Pathogen genetic variation in small-world host contact structures

Physical review, 2017

The study of complex networks, and in particular of social networks, has mostly concentrated on relational networks, abstracting the distance between nodes. Spatial networks are, however, extremely relevant in our daily lives, and a large body of research exists to show that the distances between nodes greatly influence the cost and probability of establishing and maintaining a link. A random geometric graph (RGG) is the main type of synthetic network model used to mimic the statistical properties and behavior of many social networks. We propose a model, called REDS, that extends energy-constrained RGGs to account for the synergic effect of sharing the cost of a link with our neighbors, as is observed in real relational networks. We apply both the standard Watts-Strogatz rewiring procedure and another method that conserves the degree distribution of the network. The second technique was developed to eliminate unwanted forms of spatial correlation between the degree of nodes that are affected by rewiring, limiting the effect on other properties such as clustering and assortativity. We analyze both the statistical properties of these two network types and their epidemiological behavior when used as a substrate for a standard susceptible-infected-susceptible compartmental model. We consider and discuss the differences in properties and behavior between RGGs and REDS as rewiring increases and as infection parameters are changed. We report considerable differences both between the network types and, in the case of REDS, between the two rewiring schemes. We conclude that REDS represent, with the application of these rewiring mechanisms, extremely useful and interesting tools in the study of social and epidemiological phenomena in synthetic complex networks.

A major challenge in evolutionary ecology is to explain extensive natural variation in transmission rates and virulence across pathogens. Host and pathogen ecology is a potentially important source of that variation. Theory of its effects has been developed through the study of non-spatial models, but host population spatial structure has been shown to influence evolutionary outcomes. To date, the effects of basic host and pathogen demography on pathogen evolution have not been thoroughly explored in a spatial context. Here we use simulations to show that space produces novel predictions of the influence of the shape of the pathogen's transmission-virulence tradeoff, as well as host reproduction and mortality, on the pathogen's evolutionary stable transmission rate. Importantly, nonspatial models predict that neither the slope of linear transmission-virulence relationships, nor the host reproduction rate will influence pathogen evolution, and that host mortality will only influence it when there is a transmission-virulence tradeoff. We show that this is not the case in a spatial context, and identify the ecological conditions under which spatial effects are most influential. Thus, these results may help explain observed natural variation among pathogens unexplainable by non-spatial models, and provide guidance about when space should be considered. We additionally evaluate the ability of existing analytical approaches to predict the influence of ecology, namely spatial moment equations closed with an improved pair approximation (IPA). The IPA is known to have limited accuracy, but here we show that in the context of pathogens the limitations are substantial: in many cases, IPA incorrectly predicts evolution to pathogen-driven extinction. Despite these limitations, we suggest that the impact of ecology can still be understood within the conceptual framework arising from spatial moment equations, that of ''self-shading'', whereby the spread of highly transmissible pathogens is impeded by local depletion of susceptible hosts.

PLoS Computational Biology, 2011

The spread of infectious diseases fundamentally depends on the pattern of contacts between individuals. Although studies of contact networks have shown that heterogeneity in the number of contacts and the duration of contacts can have farreaching epidemiological consequences, models often assume that contacts are chosen at random and thereby ignore the sociological, temporal and/or spatial clustering of contacts. Here we investigate the simultaneous effects of heterogeneous and clustered contact patterns on epidemic dynamics. To model population structure, we generalize the configuration model which has a tunable degree distribution (number of contacts per node) and level of clustering (number of three cliques). To model epidemic dynamics for this class of random graph, we derive a tractable, low-dimensional system of ordinary differential equations that accounts for the effects of network structure on the course of the epidemic. We find that the interaction between clustering and the degree distribution is complex. Clustering always slows an epidemic, but simultaneously increasing clustering and the variance of the degree distribution can increase final epidemic size. We also show that bond percolation-based approximations can be highly biased if one incorrectly assumes that infectious periods are homogeneous, and the magnitude of this bias increases with the amount of clustering in the network. We apply this approach to model the high clustering of contacts within households, using contact parameters estimated from survey data of social interactions, and we identify conditions under which network models that do not account for household structure will be biased.

