A minimal model for multigroup adaptive SIS epidemics

We propose a generalization of the adaptive N-Intertwined Mean-Field Approximation (aNIMFA) model studied in Achterberg and Sensi [Nonlinear Dyn. 111, 12657-12670 (2023)] to a heterogeneous network of communities. In particular, the multigroup aNIMFA model describes the impact of both local and global disease awareness on the spread of a disease in a network. We obtain results on the existence and stability of the equilibria of the system, in terms of the basic reproduction number R0. Assuming individuals have no reason to decrease their contacts in the absence of disease, we show that the basic reproduction number R0 is equivalent to the basic reproduction number of the NIMFA model on static networks. Based on numerical simulations, we demonstrate that with just two communities periodic behavior can occur, which contrasts the case with only a single community, in which periodicity was ruled out analytically. We also find that breaking connections between communities is more fruitful compared to breaking connections within communities to reduce the disease outbreak on dense networks, but both strategies are viable in networks with fewer links. Finally, we emphasize that our method of modeling adaptivity is not limited to Susceptible-Infected-Susceptible models, but has huge potential to be applied in other compartmental models in epidemiology.

A minimal model for adaptive SIS epidemics

The interplay between disease spreading and personal risk perception is of key importance for modelling the spread of infectious diseases. We propose a planar system of ordinary differential equations (ODEs) to describe the co-evolution of a spreading phenomenon and the average link density in the personal contact network. Contrary to standard epidemic models, we assume that the contact network changes based on the current prevalence of the disease in the population, i.e. the network adapts to the current state of the epidemic. We assume that personal risk perception is described using two functional responses: one for link-breaking and one for link-creation. The focus is on applying the model to epidemics, but we also highlight other possible fields of application. We derive an explicit form for the basic reproduction number and guarantee the existence of at least one endemic equilibrium, for all possible functional responses. Moreover, we show that for all functional responses, limit cycles do not exist. This means that our minimal model is not able to reproduce consequent waves of an epidemic, and more complex disease or behavioural dynamics are required to reproduce epidemic waves.

Adaptive network modeling of social distancing interventions

The COVID-19 pandemic has proved to be one of the most disruptive public health emergencies in recent memory. Among non-pharmaceutical interventions, social distancing and lockdown measures are some of the most common tools employed by governments around the world to combat the disease. While mathematical models of COVID-19 are ubiquitous, few have leveraged network theory in a general way to explain the mechanics of social distancing. In this paper, we build on existing network models for heterogeneous, clustered networks with random link activation/deletion dynamics to put forth realistic mechanisms of social distancing using piecewise constant activation/deletion rates. We find our models are capable of rich qualitative behavior, and offer meaningful insight with relatively few intervention parameters. In particular, we find that the severity of social distancing interventions and when they begin have more impact than how long it takes for the interventions to take full effect.

Moment closure approximations of susceptible-infected-susceptible epidemics on adaptive networks

The influence of people's individual responses to the spread of contagious phenomena, like the COVID-19 pandemic, is still not well understood. We investigate the Markovian Generalized Adaptive Susceptible-Infected-Susceptible (G-ASIS) epidemic model. The G-ASIS model comprises many contagious phenomena on networks, ranging from epidemics and information diffusion to innovation spread and human brain interactions. The connections between nodes in the G-ASIS model change adaptively over time, because nodes make decisions to create or break links based on the health state of their neighbors. Our contribution is fourfold. First, we rigorously derive the first-order and second-order mean-field approximations from the continuous-time Markov chain. Second, we illustrate that the first-order mean-field approximation fails to approximate the epidemic threshold of the Markovian G-ASIS model accurately. Third, we show that the second-order mean-field approximation is a qualitative good approximation of the Markovian G-ASIS model. Finally, we discuss the Adaptive Information Diffusion (AID) model in detail, which is contained in the G-ASIS model. We show that, similar to most other instances of the G-ASIS model, the AID model possesses a unique steady state, but that in the AID model, the convergence time toward the steady state is very large. Our theoretical results are supported by numerical simulations.

The interplay between disease spreading and awareness diffusion in multiplex networks with activity-driven structure

The interplay between disease and awareness has been extensively studied in static networks. However, most networks in reality will evolve over time. Based on this, we propose a novel epidemiological model in multiplex networks. In this model, the disease spreading layer is a time-varying network generated by the activity-driven model, while the awareness diffusion layer is a static network, and the heterogeneity of individual infection and recovery ability is considered. First, we extend the microscopic Markov chain approach to analytically obtain the epidemic threshold of the model. Then, we simulate the spread of disease and find that stronger heterogeneity in the individual activities of a physical layer can promote disease spreading, while stronger heterogeneity of the virtual layer network will hinder the spread of disease. Interestingly, we find that when the individual infection ability follows Gaussian distribution, the heterogeneity of infection ability has little effect on the spread of disease, but it will significantly affect the epidemic threshold when the individual infection ability follows power-law distribution. Finally, we find the emergence of a metacritical point where the diffusion of awareness is able to control the onset of the epidemics. Our research could cast some light on exploring the dynamics of epidemic spreading in time-varying multiplex networks.

