Infection dynamics on scale-free networks

Modelling flock heterogeneity in the transmission of peste des petits ruminants virus and its impact on the effectiveness of vaccination for eradication

Peste des petits ruminants (PPR) is an acute infectious disease of small ruminants targeted for global eradication by 2030. The Global Strategy for Control and Eradication (GSCE) recommends mass vaccination targeting 70% coverage of small ruminant populations in PPR-endemic regions. These small ruminant populations are diverse with heterogeneous mixing patterns that may influence PPR virus (PPRV) transmission dynamics. This paper evaluates the impact of heterogeneous mixing on (i) PPRV transmission and (ii) the likelihood of different vaccination strategies achieving PPRV elimination, including the GSCE recommended strategy. We develop models simulating heterogeneous transmission between hosts, including a metapopulation model of PPRV transmission between villages in lowland Ethiopia fitted to serological data. Our results demonstrate that although heterogeneous mixing of small ruminant populations increases the instability of PPRV transmission-increasing the chance of fadeout in the absence of intervention-a vaccination coverage of 70% may be insufficient to achieve elimination if high-risk populations are not targeted. Transmission may persist despite very high vaccination coverage (>90% small ruminants) if vaccination is biased towards more accessible but lower-risk populations such as sedentary small ruminant flocks. These results highlight the importance of characterizing small ruminant mobility patterns and identifying high-risk populations for vaccination and support a move towards targeted, risk-based vaccination programmes in the next phase of the PPRV eradication programme. Our modelling approach also illustrates a general framework for incorporating heterogeneous mixing patterns into models of directly transmitted infectious diseases where detailed contact data are limited. This study improves understanding of PPRV transmission and elimination in heterogeneous small ruminant populations and should be used to inform and optimize the design of PPRV vaccination programmes.

Population heterogeneity in vaccine coverage impacts epidemic thresholds and bifurcation dynamics

Population heterogeneity, especially in individuals' contact networks, plays an important role in transmission dynamics of infectious diseases. For vaccine-preventable diseases, outstanding issues like vaccine hesitancy and availability of vaccines further lead to nonuniform coverage among groups, not to mention the efficacy of vaccines and the mixing pattern varying from one group to another. As the ongoing COVID-19 pandemic transitions to endemicity, it is of interest and significance to understand the impact of aforementioned population heterogeneity on the emergence and persistence of epidemics. Here we analyze epidemic thresholds and characterize bifurcation dynamics by accounting for heterogeneity caused by group-dependent characteristics, including vaccination rate and efficacy as well as disease transmissibility. Our analysis shows that increases in the difference in vaccination coverage among groups can render multiple equilibria of disease burden to exist even if the overall basic reproductive ratio is below one (also known as backward bifurcation). The presence of other heterogeneity factors such as differences in vaccine efficacy, transmission, mixing pattern, and group size can each exhibit subtle impacts on bifurcation. We find that heterogeneity in vaccine efficacy can undermine the condition for backward bifurcations whereas homophily tends to aggravate disease endemicity. Our results have practical implications for improving public health efforts by addressing the role of population heterogeneity in the spread and control of diseases.

Emergent information dynamics in many-body interconnected systems

The information implicitly represented in the state of physical systems allows for their analysis using analytical techniques from statistical mechanics and information theory. This approach has been successfully applied to complex networks, including biophysical systems such as virus-host protein-protein interactions and whole-brain models in health and disease, drawing inspiration from quantum statistical physics. Here we propose a general mathematical framework for modeling information dynamics on complex networks, where the internal node states are vector valued, allowing each node to carry multiple types of information. This setup is relevant for various biophysical and sociotechnological models of complex systems, ranging from viral dynamics on networks to models of opinion dynamics and social contagion. Instead of focusing on node-node interactions, we shift our attention to the flow of information between network configurations. We uncover fundamental differences between widely used spin models on networks, such as voter and kinetic dynamics, which cannot be detected through classical node-based analysis. We illustrate the mathematical framework further through an exemplary application to epidemic spreading on a low-dimensional network. Our model provides an opportunity to adapt powerful analytical methods from quantum many-body systems to study the interplay between structure and dynamics in interconnected systems.

Bifurcation analysis in an epidemic model on adaptive networks

In this paper, we study a delayed adaptive network epidemic model in which the local spatial connections of susceptible and susceptible individuals have time-delay effects on the rate of demographic change of local spatial connections of susceptible and susceptible individuals. We prove that the Hopf bifurcation occurs at the critical value τ₀ with delay τ as the bifurcation parameter. Then, by using the normal form method and the central manifold theory, the criteria for the bifurcation direction and stability are derived. Finally, numerical simulations are presented to show the feasibility of our results.

Impact of contact rate on epidemic spreading in complex networks

Abstract

Contact reduction is an effective strategy to mitigate the spreading of epidemic. However, the existing reaction-diffusion equations for infectious disease are unable to characterize this effect. Thus, we here propose an extended susceptible-infected-recovered model by incorporating contact rate into the standard SIR model, and concentrate on investigating its impact on epidemic transmission. We analytically derive the epidemic thresholds on homogeneous and heterogeneous networks, respectively. The effects of contact rate on spreading speed, scale and outbreak threshold are explored on ER and SF networks. Simulations results show that epidemic dissemination is significantly mitigated when contact rate is reduced. Importantly, epidemic spreads faster on heterogeneous networks while broader on homogeneous networks, and the outbreak thresholds of the former are smaller.

Graphical abstract

Understanding the impact of digital contact tracing during the COVID-19 pandemic

Digital contact tracing (DCT) applications have been introduced in many countries to aid the containment of COVID-19 outbreaks. Initially, enthusiasm was high regarding their implementation as a non-pharmaceutical intervention (NPI). However, no country was able to prevent larger outbreaks without falling back to harsher NPIs. Here, we discuss results of a stochastic infectious-disease model that provide insights in how the progression of an outbreak and key parameters such as detection probability, app participation and its distribution, as well as engagement of users impact DCT efficacy informed by results of empirical studies. We further show how contact heterogeneity and local contact clustering impact the intervention's efficacy. We conclude that DCT apps might have prevented cases on the order of single-digit percentages during single outbreaks for empirically plausible ranges of parameters, ignoring that a substantial part of these contacts would have been identified by manual contact tracing. This result is generally robust against changes in network topology with exceptions for homogeneous-degree, locally-clustered contact networks, on which the intervention prevents more infections. An improvement of efficacy is similarly observed when app participation is highly clustered. We find that DCT typically averts more cases during the super-critical phase of an epidemic when case counts are rising and the measured efficacy therefore depends on the time of evaluation.