Mathematical Biosciences, 2013

A wide range of infectious diseases are both vertically and horizontally transmitted. Such diseases are spatially transmitted via multiple species in heterogeneous environments, typically described by complex meta-population models. The reproduction number, R 0 , is a critical metric predicting whether the disease can invade the meta-population system. This paper presents the reproduction number for a generic disease vertically and horizontally transmitted among multiple species in heterogeneous networks, where nodes are locations, and links reflect outgoing or incoming movement flows. The metapopulation model for vertically and horizontally transmitted diseases is gradually formulated from two species, two-node network models. We derived an explicit expression of R 0 , which is the spectral radius of a matrix reduced in size with respect to the original next generation matrix. The reproduction number is shown to be a function of vertical and horizontal transmission parameters, and the lower bound is the reproduction number for horizontal transmission. As an application, the reproduction number and its bounds for the Rift Valley fever zoonosis, where livestock, mosquitoes, and humans are the involved species are derived. By computing the reproduction number for different scenarios through numerical simulations, we found the reproduction number is affected by livestock movement rates only when parameters are heterogeneous across nodes. To summarize, our study contributes the reproduction number for vertically and horizontally transmitted diseases in heterogeneous networks. This explicit expression is easily adaptable to specific infectious diseases, affording insights into disease evolution.

Proceedings of the Royal Society B: Biological Sciences, 2016

Most spatial models of host–parasite interactions either neglect the possibility of pathogen evolution or consider that this process is slow enough for epidemiological dynamics to reach an equilibrium on a fast timescale. Here, we propose a novel approach to jointly model the epidemiological and evolutionary dynamics of spatially structured host and pathogen populations. Starting from a multi-strain epidemiological model, we use a combination of spatial moment equations and quantitative genetics to analyse the dynamics of mean transmission and virulence in the population. A key insight of our approach is that, even in the absence of long-term evolutionary consequences, spatial structure can affect the short-term evolution of pathogens because of the build-up of spatial differentiation in mean virulence. We show that spatial differentiation is driven by a balance between epidemiological and genetic effects, and this quantity is related to the effect of kin competition discussed in pr...

2016

In this thesis we analyze the relationship between epidemiology and network theory, starting from the observation that the viral propagation between interacting agents is determined by intrinsic characteristics of the population contact network. We aim to investigate how a particular network structure can impact on the long-term behavior of epidemics. This field is way too large to be fully discussed; we limit ourselves to consider networks that are partitioned into local communities, in order to incorporate realistic contact structures into the model. The gross structure of hierarchical networks of this kind can be described by a quotient graph. The rationale of this approach is that individuals infect those belonging to the same community with higher probability than individuals in other communities. We describe the epidemic process as a continuous-time individual-based susceptible–infected–susceptible (SIS) model using a first-order mean-field approximation, both in homogeneous a...

In 1998 Watts and Strogatz introduced the concepts of small-world networks and in the process expanded our views of computer, social, and biological networks. In this project, we built epidemics on small world and other networks. Epidemic outbreaks of communicable and sexually transmitted diseases are modeled on small-world and two-node networks, respectively. The results of simulations are compared to those obtained from homogeneous mixing (mean field) epidemic models. Scaling relationships between transmission rates for epidemics on small-world, random and homogeneous mixing populations are established empirically. The transmission dynamics of gonorrhea in heterosexually active populations with multiple partners is used to illustrate the spread of disease on two-node networks. Strategies for disease control are explored.

Journal of Theoretical Biology, 2003

We examine the dynamics of evolution in a generic spatial model of a pathogen infecting a population of hosts, or an analogous predator-prey system. Previous studies of this model have found a range of interesting phenomena that differ from the well-mixed version. We extend these studies by examining the spatial and temporal dynamics of strains using genealogical tracing. When transmissibility can evolve by mutation, strains of intermediate transmissibility dominate even though high-transmissibility mutants have a short-term reproductive advantage. Mutant strains continually arise and grow rapidly for many generations but eventually go extinct before dominating the system. We find that, after a number of generations, the mutant pathogen characteristics strongly impact the spatial distribution of their local host environment, even when there are diverse types coexisting. Extinction is due to the depletion of susceptibles in the local environment of these mutant strains. Studies of spatial and genealogical relatedness reveal the self-organized spatial clustering of strains that enables their impact on the local environment. Thus, we find that selection acts against the high-transmissibility strains on long time-scales as a result of the feedback due to environmental change. Our study shows that averages over space or time should not be assumed to adequately describe the evolutionary dynamics of spatially distributed host-pathogen systems.