Cold Chain Food and COVID-19 Transmission Risk: From the Perspective of Consumption and Trade

Since the outbreak of the coronavirus disease 2019 (COVID-19), political and academic circles have focused significant attention on stopping the chain of COVID-19 transmission. In particular outbreaks related to cold chain food (CCF) have been reported, and there remains a possibility that CCF can be a carrier. Based on CCF consumption and trade matrix data, here, the "source" of COVID-19 transmission through CCF was analyzed using a complex network analysis method, informing the construction of a risk assessment model reflecting internal and external transmission dynamics. The model included the COVID-19 risk index, CCF consumption level, urbanization level, CCF trade quantity, and others. The risk level of COVID-19 transmission by CCF and the dominant risk types were analyzed at national and global scales as well as at the community level. The results were as follows. (1) The global CCF trade network is typically dominated by six core countries in six main communities, such as Indonesia, Argentina, Ukraine, Netherlands, and the USA. These locations are one of the highest sources of risk for COVID-19 transmission. (2) The risk of COVID-19 transmission by CCF in specific trade communities is higher than the global average, with the Netherlands-Germany community being at the highest level. There are eight European countries (i.e., Netherlands, Germany, Belgium, France, Spain, Britain, Italy, and Poland) and three American countries (namely the USA, Mexico, and Brazil) facing a very high level of COVID-19 transmission risk by CCF. (3) Of the countries, 62% are dominated by internal diffusion and 23% by external input risk. The countries with high comprehensive transmission risk mainly experience risks from external inputs. This study provides methods for tracing the source of virus transmission and provides a policy reference for preventing the chain of COVID-19 transmission by CCF and maintaining the security of the global food supply chain.

Epidemic threshold in pairwise models for clustered networks: closures and fast correlations

The epidemic threshold is probably the most studied quantity in the modelling of epidemics on networks. For a large class of networks and dynamics, it is well studied and understood. However, it is less so for clustered networks where theoretical results are mostly limited to idealised networks. In this paper we focus on a class of models known as pairwise models where, to our knowledge, no analytical result for the epidemic threshold exists. We show that by exploiting the presence of fast variables and using some standard techniques from perturbation theory we are able to obtain the epidemic threshold analytically. We validate this new threshold by comparing it to the threshold based on the numerical solution of the full system. The agreement is found to be excellent over a wide range of values of the clustering coefficient, transmission rate and average degree of the network. Interestingly, we find that the analytical form of the threshold depends on the choice of closure, highlighting the importance of model selection when dealing with real-world epidemics. Nevertheless, we expect that our method will extend to other systems in which fast variables are present.

Effective approach to epidemic containment using link equations in complex networks

Epidemic containment is a major concern when confronting large-scale infections in complex networks. Many studies have been devoted to analytically understand how to restructure the network to minimize the impact of major outbreaks of infections at large scale. In many cases, the strategies are based on isolating certain nodes, while less attention has been paid to interventions on the links. In epidemic spreading, links inform about the probability of carrying the contagion of the disease from infected to susceptible individuals. Note that these states depend on the full structure of the network, and its determination is not straightforward from the knowledge of nodes' states. Here, we confront this challenge and propose a set of discrete-time governing equations that can be closed and analyzed, assessing the contribution of links to spreading processes in complex networks. Our approach allows a scheme for the containment of epidemics based on deactivating the most important links in transmitting the disease. The model is validated in synthetic and real networks, yielding an accurate determination of epidemic incidence and critical thresholds. Epidemic containment based on link deactivation promises to be an effective tool to maintain functionality of networks while controlling the spread of diseases, such as disease spread through air transportation networks.

Complex contagion leads to complex dynamics in models coupling behaviour and disease

Models coupling behaviour and disease as two unique but interacting contagions have existed since the mid 2000s. In these coupled contagion models, behaviour is typically treated as a 'simple contagion'. However, the means of behaviour spread may in fact be more complex. We develop a family of disease-behaviour coupled contagion compartmental models in order to examine the effect of behavioural contagion type on disease-behaviour dynamics. Coupled contagion models treating behaviour as a simple contagion and a complex contagion are investigated, showing that behavioural contagion type can have a significant impact on dynamics. We find that a simple contagion behaviour leads to simple dynamics, while a complex contagion behaviour supports complex dynamics with the possibility of bistability and periodic orbits.

