A note on a paper by Erik Volz: SIR dynamics in random networks

Recent work by Volz (J Math Biol 56:293–310, 2008) has shown how to calculate the growth and eventual decay of an SIR epidemic on a static random network, assuming infection and recovery each happen at constant rates. This calculation allows us to account for effects due to heterogeneity and finiteness of degree that are neglected in the standard mass-action SIR equations. In this note we offer an alternate derivation which arrives at a simpler—though equivalent—system of governing equations to that of Volz. This new derivation is more closely connected to the underlying physical processes, and the resulting equations are of comparable complexity to the mass-action SIR equations. We further show that earlier derivations of the final size of epidemics on networks can be reproduced using the same approach, thereby providing a common framework for calculating both the dynamics and the final size of an epidemic spreading on a random network. Under appropriate assumptions these equations reduce to the standard SIR equations, and we are able to estimate the magnitude of the error introduced by assuming the SIR equations.     

Nonhomogeneous birth and death models for epidemic outbreak data

In this paper, generalized nonlinear models are proposed in order to incorporate the following considerations in modeling an epidemic disease outbreak statistically. (1) The dependence of the data is handled with a nonhomogeneous death or a nonhomogeneous birth process. (2) The first stage of the outbreak is described with an epidemic susceptibles-infectives-removed (SIR) model. Soon the control measures taken will dominate the process. These measures are in addition to the natural epidemic removal process. The prevalence is related to the censored infection times in such a way that the distribution function and thus the survival function satisfy approximately the first equation of the SIR model. This leads in a natural way to the Burr family of distributions. (3) The nonhomogeneous birth process handles the fact that in practice, with some delay, infecteds are registered, but not susceptibles. (4) Finally, the ending of the epidemic caused by the measures taken is incorporated through a modification of the survival function with a final-size parameter, in the same way as is done in long-term survival models. These models are applied to three outbreaks: The Dutch classical swine fever outbreak from 1997 to 1998, the foot- and-mouth disease outbreak in Great Britain from 2001, and the Dutch avian influenza (H7N7) outbreak from 2003.     

Network structure, and vaccination strategy and effort interact to affect the dynamics of influenza epidemics

There is growing interest in understanding and controlling the spread of diseases through realistically structured host populations. We investigate how network structures, ranging from circulant, through small-world networks, to random networks, and vaccination strategy and effort interact to influence the proportion of the population infected, the size and timing of the epidemic peak, and the duration of the epidemic. We found these three factors, and their higher-order interactions, significantly influenced epidemic development and extent. Increasing vaccination effort (from 0% to 90%) decreased the number of hosts infected while increasing network randomness worked to increase disease spread. On average, vaccinating hosts based on degree (hubs) resulted in the smallest epidemics while vaccinating hosts with the highest clustering coefficient resulted in the largest epidemics. In a targeted test of five vaccination strategies on a small-world network (probability of rewiring edges=5%) with 10% vaccination effort we found that vaccinating hosts preferentially with high-clustering coefficients (similar to real-world strategies) resulted in twice the number of hosts infected as random vaccinations and nearly a 30-fold higher number of cases than our strategy targeting hubs (highest degree hosts). Our model suggests how vaccinations might be implemented to minimize the extent of an epidemic (e.g., duration and total number infected) as well as the timing and number of hosts infected at a given time over a wide range of structured host networks.     

Modeling the simple epidemic with deterministic differential equations and random initial conditions

In a simple epidemic the only transition in the population is from susceptible to infected and the total population size is fixed for all time. This paper investigates the effect of random initial conditions on the deterministic model for the simple epidemic. By assuming a Beta distribution on the initial proportion of susceptibles, we define a distribution that describes the proportion of susceptibles in a population at any time during an epidemic. The mean and variance for this distribution are derived as hypergeometric functions, and the behavior of these functions is investigated. Lastly, we define a distribution to describe the time until a given proportion of the population remains susceptible. A method for finding the quantiles of this distribution is developed and used to make confidence statements regarding the time until a given proportion of the population is susceptible.     