Dynamic analysis of disturbance propagation in ecological networks with quarantine items and proportional migration

In order to study the stability of the ecosystem under external attack, we regard the ecosystem as a complex network and the species disturbance after the attack as an infectious disease. We establish an ecological network disturbance propagation model based on the infectious disease model, and analyze its dynamics with the above ideas. In this paper, the species are regarded as nodes in the network, and the predator–prey relationship is regarded as the edge of the network. When the ecosystem is attacked by external forces, the disturbance can be transmitted from a species to its predator or prey through the food chain, and the disturbed species can recover themselves and then return to a stable state. At the same time, we consider adding human quarantine and protection of disturbed species. In this way, all species in the ecosystem are divided into four states: undisturbed, disturbed, quarantine and recovered. By analyzing the dynamics of disturbance propagation, the critical threshold and equilibrium point of disturbance diffusion are determined, and the local and global stability of disease-free equilibrium and endemic equilibrium are analyzed. The results show that the existence of endemic equilibrium depends on the critical threshold of disturbance propagation, which is related to the structure of food web, the propagation proportion of disturbance and the recovery proportion of species after being attacked. The larger the propagation proportion is, the weaker the resistance stability is, and the easier the disturbance propagates in the system. The higher the recovery proportion of the disturbed species, the stronger the stability of the recovery rate, and the more difficult it is for the disturbance to propagate in the system. Then we regard human protection of species as species immunity, and choose the most effective species protection measures by comparing and analyzing the threshold changes under the three protection strategies. The results show that the moderately large neighbor nodes of the disturbed species should be protected. This kind of protection measure is the most effective and it is easier to restrain the propagation of disturbance. Finally, the food webs of 85 species in a pine forest in Otago, New Zealand is selected to analyze the propagation process of disturbance by numerical simulation.

Limited resource network modeling and its opinion diffusion dynamics

The preferential attachment of the Barabási-Albert model has been playing an important role in modeling practical complex networks. The preferential attachment mechanism describes the role of many real systems, which follows the characteristic "the rich get richer." However, there are some situations that are ignored by the preferential attachment mechanism, one of which is the existence of the limited resource. Vertices with the largest degree may not obtain new edges by the highest probability due to various factors, e.g., in social relationship networks, vertices with quite a lot of relationships may not connect to new vertices since their energy and resource are limited. Hence, the limit for degree growing is proposed in our new network model. We adjust the attachment rule in light of the population growth curve in biology, which considers both attraction and restriction of the degree. In addition, the unaware-aware-unaware opinion diffusion is studied on our proposed network. The celebrity effect is taken into consideration in the opinion diffusion process.

Logistic growth on networks: Exact solutions for the susceptible-infected model

The susceptible-infected (SI) model is the most basic of all compartmental models used to describe the spreading of information through a population. Despite its apparent simplicity, the analytic solution of this model on networks is still lacking. We address this problem here using a novel formulation inspired by the mathematical treatment of many-body quantum systems. This allows us to organize the time-dependent expectation values for the state of individual nodes in terms of contributions from subgraphs of the network. We compute these contributions systematically and find a set of symmetry relations among subgraphs of differing topologies. We use our novel approach to compute the spreading of information on three different sample networks. The exact solution, which matches with Monte Carlo simulations, visibly departs from the mean-field results.

The role of high-risk geographies in the perpetuation of the HIV epidemic in rural South Africa: A spatial molecular epidemiology study

In this study, we hypothesize that HIV geographical clusters (geospatial areas with significantly higher numbers of HIV positive individuals) can behave as the highly connected nodes in the transmission network. Using data come from one of the most comprehensive demographic surveillance systems in Africa, we found that more than 70% of the HIV transmission links identified were directly connected to an HIV geographical cluster located in a peri-urban area. Moreover, we identified a single central large community of highly connected nodes located within the HIV cluster. This module was composed by nodes highly connected among them, forming a central structure of the network that was also connected with the small sparser modules located outside of the HIV geographical cluster. Our study supports the evidence of the high level of connectivity between HIV geographical high-risk populations and the entire community.

Characterising the structure of the largest online commercial sex network in the UK: observational study with implications for STI prevention

This study analyses large-scale online data to examine the characteristics of a national commercial sex network of off-street female sex workers and their male clients to identify implications for public health policy and practice. We collected sexual contact information from the largest online community dedicated to reviewing sex workers' services in the UK. We built the sexual network using reviews reported between January 2014 and December 2017. We then quantified network parameters using social network analysis measures. The network is composed of 6477 vertices with 59% of them concentred in a giant component clustered around London and Milton Keynes. We found minimal disassortative mixing by degree between sex workers and their clients, and that a few clients and sex workers are highly connected whilst the majority only have one or few sexual contacts. Finally, our simulation models suggested that prevention strategies targeting both sex workers and clients with high centrality scores are the most effective in reducing network connectedness and average closeness centrality scores, thus limiting the transmission of STIs.

Stochastic social behavior coupled to COVID-19 dynamics leads to waves, plateaus, and an endemic state

It is well recognized that population heterogeneity plays an important role in the spread of epidemics. While individual variations in social activity are often assumed to be persistent, that is, constant in time, here we discuss the consequences of dynamic heterogeneity. By integrating the stochastic dynamics of social activity into traditional epidemiological models, we demonstrate the emergence of a new long timescale governing the epidemic, in broad agreement with empirical data. Our stochastic social activity model captures multiple features of real-life epidemics such as COVID-19, including prolonged plateaus and multiple waves, which are transiently suppressed due to the dynamic nature of social activity. The existence of a long timescale due to the interplay between epidemic and social dynamics provides a unifying picture of how a fast-paced epidemic typically will transition to an endemic state.