Edge-Based Compartmental Modelling of an SIR Epidemic on a Dual-Layer Static-Dynamic Multiplex Network with Tunable Clustering

The duration, type and structure of connections between individuals in real-world populations play a crucial role in how diseases invade and spread. Here, we incorporate the aforementioned heterogeneities into a model by considering a dual-layer static–dynamic multiplex network. The static network layer affords tunable clustering and describes an individual’s permanent community structure. The dynamic network layer describes the transient connections an individual makes with members of the wider population by imposing constant edge rewiring. We follow the edge-based compartmental modelling approach to derive equations describing the evolution of a susceptible–infected–recovered epidemic spreading through this multiplex network of individuals. We derive the basic reproduction number, measuring the expected number of new infectious cases caused by a single infectious individual in an otherwise susceptible population. We validate model equations by showing convergence to pre-existing edge-based compartmental model equations in limiting cases and by comparison with stochastically simulated epidemics. We explore the effects of altering model parameters and multiplex network attributes on resultant epidemic dynamics. We validate the basic reproduction number by plotting its value against associated final epidemic sizes measured from simulation and predicted by model equations for a number of set-ups. Further, we explore the effect of varying individual model parameters on the basic reproduction number. We conclude with a discussion of the significance and interpretation of the model and its relation to existing research literature. We highlight intrinsic limitations and potential extensions of the present model and outline future research considerations, both experimental and theoretical.

ADAPTIVE SIS EPIDEMIC MODELS ON HETEROGENEOUS NETWORKS WITH DEMOGRAPHICS AND RISK PERCEPTION

In the context of epidemic spreading on networks, there are at least three concerns that should be considered. Firstly, the demographic process can change the underlying network structure and thus affect the disease spreading; secondly, the attachment probability of newcomers (namely, newborn or immigrated nodes) to other nodes depends on the degrees and states of existing nodes in the network; and thirdly, newcomers may remove some dangerous contacts once they perceive the risk of disease. In view of these facts, we propose two high-dimensional susceptible-infectious-susceptible (SIS) epidemic models on heterogeneous networks with demographics and risk perception — one that neglects both the degree and state correlations among nodes and the other that reserves the state correlation. Then the log-normal moment closure is adopted to reduce the dimensions of models. The basic reproduction numbers of the two corresponding low-dimensional models are derived. It is found that the basic reproduction numbers of the two models are different. In spite of this, both of them show that increasing recruitment will help to inhibit the disease spreading. In addition, numerical results demonstrate that the extent to which newcomers avoid contacting with infectious nodes can affect both the epidemic threshold and the equilibrium prevalence. Finally, comparisons between numerical and stochastic simulations indicate that the model with state correlation is more accurate.

The contact process on scale-free networks evolving by vertex updating

We study the contact process on a class of evolving scale-free networks, where each node updates its connections at independent random times. We give a rigorous mathematical proof that there is a transition between a phase where for all infection rates the infection survives for a long time, at least exponential in the network size, and a phase where for sufficiently small infection rates extinction occurs quickly, at most polynomially in the network size. The phase transition occurs when the power-law exponent crosses the value four. This behaviour is in contrast with that of the contact process on the corresponding static model, where there is no phase transition, as well as that of a classical mean-field approximation, which has a phase transition at power-law exponent three. The new observation behind our result is that temporal variability of networks can simultaneously increase the rate at which the infection spreads in the network, and decrease the time at which the infection spends in metastable states.

Using Social Network Measures in Wildlife Disease Ecology, Epidemiology, and Management

Contact networks, behavioral interactions, and shared use of space can all have important implications for the spread of disease in animals. Social networks enable the quantification of complex patterns of interactions; therefore, network analysis is becoming increasingly widespread in the study of infectious disease in animals, including wildlife. We present an introductory guide to using social-network-analytical approaches in wildlife disease ecology, epidemiology, and management. We focus on providing detailed practical guidance for the use of basic descriptive network measures by suggesting the research questions to which each technique is best suited and detailing the software available for each. We also discuss how using network approaches can be used beyond the study of social contacts and across a range of spatial and temporal scales. Finally, we integrate these approaches to examine how network analysis can be used to inform the implementation and monitoring of effective disease management strategies.