This paper presents a model to approach the dynamics of infectious diseases expansion. Our model aims to establish a link between traditional simulation of the Susceptible-Infectious (SI) model of disease expansion based on ordinary differential equations (ODE), and a very simple approach based on both connectivity between people and elementary binary rules that define the result of these contacts. The SI deterministic compartmental model has been analysed and successfully modelled by our method, in the case of 4-connected neighbourhood.

Physical Review E, 2013

The interplay between disease dynamics on a network and the dynamics of the structure of that network characterizes many real-world systems of contacts. A continuous-time adaptive Susceptible-Infectious-Susceptible (ASIS) model is introduced in order to investigate this interaction, where a susceptible node avoids infections by breaking its links to its infected neighbors while it enhances the connections with other susceptible nodes by creating links to them. When the initial topology of the network is a complete graph, an exact solution to the average metastable state fraction of infected nodes is derived without resorting to any mean-field approximation. A linear scaling law of the epidemic threshold τc as a function of the effective link-breaking rate ω is found. Furthermore, the bifurcation nature of the metastable fraction of infected nodes of the ASIS model is explained. The metastable state topology shows high connectivity and low modularity in two regions of the τ, ω-plane for any effective infection rate τ > τc: (i) a "strongly adaptive" region with very high ω, and (ii) a "weakly adaptive" region with very low ω. These two regions are separated from the other half-open elliptical-like regions of low connectivity and high modularity in a contour-line-like way. Our results indicate that the adaptation of the topology in response to disease dynamics suppresses the infection, while it promotes the network evolution towards a topology that exhibits assortative-mixing, modularity and a binomial-like degree distribution.

2015

The main purpose of this document is to give a brief but mathematically rigorous description of the network-based models of transmission of infectious diseases that are studied on this web site1. Readers will be able to find a much more detailed development of this material in our book chapters [5, 6]. We also briefly describe how the network-based models that are defined here are related to compartment-level models that are used in most on the literature on mathematical epidemiology.