Effects of heterogeneous and clustered contact patterns on infectious disease dynamics

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 far-reaching 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.     

Biological network comparison using graphlet degree distribution

MOTIVATION: Analogous to biological sequence comparison, comparing cellular networks is an important problem that could provide insight into biological understanding and therapeutics. For technical reasons, comparing large networks is computationally infeasible, and thus heuristics, such as the degree distribution, clustering coefficient, diameter, and relative graphlet frequency distribution have been sought. It is easy to demonstrate that two networks are different by simply showing a short list of properties in which they differ. It is much harder to show that two networks are similar, as it requires demonstrating their similarity in all of their exponentially many properties. Clearly, it is computationally prohibitive to analyze all network properties, but the larger the number of constraints we impose in determining network similarity, the more likely it is that the networks will truly be similar. RESULTS: We introduce a new systematic measure of a network's local structure that imposes a large number of similarity constraints on networks being compared. In particular, we generalize the degree distribution, which measures the number of nodes 'touching' k edges, into distributions measuring the number of nodes 'touching' k graphlets, where graphlets are small connected non-isomorphic subgraphs of a large network. Our new measure of network local structure consists of 73 graphlet degree distributions of graphlets with 2-5 nodes, but it is easily extendible to a greater number of constraints (i.e. graphlets), if necessary, and the extensions are limited only by the available CPU. Furthermore, we show a way to combine the 73 graphlet degree distributions into a network 'agreement' measure which is a number between 0 and 1, where 1 means that networks have identical distributions and 0 means that they are far apart. Based on this new network agreement measure, we show that almost all of the 14 eukaryotic PPI networks, including human, resulting from various high-throughput experimental techniques, as well as from curated databases, are better modeled by geometric random graphs than by Erd?s-Rény, random scale-free, or Barabási-Albert scale-free networks. AVAILABILITY: Software executables are available upon request.     

A Simulation Study Comparing Epidemic Dynamics on Exponential Random Graph and Edge-Triangle Configuration Type Contact Network Models

We compare two broad types of empirically grounded random network models in terms of their abilities to capture both network features and simulated Susceptible-Infected-Recovered (SIR) epidemic dynamics. The types of network models are exponential random graph models (ERGMs) and extensions of the configuration model. We use three kinds of empirical contact networks, chosen to provide both variety and realistic patterns of human contact: a highly clustered network, a bipartite network and a snowball sampled network of a “hidden population”. In the case of the snowball sampled network we present a novel method for fitting an edge-triangle model. In our results, ERGMs consistently capture clustering as well or better than configuration-type models, but the latter models better capture the node degree distribution. Despite the additional computational requirements to fit ERGMs to empirical networks, the use of ERGMs provides only a slight improvement in the ability of the models to recreate epidemic features of the empirical network in simulated SIR epidemics. Generally, SIR epidemic results from using configuration-type models fall between those from a random network model (i.e., an Erdős-Rényi model) and an ERGM. The addition of subgraphs of size four to edge-triangle type models does improve agreement with the empirical network for smaller densities in clustered networks. Additional subgraphs do not make a noticeable difference in our example, although we would expect the ability to model cliques to be helpful for contact networks exhibiting household structure.     

On the dynamics of predation risk perception for a vigilant forager

There has been much recent interest in modelling epidemics on networks, particularly in the presence of substantial clustering. Here, we develop pairwise methods to answer questions that are often addressed using epidemic models, in particular: on the basis of potential observations early in an outbreak, what can be predicted about the epidemic outcomes and the levels of intervention necessary to control the epidemic? We find that while some results are independent of the level of clustering (early growth predicts the level of ‘leaky’ vaccine needed for control and peak time, while the basic reproductive ratio predicts the random vaccination threshold) the relationship between other quantities is very sensitive to clustering.     