A Network Epidemic Model with Preventive Rewiring: Comparative Analysis of the Initial Phase

This paper is concerned with stochastic SIR and SEIR epidemic models on random networks in which individuals may rewire away from infected neighbors at some rate [Formula: see text] (and reconnect to non-infectious individuals with probability [Formula: see text] or else simply drop the edge if [Formula: see text]), so-called preventive rewiring. The models are denoted SIR-[Formula: see text] and SEIR-[Formula: see text], and we focus attention on the early stages of an outbreak, where we derive the expressions for the basic reproduction number [Formula: see text] and the expected degree of the infectious nodes [Formula: see text] using two different approximation approaches. The first approach approximates the early spread of an epidemic by a branching process, whereas the second one uses pair approximation. The expressions are compared with the corresponding empirical means obtained from stochastic simulations of SIR-[Formula: see text] and SEIR-[Formula: see text] epidemics on Poisson and scale-free networks. Without rewiring of exposed nodes, the two approaches predict the same epidemic threshold and the same [Formula: see text] for both types of epidemics, the latter being very close to the mean degree obtained from simulated epidemics over Poisson networks. Above the epidemic threshold, pairwise models overestimate the value of [Formula: see text] computed from simulations, which turns out to be very close to the one predicted by the branching process approximation. When exposed individuals also rewire with [Formula: see text] (perhaps unaware of being infected), the two approaches give different epidemic thresholds, with the branching process approximation being more in agreement with simulations.

Compact pairwise models for epidemics with multiple infectious stages on degree heterogeneous and clustered networks

This paper presents a compact pairwise model describing the spread of multi-stage epidemics on networks. The multi-stage model corresponds to a gamma-distributed infectious period which interpolates between the classical Markovian models with exponentially distributed infectious period and epidemics with a constant infectious period. We show how the compact approach leads to a system of equations whose size is independent of the range of node degrees, thus significantly reducing the complexity of the model. Network clustering is incorporated into the model to provide a more accurate representation of realistic contact networks, and the accuracy of proposed closures is analysed for different levels of clustering and number of infection stages. Our results support recent findings that standard closure techniques are likely to perform better when the infectious period is constant.

Impact of Network Activity on the Spread of Infectious Diseases through the German Pig Trade Network

The trade of livestock is an important and growing economic sector, but it is also a major factor in the spread of diseases. The spreading of diseases in a trade network is likely to be influenced by how often existing trade connections are active. The activity α is defined as the mean frequency of occurrences of existing trade links, thus 0 < α ≤ 1. The observed German pig trade network had an activity of α = 0.11, thus each existing trade connection between two farms was, on average, active at about 10% of the time during the observation period 2008–2009. The aim of this study is to analyze how changes in the activity level of the German pig trade network influence the probability of disease outbreaks, size, and duration of epidemics for different disease transmission probabilities. Thus, we want to investigate the question, whether it makes a difference for a hypothetical spread of an animal disease to transport many animals at the same time or few animals at many times. A SIR model was used to simulate the spread of a disease within the German pig trade network. Our results show that for transmission probabilities <1, the outbreak probability increases in the case of a decreased frequency of animal transports, peaking range of α from 0.05 to 0.1. However, for the final outbreak size, we find that a threshold exists such that finite outbreaks occur only above a critical value of α, which is ~0.1, and therefore in proximity of the observed activity level. Thus, although the outbreak probability increased when decreasing α, these outbreaks affect only a small number of farms. The duration of the epidemic peaks at an activity level in the range of α = 0.2–0.3. Additionally, the results of our simulations show that even small changes in the activity level of the German pig trade network would have dramatic effects on outbreak probability, outbreak size, and epidemic duration. Thus, we can conclude and recommend that the network activity is an important aspect, which should be taken into account when modeling the spread of diseases within trade networks.

Solvable non-Markovian dynamic network

Non-Markovian processes are widespread in natural and human-made systems, yet explicit modeling and analysis of such systems is underdeveloped. We consider a non-Markovian dynamic network with random link activation and deletion (RLAD) and heavy-tailed Mittag-Leffler distribution for the interevent times. We derive an analytically and computationally tractable system of Kolmogorov-like forward equations utilizing the Caputo derivative for the probability of having a given number of active links in the network and solve them. Simulations for the RLAD are also studied for power-law interevent times and we show excellent agreement with the Mittag-Leffler model. This agreement holds even when the RLAD network dynamics is coupled with the susceptible-infected-susceptible spreading dynamics. Thus, the analytically solvable Mittag-Leffler model provides an excellent approximation to the case when the network dynamics is characterized by power-law-distributed interevent times. We further discuss possible generalizations of our result.