Global and local community memberships for estimating spreading capability of nodes in social networks

The analysis of spreading processes within complex networks can offer many important insights for the application in contexts such as epidemics, information dissemination or rumours. Particularly, structural factors of the network which either contribute or hinder the spreading are of interest, as they can be used to control or predict such processes. In social networks, the community structure is especially relevant, as actors usually participate in different densely connected social groups which emerge from various contexts, potentially allowing them to inject the spreading process into many different communities quickly. This paper extends our recent findings on the community membership of nodes and how it can be used to predict their individual spreading capability (Krukowski and Hecking, in: Benito, Cherifi, Cherifi, Moro, Rocha, Sales-Pardo (eds) Complex networks & their applications IX. Springer, Cham, pp 408-419, 2021) by further evaluating it on additional networks (both real-world networks and artificially generated networks), while additionally introducing a new local measure to identify influential spreaders that-in contrast to most other measures, does not rely on knowledge of the global network structure. The results confirm our recent findings, showing that the community membership of nodes can be used as a predictor for their spreading capability, while also showing that especially the local measure proves to be a good predictor, effectively outperforming the global measure in many cases. The results are discussed with regard to real-world use cases, where knowledge of the global structure is often not given, yet a prediction regarding the spreading capability highly desired (e.g., contact-tracing apps).

A Multi-Scale Model for the Spread of HIV in a Population Considering the Immune Status of People

A multi-scale mathematical model is proposed, seeking to describe the propagation of Human Immunodeficiency Virus (HIV) in a group of young people between 15 and 24 years of age, through unprotected sexual contact. The uses of antiretroviral therapy (ART) and therapeutic failure are considered to show how the rate of propagation and prevalence are affected. The model consists of a complex network modeling the interactions on the population scale, coupled with the immunological dynamics of each individual, which is modeled by a system of differential equations. The immunological model allows to observe some known facts from the literature, such as to control HIV infection in the immune system, it is necessary to reduce the probability of healthy CD4 T cells becoming infected or increase the probability at which cells of the specific cell response against HIV eliminate infected CD4 T cells. At the population level, it is shown that, to have a high prevalence, it is not necessary for the virus to spread rapidly at the beginning of the simulation time. Additionally, it is observed that a greater number of sexual partners induces higher HIV prevalence. Using ART in the immune system reduces the number of infected CD4 T cells and, consequently, helps to reduce the spread of infection at the population scale. An important result observed in simulations is that the average number of HIV carriers who abandon ART is greater than those who access it. The study adds to the available literature an original simulation model that describes the dynamics of HIV propagation in a population, considering the immune state of people within that population, and serves as a basis for future research involving more detailed aspects aiming for a model closest to reality.

Immune amnesia induced by measles and its effects on concurrent epidemics

It has been recently discovered that the measles virus can damage pre-existing immunological memory, destroying B lymphocytes and reducing the diversity of non-specific B cells of the infected host. In particular, this implies that previously acquired immunization from vaccination or direct exposition to other pathogens could be partially erased in a phenomenon named ‘immune amnesia’, whose effects can become particularly worrisome given the actual rise of anti-vaccination movements. Here, we present the first attempt to incorporate immune amnesia into standard models of epidemic spreading by proposing a simple model for the spreading of two concurrent pathogens causing measles and another generic disease. Different analyses confirm that immune amnesia can have important consequences for epidemic spreading, significantly altering the vaccination coverage required to reach herd immunity. We also uncover the existence of novel propagating and endemic phases induced by immune amnesia. Finally, we discuss the meaning and consequences of our results and their relation with, e.g. immunization strategies, together with the possibility that explosive types of transitions may emerge, making immune-amnesia effects particularly dramatic. This work opens the door to further developments and analyses of immune-amnesia effects, contributing also to the theory of interacting epidemics on complex networks.

Containing pandemics through targeted testing of households

Background

While invasive social distancing measures have proven efficient to control the spread of pandemics failing wide-scale deployment of vaccines, they carry vast societal costs. The development of a diagnostic methodology for identifying COVID-19 infection through simple testing was a reality only a few weeks after the novel virus was officially announced. Thus, we were interested in exploring the ability of regular testing of non-symptomatic people to reduce cases and thereby offer a non-pharmaceutical tool for controlling the spread of a pandemic.

Methods

We developed a data-driven individual-based epidemiological network model in order to investigate epidemic countermeasures. This models is based on high-resolution demographic data for each municipality in Norway, and each person in the model is subject to Susceptible-Exposed-Infectious-Recovered (SEIR) dynamics. The model was calibrated against hospitalization data in Oslo, Norway, a city with a population of 700k which we have used as the simulations focus.

Results

Finding that large households function as hubs for the propagation of COVID-19, we assess the intervention efficiency of targeted pooled household testing (TPHT) repeatedly. For an outbreak with reproductive number R=1.4, we find that weekly TPHT of the 25% largest households brings R below unity. For the case of R=1.2, our results suggest that TPHT with the largest 25% of households every three days in an urban area is as effective as a lockdown in curbing the outbreak. Our investigations of different disease parameters suggest that these results are markedly improved for disease variants that more easily infect young people, and when compliance with self-isolation rules is less than perfect among suspected symptomatic cases. These results are quite robust to changes in the testing frequency, city size, and the household-size distribution. Our results are robust even with only 50% of households willing to participate in TPHT, provided the total number of tests stay unchanged.

Conclusions

Pooled and targeted household testing appears to be a powerful non-pharmaceutical alternative to more invasive social-distancing and lock-down measures as a localized early response to contain epidemic outbreaks.

Supplementary Information

The online version contains supplementary material available at (10.1186/s12879-021-06256-8).

Network models and the interpretation of prolonged infection plateaus in the COVID19 pandemic

Non-pharmaceutical intervention measures, such as social distancing, have so far been the only means to slow the spread of SARS-CoV-2. In the United States, strict social distancing during the first wave of virus spread has resulted in different types of infection dynamics. In some states, such as New York, extensive infection spread was followed by a pronounced decline of infection levels. In other states, such as California, less infection spread occurred before strict social distancing, and a different pattern was observed. Instead of a pronounced infection decline, a long-lasting plateau is evident, characterized by similar daily new infection levels. Here we show that network models, in which individuals and their social contacts are explicitly tracked, can reproduce the plateau if network connections are cut due to social distancing measures. The reason is that in networks characterized by a 2D spatial structure, infection tends to spread quadratically with time, but as edges are randomly removed, the infection spreads along nearly one-dimensional infection “corridors”, resulting in plateau dynamics. Further, we show that plateau dynamics are observed only if interventions start sufficiently early; late intervention leads to a “peak and decay” pattern. Interestingly, the plateau dynamics are predicted to eventually transition into an infection decline phase without any further increase in social distancing measures. Additionally, the models suggest that a second wave becomes significantly less pronounced if social distancing is only relaxed once the dynamics have transitioned to the decline phase. The network models analyzed here allow us to interpret and reconcile different infection dynamics during social distancing observed in various US states.