Network epidemic models with two levels of mixing

The study of epidemics on social networks has attracted considerable attention recently. In this paper, we consider a stochastic SIR (susceptible-->infective-->removed) model for the spread of an epidemic on a finite network, having an arbitrary but specified degree distribution, in which individuals also make casual contacts, i.e. with people chosen uniformly from the population. The behaviour of the model as the network size tends to infinity is investigated. In particular, the basic reproduction number R(0), that governs whether or not an epidemic with few initial infectives can become established is determined, as are the probability that an epidemic becomes established and the proportion of the population who are ultimately infected by such an epidemic. For the case when the infectious period is constant and all individuals in the network have the same degree, the asymptotic variance and a central limit theorem for the size of an epidemic that becomes established are obtained. Letting the rate at which individuals make casual contacts decrease to zero yields, heuristically, corresponding results for the model without casual contacts, i.e. for the standard SIR network epidemic model. A deterministic model that approximates the spread of an epidemic that becomes established in a large population is also derived. The theory is illustrated by numerical studies, which demonstrate that the asymptotic approximations work well, even for only moderately sized networks, and that the degree distribution and the inclusion of casual contacts can each have a major impact on the outcome of an epidemic.     

Epidemic threshold and network structure: The interplay of probability of transmission and of persistence in small-size directed networks

A growing number of studies are investigating the effect of contact structure on the dynamics of epidemics in large-scale complex networks. Whether findings thus obtained apply also to networks of small size, and thus to many real-world biological applications, is still an open question. We use numerical simulations of disease spread in directed networks of 100 individual nodes with a constant number of links. We show that, no matter the type of network structure (local, small-world, random and scale-free), there is a linear threshold determined by the probability of infection transmission between connected nodes and the probability of infection persistence in an infected node. The threshold is significantly lower for scale-free networks compared to local, random and small-world ones only if super-connected nodes have a higher number of links both to and from other nodes. The starting point, the node at which the epidemic starts, does not affect the threshold conditions, but has a marked influence on the final size of the epidemic in all kinds of network. There is evidence that contact structure has an influence on the average final size of an epidemic across all starting nodes, with significantly lower values in scale-free networks at equilibrium. Simulations in scale-free networks show a distinctive time-series pattern, which, if found in a real epidemic, can be used to infer the underlying network structure. The findings have relevance also for meta-population ecology and species conservation.     

The relationship between real-time and discrete-generation models of epidemic spread

Many important results in stochastic epidemic modelling are based on the Reed-Frost model or on other similar models that are characterised by unrealistic temporal dynamics. Nevertheless, they can be extended to many other more realistic models thanks to an argument first provided by Ludwig [Final size distributions for epidemics, Math. Biosci. 23 (1975) 33-46], that states that, for a disease leading to permanent immunity after recovery, under suitable conditions, a continuous-time infectious process has the same final size distribution as another more tractable discrete-generation contact process; in other words, the temporal dynamics of the epidemic can be neglected without affecting the final size distribution. Despite the importance of such an argument, its presence behind many results is often not clearly stated or hidden in references to previous results. In this paper, we reanalyse Ludwig’s result, highlighting some of the conditions under which it does not hold and providing a general framework to examine the differences between the continuous-time and the discrete-generation process.     

Small amplitude,long period outbreaks in seasonally driven epidemics

It is now documented that childhood diseases such as measles, mumps, and chickenpox exhibit a wide range of recurrent behavior (periodic as well as chaotic) in large population centers in the first world. Mathematical models used in the past (such as the SEIR model with seasonal forcing) have been able to predict the onset of both periodic and chaotic sustained epidemics using parameters of childhood diseases. Although these models possess stable solutions which appear to have the correct frequency content, the corresponding outbreaks require extremely large populations to support the epidemic. This paper shows that by relaxing the assumption of uniformity in the supply of susceptibles, simple models predict stable long period oscillatory epidemics having small amplitude. Both coupled and single population models are considered.     