Oscillating epidemics in a dynamic network model: stochastic and mean-field analysis

An adaptive network model using SIS epidemic propagation with link-type-dependent link activation and deletion is considered. Bifurcation analysis of the pairwise ODE approximation and the network-based stochastic simulation is carried out, showing that three typical behaviours may occur; namely, oscillations can be observed besides disease-free or endemic steady states. The oscillatory behaviour in the stochastic simulations is studied using Fourier analysis, as well as through analysing the exact master equations of the stochastic model. By going beyond simply comparing simulation results to mean-field models, our approach yields deeper insights into the observed phenomena and help better understand and map out the limitations of mean-field models.

Impact of constrained rewiring on network structure and node dynamics

In this paper, we study an adaptive spatial network. We consider a susceptible-infected-susceptible (SIS) epidemic on the network, with a link or contact rewiring process constrained by spatial proximity. In particular, we assume that susceptible nodes break links with infected nodes independently of distance and reconnect at random to susceptible nodes available within a given radius. By systematically manipulating this radius we investigate the impact of rewiring on the structure of the network and characteristics of the epidemic. We adopt a step-by-step approach whereby we first study the impact of rewiring on the network structure in the absence of an epidemic, then with nodes assigned a disease status but without disease dynamics, and finally running network and epidemic dynamics simultaneously. In the case of no labeling and no epidemic dynamics, we provide both analytic and semianalytic formulas for the value of clustering achieved in the network. Our results also show that the rewiring radius and the network's initial structure have a pronounced effect on the endemic equilibrium, with increasingly large rewiring radiuses yielding smaller disease prevalence.

SI infection on a dynamic partnership network: characterization of R0

We model the spread of an SI (Susceptible → Infectious) sexually transmitted infection on a dynamic homosexual network. The network consists of individuals with a dynamically varying number of partners. There is demographic turnover due to individuals entering the population at a constant rate and leaving the population after an exponentially distributed time. Infection is transmitted in partnerships between susceptible and infected individuals. We assume that the state of an individual in this structured population is specified by its disease status and its numbers of susceptible and infected partners. Therefore the state of an individual changes through partnership dynamics and transmission of infection. We assume that an individual has precisely n 'sites' at which a partner can be bound, all of which behave independently from one another as far as forming and dissolving partnerships are concerned. The population level dynamics of partnerships and disease transmission can be described by a set of (n +1)(n +2) differential equations. We characterize the basic reproduction ratio R0 using the next-generation-matrix method. Using the interpretation of R0 we show that we can reduce the number of states-at-infection n to only considering three states-at-infection. This means that the stability analysis of the disease-free steady state of an (n +1)(n +2)-dimensional system is reduced to determining the dominant eigenvalue of a 3 × 3 matrix. We then show that a further reduction to a 2 × 2 matrix is possible where all matrix entries are in explicit form. This implies that an explicit expression for R0 can be found for every value of n.

Epidemics on networks with large initial conditions or changing structure

In this paper we extend previous work deriving dynamic equations governing infectious disease spread on networks. The previous work has implicitly assumed that the disease is initialized by an infinitesimally small proportion of the population. Our modifications allow us to account for an arbitrarily large initial proportion infected. This helps resolve an apparent paradox in earlier work whereby the number of susceptible individuals could increase if too many individuals were initially infected. It also helps explain an apparent small deviation that has been observed between simulation and theory. An advantage of this modification is that it allows us to account for changes in the structure or behavior of the population during the epidemic.

Asymptotically inspired moment-closure approximation for adaptive networks

Adaptive social networks, in which nodes and network structure coevolve, are often described using a mean-field system of equations for the density of node and link types. These equations constitute an open system due to dependence on higher-order topological structures. We propose a new approach to moment closure based on the analytical description of the system in an asymptotic regime. We apply the proposed approach to two examples of adaptive networks: recruitment to a cause model and adaptive epidemic model. We show a good agreement between the improved mean-field prediction and simulations of the full network system.

Epidemics in adaptive social networks with temporary link deactivation

Disease spread in a society depends on the topology of the network of social contacts. Moreover, individuals may respond to the epidemic by adapting their contacts to reduce the risk of infection, thus changing the network structure and affecting future disease spread. We propose an adaptation mechanism where healthy individuals may choose to temporarily deactivate their contacts with sick individuals, allowing reactivation once both individuals are healthy. We develop a mean-field description of this system and find two distinct regimes: slow network dynamics, where the adaptation mechanism simply reduces the effective number of contacts per individual, and fast network dynamics, where more efficient adaptation reduces the spread of disease by targeting dangerous connections. Analysis of the bifurcation structure is supported by numerical simulations of disease spread on an adaptive network. The system displays a single parameter-dependent stable steady state and non-monotonic dependence of connectivity on link deactivation rate.