Time-dependent heterogeneity leads to transient suppression of the COVID-19 epidemic, not herd immunity

Epidemics generally spread through a succession of waves that reflect factors on multiple timescales. On short timescales, superspreading events lead to burstiness and overdispersion, whereas long-term persistent heterogeneity in susceptibility is expected to lead to a reduction in both the infection peak and the herd immunity threshold (HIT). Here, we develop a general approach to encompass both timescales, including time variations in individual social activity, and demonstrate how to incorporate them phenomenologically into a wide class of epidemiological models through reparameterization. We derive a nonlinear dependence of the effective reproduction number [Formula: see text] on the susceptible population fraction S. We show that a state of transient collective immunity (TCI) emerges well below the HIT during early, high-paced stages of the epidemic. However, this is a fragile state that wanes over time due to changing levels of social activity, and so the infection peak is not an indication of long-lasting herd immunity: Subsequent waves may emerge due to behavioral changes in the population, driven by, for example, seasonal factors. Transient and long-term levels of heterogeneity are estimated using empirical data from the COVID-19 epidemic and from real-life face-to-face contact networks. These results suggest that the hardest hit areas, such as New York City, have achieved TCI following the first wave of the epidemic, but likely remain below the long-term HIT. Thus, in contrast to some previous claims, these regions can still experience subsequent waves.

Vaccination strategies against COVID-19 and the diffusion of anti-vaccination views

Misinformation is usually adjusted to fit distinct narratives and propagates rapidly through social networks. False beliefs, once adopted, are rarely corrected. Amidst the COVID-19 crisis, pandemic-deniers and people who oppose wearing face masks or quarantine have already been a substantial aspect of the development of the pandemic. With the vaccine for COVID-19, different anti-vaccine narratives are being created and are probably being adopted by large population groups with critical consequences. Assuming full adherence to vaccine administration, we use a diffusion model to analyse epidemic spreading and the impact of different vaccination strategies, measured with the average years of life lost, in three network topologies (a proximity, a scale-free and a small-world network). Then, using a similar diffusion model, we consider the spread of anti-vaccine views in the network, which are adopted based on a persuasiveness parameter of anti-vaccine views. Results show that even if anti-vaccine narratives have a small persuasiveness, a large part of the population will be rapidly exposed to them. Assuming that all individuals are equally likely to adopt anti-vaccine views after being exposed, more central nodes in the network, which are more exposed to these views, are more likely to adopt them. Comparing years of life lost, anti-vaccine views could have a significant cost not only on those who share them, since the core social benefits of a limited vaccination strategy (reduction of susceptible hosts, network disruptions and slowing the spread of the disease) are substantially shortened.

#stayhome to contain Covid-19: Neuro-SIR - Neurodynamical epidemic modeling of infection patterns in social networks

An innovative neurodynamical model of epidemics in social networks - the Neuro-SIR - is introduced. Susceptible-Infected-Removed (SIR) epidemic processes are mechanistically modeled as analogous to the activity propagation in neuronal populations. The workings of infection transmission from individual to individual through a network of social contacts, is driven by the dynamics of the threshold mechanism of leaky integrate-and-fire neurons. Through this approach a dynamically evolving landscape of the susceptibility of a population to a disease is formed. In this context, epidemics with varying velocities and scales are triggered by a small fraction of infected individuals according to the configuration of various endogenous and exogenous factors representing the individuals' vulnerability, the infectiousness of a pathogen, the density of a contact network, and environmental conditions. Adjustments in the length of immunity (if any) after recovery, enable the modeling of the Susceptible-Infected-Recovered-Susceptible (SIRS) process of recurrent epidemics. Neuro-SIR by supporting an impressive level of heterogeneities in the description of a population, contagiousness of a disease, and external factors, allows a more insightful investigation of epidemic spreading in comparison with existing approaches. Through simulation experiments with Neuro-SIR, we demonstrate the effectiveness of the #stayhome strategy for containing Covid-19, and successfully validate the simulation results against the classical epidemiological theory. Neuro-SIR is applicable in designing and assessing prevention and control strategies for spreading diseases, as well as in predicting the evolution pattern of epidemics.

Animal movement in a pastoralist population in the Maasai Mara Ecosystem in Kenya and implications for pathogen spread and control

Livestock movements are important drivers for infectious disease transmission. However, paucity of such data in pastoralist communities in rangeland ecosystems limits our understanding of their dynamics and hampers disease surveillance and control. The aim of this study was to investigate animal movement networks in a pastoralist community in Kenya, and assess network-based strategies for disease control. We used network analysis to characterize five types of between-village animal movement networks. We then evaluated implications of these networks for disease spread and control by quantifying topological changes in the network associated with targeted and random removal of nodes. To construct these networks, data were collected using standardized questionnaires (N = 165 households) from communities living within the Maasai Mara Ecosystem in southwestern Kenya. Our analyses show that the Maasai Mara National Reserve (MMNR), a protected wildlife area, was critical for maintaining village connectivity in the agistment network (dry season grazing), with MMNR-adjacent villages being highly utilized during the dry season. In terms of disease dynamics, the network-based basic reproduction number, R₀, was sufficient to allow disease invasion in all the five networks, and removal of villages based on degree or betweenness was not efficient in reducing R₀. However, we show that villages with high degree or betweenness may play an important role in maintaining network connectivity, which may not be captured by assessment of R₀ alone. Such villages may function as potential "firebreaks." For example, targeted removal of highly connected village nodes was more effective at fragmenting each network than random removal of nodes, indicating that network-based targeting of interventions such as vaccination could potentially disrupt transmission pathways in the ecosystem. In conclusion, this work shows that animal movements have the potential to shape patterns of disease transmission in this ecosystem, with targeted interventions being a practical and efficient measure for disease control.

