic consequences (1 to 20) are increasing, to the detriment of purely theoretical explorations. Unfortunately, terminology used to describe IBMs is inconsistent and ambiguous. We retrieved 24 studies on a vaccine-preventable childhood disease (covering 7 different diseases), with publication frequency increasing from the first such study published in 2008. IBMs have been useful to explore heterogeneous between- and within-host interactions, but combined applications are still sparse. The amount of missing information on model characteristics and study design is remarkable.

Conclusions

IBMs are suited to combine heterogeneous within- and between-host interactions, which offers many opportunities, especially to analyze targeted interventions for endemic infections. We advocate the exchange of (open-source) platforms and stress the need for consistent “branding”. Using (existing) conventions and reporting protocols would stimulate cross-fertilization between research groups and fields, and ultimately policy making in decades to come.

Electronic supplementary material

The online version of this article (doi:10.1186/s12879-017-2699-8) contains supplementary material, which is available to authorized users.

Reproduction numbers of infectious disease models

This primer article focuses on the basic reproduction number, ℛ 0 , for infectious diseases, and other reproduction numbers related to ℛ 0 that are useful in guiding control strategies. Beginning with a simple population model, the concept is developed for a threshold value of ℛ 0 determining whether or not the disease dies out. The next generation matrix method of calculating ℛ 0 in a compartmental model is described and illustrated. To address control strategies, type and target reproduction numbers are defined, as well as sensitivity and elasticity indices. These theoretical ideas are then applied to models that are formulated for West Nile virus in birds (a vector-borne disease), cholera in humans (a disease with two transmission pathways), anthrax in animals (a disease that can be spread by dead carcasses and spores), and Zika in humans (spread by mosquitoes and sexual contacts). Some parameter values from literature data are used to illustrate the results. Finally, references for other ways to calculate ℛ 0 are given. These are useful for more complicated models that, for example, take account of variations in environmental fluctuation or stochasticity.