Inferring epidemic contact structure from phylogenetic trees

Contact structure is believed to have a large impact on epidemic spreading and consequently using networks to model such contact structure continues to gain interest in epidemiology. However, detailed knowledge of the exact contact structure underlying real epidemics is limited. Here we address the question whether the structure of the contact network leaves a detectable genetic fingerprint in the pathogen population. To this end we compare phylogenies generated by disease outbreaks in simulated populations with different types of contact networks. We find that the shape of these phylogenies strongly depends on contact structure. In particular, measures of tree imbalance allow us to quantify to what extent the contact structure underlying an epidemic deviates from a null model contact network and illustrate this in the case of random mixing. Using a phylogeny from the Swiss HIV epidemic, we show that this epidemic has a significantly more unbalanced tree than would be expected from random mixing.     

Effectiveness of realistic vaccination strategies for contact networks of various degree distributions

A "contact network" that models infection transmission comprises nodes (or individuals) that are linked when they are in contact and can potentially transmit an infection. Through analysis and simulation, we studied the influence of the distribution of the number of contacts per node, defined as degree, on infection spreading and its control by vaccination. Three random contact networks of various degree distributions were examined. In a scale-free network, the frequency of high-degree nodes decreases as the power of the degree (the case of the third power is studied here); the decrease is exponential in an exponential network, whereas all nodes have the same degree in a constant network. Aiming for containment at a very early stage of an epidemic, we measured the sustainability of a specific network under a vaccination strategy by employing the critical transmissibility larger than which the epidemic would occur. We examined three vaccination strategies: mass, ring, and acquaintance. Irrespective of the networks, mass preventive vaccination increased the critical transmissibility inversely proportional to the unvaccinated rate of the population. Ring post-outbreak vaccination increased the critical transmissibility inversely proportional to the unvaccinated rate, which is the rate confined to the targeted ring comprising the neighbors of an infected node; however, the total number of vaccinated nodes could mostly be fewer than 100 nodes at the critical transmissibility. In combination, mass and ring vaccinations decreased the pathogen's "effective" transmissibility each by the factor of the unvaccinated rate. The amount of vaccination used in acquaintance preventive vaccination was lesser than the mass vaccination, particularly under a highly heterogeneous degree distribution; however, it was not as less as that used in ring vaccination. Consequently, our results yielded a quantitative assessment of the amount of vaccination necessary for infection containment, which is universally applicable to contact networks of various degree distributions.     

Network frailty and the geometry of herd immunity

The spread of infectious disease through communities depends fundamentally on the underlying patterns of contacts between individuals. Generally, the more contacts one individual has, the more vulnerable they are to infection during an epidemic. Thus, outbreaks disproportionately impact the most highly connected demographics. Epidemics can then lead, through immunization or removal of individuals, to sparser networks that are more resistant to future transmission of a given disease. Using several classes of contact networks-Poisson, scale-free and small-world-we characterize the structural evolution of a network due to an epidemic in terms of frailty (the degree to which highly connected individuals are more vulnerable to infection) and interference (the extent to which the epidemic cuts off connectivity among the susceptible population that remains following an epidemic). The evolution of the susceptible network over the course of an epidemic differs among the classes of networks; frailty, relative to interference, accounts for an increasing component of network evolution on networks with greater variance in contacts. The result is that immunization due to prior epidemics can provide greater community protection than random vaccination on networks with heterogeneous contact patterns, while the reverse is true for highly structured populations.     