SIR Dynamics with Vaccination in a Large Configuration Model

We consider an SIR model with vaccination strategy on a sparse configuration model random graph. We show the convergence of the system when the number of nodes grows and characterize the scaling limits. Then, we prove the existence of optimal controls for the limiting equations formulated in the framework of game theory, both in the centralized and decentralized setting. We show how the characteristics of the graph (degree distribution) influence the vaccination efficiency for optimal strategies, and we compute the limiting final size of the epidemic depending on the degree distribution of the graph and the parameters of infection, recovery and vaccination. We also present several simulations for two types of vaccination, showing how the optimal controls allow to decrease the number of infections and underlining the crucial role of the network characteristics in the propagation of the disease and the vaccination program.

Development and dissemination of infectious disease dynamic transmission models during the COVID-19 pandemic: what can we learn from other pathogens and how can we move forward?

The current COVID-19 pandemic has resulted in the unprecedented development and integration of infectious disease dynamic transmission models into policy making and public health practice. Models offer a systematic way to investigate transmission dynamics and produce short-term and long-term predictions that explicitly integrate assumptions about biological, behavioural, and epidemiological processes that affect disease transmission, burden, and surveillance. Models have been valuable tools during the COVID-19 pandemic and other infectious disease outbreaks, able to generate possible trajectories of disease burden, evaluate the effectiveness of intervention strategies, and estimate key transmission variables. Particularly given the rapid pace of model development, evaluation, and integration with decision making in emergency situations, it is necessary to understand the benefits and pitfalls of transmission models. We review and highlight key aspects of the history of infectious disease dynamic models, the role of rigorous testing and evaluation, the integration with data, and the successful application of models to guide public health. Rather than being an expansive history of infectious disease models, this Review focuses on how the integration of modelling can continue to be advanced through policy and practice in appropriate and conscientious ways to support the current pandemic response.

Stay with your community: Bridges between clusters trigger expansion of COVID-19

In this study, the spread of virus infection was simulated using artificial human networks. Here, real-space urban life was modeled as a modified scale-free network with constraints. To date, the scale-free network has been adopted for modeling online communities in several studies. However, in the present study, it has been modified to represent the social behaviors of people where the generated communities are restricted and reflect spatiotemporal constraints in real life. Furthermore, the networks have been extended by introducing multiple cliques in the initial step of network construction and enabling people to contact hidden (zero-degree) as well as popular (large-degree) people. Consequently, four findings and a policy proposal were obtained. First, "second waves" were observed in some cases of the simulations even without external influence or constraints on people's social contacts or the releasing of the constraints. These waves tend to be lower than the first wave and occur in "fresh" clusters, that is, via the infection of people who are connected in the network but have not been infected previously. This implies that the bridge between infected and fresh clusters may trigger a new spread of the virus. Second, if the network changes its structure on the way of infection spread or after its suppression, a second wave larger than the first can occur. Third, the peak height in the time series of the number of infected cases depends on the difference between the upper bound of the number of people each member actually meets and the number of people they choose to meet during the period of infection spread. This tendency is observed for the two kinds of artificial networks introduced here and implies the impact of bridges between communities on the virus spreading. Fourth, the release of a previously imposed constraint may trigger a second wave higher than the peak of the time series without introducing any constraint so far previously, if the release is introduced at a time close to the peak. Thus, overall, both the government and individuals should be careful in returning to society where people enjoy free inter-community contact.

Strong spatial embedding of social networks generates nonstandard epidemic dynamics independent of degree distribution and clustering

Some directly transmitted human pathogens, such as influenza and measles, generate sustained exponential growth in incidence and have a high peak incidence consistent with the rapid depletion of susceptible individuals. Many do not. While a prolonged exponential phase typically arises in traditional disease-dynamic models, current quantitative descriptions of nonstandard epidemic profiles are either abstract, phenomenological, or rely on highly skewed offspring distributions in network models. Here, we create large socio-spatial networks to represent contact behavior using human population-density data, a previously developed fitting algorithm, and gravity-like mobility kernels. We define a basic reproductive number [Formula: see text] for this system, analogous to that used for compartmental models. Controlling for [Formula: see text], we then explore networks with a household-workplace structure in which between-household contacts can be formed with varying degrees of spatial correlation, determined by a single parameter from the gravity-like kernel. By varying this single parameter and simulating epidemic spread, we are able to identify how more frequent local movement can lead to strong spatial correlation and, thus, induce subexponential outbreak dynamics with lower, later epidemic peaks. Also, the ratio of peak height to final size was much smaller when movement was highly spatially correlated. We investigate the topological properties of our networks via a generalized clustering coefficient that extends beyond immediate neighborhoods, identifying very strong correlations between fourth-order clustering and nonstandard epidemic dynamics. Our results motivate the observation of both incidence and socio-spatial human behavior during epidemics that exhibit nonstandard incidence patterns.

A modified Ising model of Barabási-Albert network with gene-type spins

The central question of systems biology is to understand how individual components of a biological system such as genes or proteins cooperate in emerging phenotypes resulting in the evolution of diseases. As living cells are open systems in quasi-steady state type equilibrium in continuous exchange with their environment, computational techniques that have been successfully applied in statistical thermodynamics to describe phase transitions may provide new insights to the emerging behavior of biological systems. Here we systematically evaluate the translation of computational techniques from solid-state physics to network models that closely resemble biological networks and develop specific translational rules to tackle problems unique to living systems. We focus on logic models exhibiting only two states in each network node. Motivated by the apparent asymmetry between biological states where an entity exhibits boolean states i.e. is active or inactive, we present an adaptation of symmetric Ising model towards an asymmetric one fitting to living systems here referred to as the modified Ising model with gene-type spins. We analyze phase transitions by Monte Carlo simulations and propose a mean-field solution of a modified Ising model of a network type that closely resembles a real-world network, the Barabási–Albert model of scale-free networks. We show that asymmetric Ising models show similarities to symmetric Ising models with the external field and undergoes a discontinuous phase transition of the first-order and exhibits hysteresis. The simulation setup presented herein can be directly used for any biological network connectivity dataset and is also applicable for other networks that exhibit similar states of activity. The method proposed here is a general statistical method to deal with non-linear large scale models arising in the context of biological systems and is scalable to any network size.