Scientific Reports, 2022

Indiscriminate suppression is unsustainable long term and presupposes that all interactions carry equal importance. Instead, transmission within a social network has been shown to be determined by its topology. In this paper, we deploy simulations to understand and quantify the impact on disease transmission of a set of topological network features, building a dataset of 9000 interaction graphs using generators of different types of synthetic social networks. Independently of the topology of the network, we maintain constant the total volume of social interactions in our simulations, to show how even with the same social contact some network structures are more or less resilient to the spread. We find a suitable intervention to be specific suppression of unfamiliar and casual interactions that contribute to the network's global efficiency. This is, pathogen spread is significantly reduced by limiting specific kinds of contact rather than their global number. Our numerical studies might inspire further investigation in connection to public health, as an integrative framework to craft and evaluate social interventions in communicable diseases with different social graphs or as a highlight of network metrics that should be captured in social studies. The contact patterns that underlie disease transmission naturally form a network, where links join individuals that interact, and disease spreads along links. Computational models of infectious disease can support scenario analysis during epidemic outbreaks necessary to devise effective public health interventions. All epidemiological models make assumptions about the underlying network of interactions. Contact network models mathematically formalize this intuitive concept, so that epidemiological calculations explicitly consider complex patterns of interactions 1 . Recent studies 2 have shown that networks with equal number of nodes and edges, but different network structure (e.g. path length and clustering) lead to different infection curves. Most of the theoretical literature on this topic focuses on the effect on pathogen spread of individual network properties (e.g. degree distribution, assortative mixing or clustering), typically analyzed by controlled numerical experiments or analytical calculations, where only the property of interest is varied. However, in a realistic context, altering the structure of a network means simultaneously changing different network characteristics. This is because when one perturbs a network, increasing the variance of the degree distribution, for example, many other network metrics can also change, entangling the effect of different network metrics on spread. Even more, the size and complexity of the space of possible network characteristics makes the derivation of optimal metrics of spread from empirical data infeasible, for any candidate model is bound to be under determined by the scale and fidelity of available data. Rather we need large-scale simulations spanning the horizon of network parameters within which a network topology is bound to lie. Our simulations of the spread of a pathogen using a contact network model on more than 9000 synthetic social networks are a first step in this direction. Although all of the networks considered are synthetic, the families from which they are sampled have been shown to be representative in some instances of human interaction networks. Importantly, the number of social interactions per time step is kept constant for all of our simulations, independently of the network topology and average degree. That is, the same number of social interactions occur on average at each time step of the simulations (i.e., equal to the number of individuals). Note, however, that there are differences across individuals (we assume a linear relationship between the individual average degree and their number of interactions per time step) 9 . Thus, the differences reported across our simulations stem from which individuals in the population interact, rather than the total number of interactions. The analysis of our large-scale simulations demonstrate that metrics that were found to be predictive of the spread in the literature (through experiments that only varied the metric of interest), are not predictive when one considers a larger variety of network topologies. Instead, we show that specific suppression of unfamiliar and casual interactions that contribute to the network's global efficiency is a well suited intervention. Our conclusions

Mathematical Biosciences, 2002

We consider a spatial model related to bond percolation for the spread of a disease that includes variation in the susceptibility to infection. We work on a lattice with random bond strengths and show that with strong heterogeneity, i.e. a wide range of variation of susceptibility, patchiness in the spread of the epidemic is very likely, and the criterion for epidemic outbreak depends strongly on the heterogeneity. These results are qualitatively different from those of standard models in epidemiology, but correspond to real effects. We suggest that heterogeneity in the epidemic will affect the phylogenetic distance distribution of the disease-causing organisms. We also investigate small world lattices, and show that the effects mentioned above are even stronger.

PLoS Computational Biology, 2010

Human societies are organized in complex webs that are constantly reshaped by a social dynamic which is influenced by the information individuals have about others. Similarly, epidemic spreading may be affected by local information that makes individuals aware of the health status of their social contacts, allowing them to avoid contact with those infected and to remain in touch with the healthy. Here we study disease dynamics in finite populations in which infection occurs along the links of a dynamical contact network whose reshaping may be biased based on each individual's health status. We adopt some of the most widely used epidemiological models, investigating the impact of the reshaping of the contact network on the disease dynamics. We derive analytical results in the limit where network reshaping occurs much faster than disease spreading and demonstrate numerically that this limit extends to a much wider range of time scales than one might anticipate. Specifically, we show that from a population-level description, disease propagation in a quickly adapting network can be formulated equivalently as disease spreading on a well-mixed population but with a rescaled infectiousness. We find that for all models studied here -SI, SIS and SIR -the effective infectiousness of a disease depends on the population size, the number of infected in the population, and the capacity of healthy individuals to sever contacts with the infected. Importantly, we indicate how the use of available information hinders disease progression, either by reducing the average time required to eradicate a disease (in case recovery is possible), or by increasing the average time needed for a disease to spread to the entire population (in case recovery or immunity is impossible).

Physical Review E, 2001

Statistical properties and dynamical disease propagation have been studied numerically using a percolation model in a one dimensional small world network. The parameters chosen correspond to a realistic network of school age children. It has been found that percolation threshold decreases as a power law as the shortcut fluctuations increase. It has also been found that the number of infected sites grows exponentially with time and its rate depends logarithmically on the density of susceptibles. This behavior provides an interesting way to estimate the serology for a given population from the measurement of the disease growing rate during an epidemic phase. The case in which the infection probability of nearest neighbors is different from that of short cuts has also been examined. A double diffusion behavior with a slower diffusion between the characteristic times has been found.