Identifying epidemic spreading dynamics of COVID-19 by pseudocoevolutionary simulated annealing optimizers

At the end of 2019, a new coronavirus (COVID-19) epidemic has triggered global public health concern. Here, a model integrating the daily intercity migration network, which constructed from real-world migration records and the Susceptible–Exposed–Infected–Removed model, is utilized to predict the epidemic spreading of the COVID-19 in more than 300 cities in China. However, the model has more than 1800 unknown parameters, which is a challenging task to estimate all unknown parameters from historical data within a reasonable computation time. In this article, we proposed a pseudocoevolutionary simulated annealing (SA) algorithm for identifying these unknown parameters. The large volume of unknown parameters of this model is optimized through three procedures co-adapted SA-based optimization processes, respectively. Our results confirm that the proposed method is both efficient and robust. Then, we use the identified model to predict the trends of the epidemic spreading of the COVID-19 in these cities. We find that the number of infections in most cities in China has reached their peak from February 29, 2020, to March 15, 2020. For most cities outside Hubei province, the total number of infected individuals would be less than 100, while for most cities in Hubei province (exclude Wuhan), the total number of infected individuals would be less than 3000.

Critical network cascades with re-excitable nodes: Why treelike approximations usually work, when they break down, and how to correct them

Network science is a rapidly expanding field, with a large and growing body of work on network-based dynamical processes. Most theoretical results in this area rely on the so-called locally treelike approximation. This is, however, usually an "uncontrolled" approximation, in the sense that the magnitudes of the error are typically unknown, although numerical results show that this error is often surprisingly small. In this paper we place this approximation on more rigorous footing by calculating the magnitude of deviations away from tree-based theories in the context of discrete-time critical network cascades with re-excitable nodes. We discuss the conditions under which tree-like approximations give good results for calculating network criticality, and also explain the reasons for deviation from this approximation, in terms of the density of certain kinds of network motifs. Using this understanding, we derive results for network criticality that apply to general networks that explicitly do not satisfy the locally treelike approximation. In particular, we focus on the biparallel motif, the smallest motif relevant to the failure of a tree-based theory in this context, and we derive the corrections due to such motifs on the conditions for criticality. We verify our claims on computer-generated networks, and we confirm that our theory accurately predicts the observed deviations from criticality. Using our theory, we explain why numerical simulations often show that deviations from a tree-based theory are surprisingly small. More specifically, we show that these deviations are negligible for networks whose average degree is even modestly large compared to one, justifying why tree-based theories appear to work well for most real-world networks.

Centrality metric for the vulnerability of urban networks to toxic releases

The dispersion of airborne pollutants in the urban atmosphere is a complex, canopy-driven process. The intricate structure of the city, the high number of potential sources, and the large spatial domain make it difficult to predict dispersion patterns, to simulate a great number of scenarios, and to identify the high-impact emission areas. Here we show that these complex transport dynamics can be efficiently characterized by adopting a complex network approach. The urban canopy layer is represented as a complex network. Street canyons and their intersections shape the spatial structure of the network. The direction and the transport capacity of the flow in the streets define the direction and the weight of the links. Within this perspective, pollutant contamination from a source is modeled as a spreading process on a network, and the most dangerous areas in a city are identified as the best spreading nodes. To this aim, we derive a centrality metric tailored to mass transport in flow networks. By means of the proposed approach, vulnerability maps of cities are rapidly depicted, revealing the nontrivial relation between urban topology, transport capacity of the street canyons, and forcing of the external wind. The network formalism provides promising insight in the comprehensive analysis of the fragility of cities to air pollution.

Epidemics with mutating infectivity on small-world networks

Epidemics and evolution of many pathogens occur on similar timescales so that their dynamics are often entangled. Here, in a first step to study this problem theoretically, we analyze mutating pathogens spreading on simple SIR networks with grid-like connectivity. We have in mind the spatial aspect of epidemics, which often advance on transport links between hosts or groups of hosts such as cities or countries. We focus on the case of mutations that enhance an agent’s infection rate. We uncover that the small-world property, i.e., the presence of long-range connections, makes the network very vulnerable, supporting frequent supercritical mutations and bringing the network from disease extinction to full blown epidemic. For very large numbers of long-range links, however, the effect reverses and we find a reduced chance for large outbreaks. We study two cases, one with discrete number of mutational steps and one with a continuous genetic variable, and we analyze various scaling regimes. For the continuous case we derive a Fokker-Planck-like equation for the probability density and solve it for small numbers of shortcuts using the WKB approximation. Our analysis supports the claims that a potentiating mutation in the transmissibility might occur during an epidemic wave and not necessarily before its initiation.

Global transmission network of SARS-CoV-2: from outbreak to pandemic

Background

The COVID-19 pandemic caused by the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) is straining health systems around the world. Although the Chinese government implemented a number of severe restrictions on people's movement in an attempt to contain its local and international spread, the virus had already reached many areas of the world in part due to its potent transmissibility and the fact that a substantial fraction of infected individuals develop little or no symptoms at all. Following its emergence, the virus started to generate sustained transmission in neighboring countries in Asia, Western Europe, Australia, Canada and the United States, and finally in South America and Africa. As the virus continues its global spread, a clear and evidence-based understanding of properties and dynamics of the global transmission network of SARS-CoV-2 is essential to design and put in place efficient and globally coordinated interventions.

Methods

We employ molecular surveillance data of SARS-CoV-2 epidemics for inference and comprehensive analysis of its global transmission network before the pandemic declaration. Our goal was to characterize the spatial-temporal transmission pathways that led to the establishment of the pandemic. We exploited a network-based approach specifically tailored to emerging outbreak settings. Specifically, it traces the accumulation of mutations in viral genomic variants via mutation trees, which are then used to infer transmission networks, revealing an up-to-date picture of the spread of SARS-CoV-2 between and within countries and geographic regions.

Results and Conclusions

The analysis suggest multiple introductions of SARS-CoV-2 into the majority of world regions by means of heterogeneous transmission pathways. The transmission network is scale-free, with a few genomic variants responsible for the majority of possible transmissions. The network structure is in line with the available temporal information represented by sample collection times and suggest the expected sampling time difference of few days between potential transmission pairs. The inferred network structural properties, transmission clusters and pathways and virus introduction routes emphasize the extent of the global epidemiological linkage and demonstrate the importance of internationally coordinated public health measures.

Topological dynamics of the 2015 South Korea MERS-CoV spread-on-contact networks

Network analysis to examine infectious contact relations provides an important means to uncover the topologies of individual infectious contact networks. This study aims to investigate the spread of diseases among individuals over contact networks by exploring the 2015 Middle East Respiratory Syndrome (MERS) outbreak in Korea. We present several distinct features of MERS transmission by employing a comprehensive approach in network research to examine both the traced relationship matrix of infected individuals and their bipartite transmission routes among healthcare facilities visited for treatment. The results indicate that a few super-spreaders were more likely to hold certain structural advantages by linking to an exceptional number of other individuals, causing several ongoing transmission events in neighbourhoods without the aid of any intermediary. Thus, the infectious contact network exhibited small-world dynamics characterised by locally clustered contacts exposed to transmission paths via short path lengths. In addition, nosocomial infection analysis shows the pattern of a common-source outbreak followed by secondary person-to-person transmission of the disease. Based on the results, we suggest policy implications related to the redesign of prevention and control strategies against the spread of epidemics.

Localized spatial distributions of disease phases yield long-term persistence of infection

We explore the emergence of persistent infection in two patches where the phases of disease progression of the individuals is given by the well known SIRS cycle modelling non-fatal communicable diseases. We find that a population structured into two patches with significantly different initial states, yields persistent infection, though interestingly, the infection does not persist in a homogeneous population having the same average initial composition as the average of the initial states of the two patches. This holds true for inter-patch links ranging from a single connection to connections across the entire inter-patch boundary. So a population with spatially uniform distribution of disease phases leads to disease extinction, while a population spatially separated into distinct patches aids the long-term persistence of disease. After transience, even very dissimilar patches settle down to the same average infected sub-population size. However the patterns of disease spreading in the patches remain discernibly dissimilar, with the evolution of the total number of infecteds in the two patches displaying distinct periodic wave forms, having markedly different amplitudes, though identical frequencies. We quantify the persistent infection through the size of the asymptotic infected set. We find that the number of inter-patch links does not affect the persistence in any significant manner. The most important feature determining persistence of infection is the disparity in the initial states of the patches, and it is clearly evident that persistence increases with increasing difference in the constitution of the patches. So we conclude that populations with very non-uniform distributions, where the individuals in different phases of disease are strongly compartmentalized spatially, lead to sustained persistence of disease in the entire population.

Epidemics on networks: Reducing disease transmission using health emergency declarations and peer communication

Understanding individual decisions in a world where communications and information move instantly via cell phones and the internet, contributes to the development and implementation of policies aimed at stopping or ameliorating the spread of diseases. In this manuscript, the role of official social network perturbations generated by public health officials to slow down or stop a disease outbreak are studied over distinct classes of static social networks. The dynamics are stochastic in nature with individuals (nodes) being assigned fixed levels of education or wealth. Nodes may change their epidemiological status from susceptible, to infected and to recovered. Most importantly, it is assumed that when the prevalence reaches a pre-determined threshold level,P * , information, called awareness in our framework, starts to spread, a process triggered by public health authorities. Information is assumed to spread over the same static network and whether or not one becomes a temporary informer, is a function of his/her level of education or wealth and epidemiological status. Stochastic simulations show that threshold selectionP * and the value of the average basic reproduction number impact the final epidemic size differentially. For the Erdős-Rényi and Small-world networks, an optimal choice forP * that minimize the final epidemic size can be identified under some conditions while for Scale-free networks this is not case.

Misinformation spreading on correlated multiplex networks

The numerous expanding online social networks offer fast channels for misinformation spreading, which could have a serious impact on socioeconomic systems. Researchers across multiple areas have paid attention to this issue with a view of addressing it. However, no systematical theoretical study has been performed to date on observing misinformation spreading on correlated multiplex networks. In this study, we propose a multiplex network-based misinformation spreading model, considering the fact that each individual can obtain misinformation from multiple platforms. Subsequently, we develop a heterogeneous edge-based compartmental theory to comprehend the spreading dynamics of our proposed model. In addition, we establish an analytical method based on stability analysis to obtain the misinformation outbreak threshold. On the basis of these theories, we finally analyze the influence of different dynamical and structural parameters on the misinformation spreading dynamics. Results show that the misinformation outbreak size R(∞) grows continuously with the effective transmission probability β once β exceeds a certain value, that is, the outbreak threshold β_c. Large average degrees, strong degree heterogeneity, or positive interlayer correlation will reduce β_c, accelerating the outbreak of misinformation. Besides, increasing the degree heterogeneity or a more positive interlayer correlation will enlarge (reduce) R(∞) for small (large) values of β. Our systematic theoretical analysis results agree well with the numerical simulation results. Our proposed model and accurate theoretical analysis will serve as a useful framework to understand and predict the spreading dynamics of misinformation on multiplex networks and thereby pave the way to address this serious issue.

Information diffusion in signed networks

Information diffusion has been widely discussed in various disciplines including sociology, economics, physics or computer science. In this paper, we generalize the linear threshold model in signed networks consisting of both positive and negative links. We analyze the dynamics of the spread of information based on balance theory, and find that a signed network can generate path dependence while structural balance can help remove the path dependence when seeded with balanced initialized active nodes. Simulation shows that the diffusion of information based on positive links contradicts that based on negative links. More positive links in signed networks are more likely to activate nodes and remove path dependence, but they can reduce predictability that is based on active states. We also find that a balanced structure can facilitate both the magnitude and speed of information diffusion, remove the path dependence, and cause polarization.

Network structure and risk-based surveillance algorithms for live shrimp movements in Thailand

Live shrimp movements pose a potential route for site-to-site transmission of acute hepatopancreatic necrosis disease (AHPND) and other shrimp diseases. We present the first application of network theory to study shrimp epizootiology, providing quantitative information about the live shrimp movement network of Thailand (LSMN), and supporting practical and policy implementations of disease surveillance and control measures. We examined the LSMN over a 13-month period from March 2013 to March 2014, with data obtained from the Thailand Department of Fisheries. The LSMN had a mixture of characteristics both limiting and facilitating disease spread. Importantly, the LSMN exhibited power-law distributions of in and out degrees with exponents of 2.87 and 2.17, respectively. This characteristic indicates that the LSMN behaves like a scale-free network and suggests that an effective strategy to control disease spread in the Thai shrimp farming sector can be achieved by removing a small number of targeted inter-site connections (arcs between nodes). Specifically, a disease-control algorithm based on betweenness centrality (defined as the number of shortest paths between node pairs that traverse a given arc) is proposed here to prioritize targets for disease surveillance and control measures.

Contact-Based Model for Epidemic Spreading on Temporal Networks

We present a contact-based model to study the spreading of epidemics by means of extending the dynamic message-passing approach to temporal networks. The shift in perspective from node- to edge-centric quantities enables accurate modeling of Markovian susceptible-infected-recovered outbreaks on time-varying trees, i.e., temporal networks with a loop-free underlying topology. On arbitrary graphs, the proposed contact-based model incorporates potential structural and temporal heterogeneities of the contact network and improves analytic estimations with respect to the individual-based (node-centric) approach at a low computational and conceptual cost. Within this new framework, we derive an analytical expression for the epidemic threshold on temporal networks and demonstrate the feasibility of this method on empirical data.

The spread of infection, information, computer malware, or any contagionlike process is often described by disease models on complex networks with a time-varying topology. Recurrent, or flulike, spreading can be modeled accurately by taking an “individual-based” approach that focuses on nodes in a network. Here, we instead focus on the interactions—the links in a network—and present a contact-based model that accurately describes a second group of contagion processes: those that lead to permanent immunization. Taking this new perspective, we derive a criterion that separates local outbreaks from global epidemics, a crucial tool for risk assessment and control of, for instance, viral marketing.

To develop our model, we integrate time-varying network topologies into dynamic message passing, a widely used approach to describe unidirectional contagion processes. Based on this generalized model, we derive a spectral criterion for the stability of the disease-free solution, which determines the critical disease parameters. Through numerous numerical studies, we provide evidence that the contact-based perspective improves the individual-based approach. Finally, we investigate the epidemic risk based on the German cattle-trade network with over 180 000 nodes. Results from the individual-based and contact-based approaches deviate considerably, and thus justify this paradigmatic shift.

Our contact-based model is conceptually similar to those that focus on individuals, so we expect that numerous individual-based findings as well as results from networks with a static topology can be transferred in the future. These may include general epidemic models with a non-Poissonian transition process that go beyond the assumption of treelike topologies, stochastic effects, and temporal networks that evolve continuously in time.

Indirect interactions influence contact network structure and diffusion dynamics

Interaction patterns at the individual level influence the behaviour of diffusion over contact networks. Most of the current diffusion models only consider direct interactions, capable of transferring infectious items among individuals, to build transmission networks of diffusion. However, delayed indirect interactions, where a susceptible individual interacts with infectious items after the infected individual has left the interaction space, can also cause transmission events. We define a diffusion model called the same place different time transmission (SPDT)-based diffusion that considers transmission links for these indirect interactions. Our SPDT model changes the network dynamics where the connectivity among individuals varies with the decay rates of link infectivity. We investigate SPDT diffusion behaviours by simulating airborne disease spreading on data-driven contact networks. The SPDT model significantly increases diffusion dynamics with a high rate of disease transmission. By making the underlying connectivity denser and stronger due to the inclusion of indirect transmissions, SPDT models are more realistic than same place same time transmission (SPST)-based models for the study of various airborne disease outbreaks. Importantly, we also find that the diffusion dynamics including indirect links are not reproducible by the current SPST models based on direct links, even if both SPDT and SPST networks assume the same underlying connectivity. This is because the transmission dynamics of indirect links are different from those of direct links. These outcomes highlight the importance of the indirect links for predicting outbreaks of airborne diseases.

Comparing the effects of non-homogenous mixing patterns on epidemiological outcomes in equine populations: A mathematical modelling study

Disease transmission models often assume homogenous mixing. This assumption, however, has the potential to misrepresent the disease dynamics for populations in which contact patterns are non-random. A disease transmission model with an SEIR structure was used to compare the effect of weighted and unweighted empirical equine contact networks to weighted and unweighted theoretical networks generated using random mixing. Equine influenza was used as a case study. Incidence curves generated with the unweighted empirical networks were similar in epidemic duration (5–8 days) and peak incidence (30.8–46.4%). In contrast, the weighted empirical networks resulted in a more pronounced difference between the networks in terms of the epidemic duration (8–15 days) and the peak incidence (5–25%). The incidence curves for the empirical networks were bimodal, while the incidence curves for the theoretical networks were unimodal. The incorporation of vaccination and isolation in the model caused a decrease in the cumulative incidence for each network, however, this effect was only seen at high levels of vaccination and isolation for the complete network. This study highlights the importance of using empirical networks to describe contact patterns within populations that are unlikely to exhibit random mixing such as equine populations.

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.

Tuning the overlap and the cross-layer correlations in two-layer networks: Application to a susceptible-infectious-recovered model with awareness dissemination

We study the properties of the potential overlap between two networks A,B sharing the same set of N nodes (a two-layer network) whose respective degree distributions pA(k),pB(k) are given. Defining the overlap coefficient α as the Jaccard index, we prove that α is very close to 0 when A and B are random and independently generated. We derive an upper bound αM for the maximum overlap coefficient permitted in terms of pA(k), pB(k), and N. Then we present an algorithm based on cross rewiring of links to obtain a two-layer network with any prescribed α inside the range (0,αM). A refined version of the algorithm allows us to minimize the cross-layer correlations that unavoidably appear for values of α beyond a critical overlap αc<αM. Finally, we present a very simple example of a susceptible-infectious-recovered epidemic model with information dissemination and use the algorithms to determine the impact of the overlap on the final outbreak size predicted by the model.

Ecological Complexity in Plant Virus Host Range Evolution

The host range of a plant virus is the number of species in which it can reproduce. Most studies of plant virus host range evolution have focused on the genetics of host-pathogen interactions. However, the distribution and abundance of plant viruses and their hosts do not always overlap, and these spatial and temporal discontinuities in plant virus-host interactions can result in various ecological processes that shape host range evolution. Recent work shows that the distributions of pathogenic and resistant genotypes, vectors, and other resources supporting transmission vary widely in the environment, producing both expected and unanticipated patterns. The distributions of all of these factors are influenced further by competitive effects, natural enemies, anthropogenic disturbance, the abiotic environment, and herbivory to mention some. We suggest the need for further development of approaches that (i) explicitly consider resource use and the abiotic and biotic factors that affect the strategies by which viruses exploit resources; and (ii) are sensitive across scales. Host range and habitat specificity will largely determine which phyla are most likely to be new hosts, but predicting which host and when it is likely to be infected is enormously challenging because it is unclear how environmental heterogeneity affects the interactions of viruses and hosts.