Community Size Effects on Epidemic Spreading in Multiplex Social Networks

The dynamical process of epidemic spreading has drawn much attention of the complex network community. In the network paradigm, diseases spread from one person to another through the social ties amongst the population. There are a variety of factors that govern the processes of disease spreading on the networks. A common but not negligible factor is people’s reaction to the outbreak of epidemics. Such reaction can be related information dissemination or self-protection. In this work, we explore the interactions between disease spreading and population response in terms of information diffusion and individuals’ alertness. We model the system by mapping multiplex networks into two-layer networks and incorporating individuals’ risk awareness, on the assumption that their response to the disease spreading depends on the size of the community they belong to. By comparing the final incidence of diseases in multiplex networks, we find that there is considerable mitigation of diseases spreading for full phase of spreading speed when individuals’ protection responses are introduced. Interestingly, the degree of community overlap between the two layers is found to be critical factor that affects the final incidence. We also analyze the consequences of the epidemic incidence in communities with different sizes and the impacts of community overlap between two layers. Specifically, as the diseases information makes individuals alert and take measures to prevent the diseases, the effective protection is more striking in small community. These phenomena can be explained by the multiplexity of the networked system and the competition between two spreading processes.     

The number of links to and from the starting node as a predictor of epidemic size in small-size directed networks

Much recent modelling is focusing on epidemics in large-scale complex networks. Whether or not findings of these investigations also apply to networks of small size is still an open question. This is an important gap for many biological applications, including the spread of the oomycete pathogen Phytophthora ramorum in networks of plant nurseries. We use numerical simulations of disease spread and establishment in directed networks of 100 individual nodes at four levels of connectivity. Factors governing epidemic spread are network structure (local, small-world, random, scale-free) and the probabilities of infection persistence in a node and of infection transmission between connected nodes. Epidemic final size at equilibrium varies widely depending on the starting node of infection, although the latter does not affect the threshold condition for spread. The number of links from (out-degree) but not the number of links to (in-degree) the starting node of the epidemic explains a substantial amount of variation in final epidemic size at equilibrium irrespective of the structure of the network. The proportion of variance in epidemic size explained by the out-degree of the starting node increases with the level of connectivity. Targeting highly connected nodes is thus likely to make disease control more effective also in case of small-size populations, a result of relevance not just for the horticultural trade, but for epidemiology in general.     

An assessment of preferential attachment as a mechanism for human sexual network formation

Recent research into the properties of human sexual-contact networks has suggested that the degree distribution of the contact graph exhibits power-law scaling. One notable property of this power-law scaling is that the epidemic threshold for the population disappears when the scaling exponent rho is in the range 2 < rho < or = 3. This property is of fundamental significance for the control of sexually transmitted diseases (STDs) such as HIV/AIDS since it implies that an STD can persist regardless of its transmissibility. A stochastic process, known as preferential attachment, that yields one form of power-law scaling has been suggested to underlie the scaling of sexual degree distributions. The limiting distribution of this preferential attachment process is the Yule distribution, which we fit using maximum likelihood to local network data from samples of three populations: (i) the Rakai district, Uganda; (ii) Sweden; and (iii) the USA. For all local networks but one, our interval estimates of the scaling parameters are in the range where epidemic thresholds exist. The estimate of the exponent for male networks in the USA is close to 3, but the preferential attachment model is a very poor fit to these data. We conclude that the epidemic thresholds implied by this model exist in both single-sex and two-sex epidemic model formulations. A strong conclusion that we derive from these results is that public health interventions aimed at reducing the transmissibility of STD pathogens, such as implementing condom use or high-activity anti-retroviral therapy, have the potential to bring a population below the epidemic transition, even in populations exhibiting large degrees of behavioural heterogeneity.     

Assessment of the prevalence of vCJD through testing tonsils and appendices for abnormal prion protein

The objective of this study was to determine the age group or groups which will provide the most information on the potential size of the vCJD epidemic in Great Britain via the sampling of tonsil and appendix material to detect the presence of abnormal prion protein (PrP(Sc)). A subsidiary aim was to determine the degree to which such an anonymous age-stratified testing programme will reduce current uncertainties in the size of the epidemic in future years. A cohort- and time-stratified model was used to generate epidemic scenarios consistent with the observed vCJD case incidence. These scenarios, together with data on the age distribution of tonsillectomies and appendectomies, were used to evaluate the optimal age group and calendar time for undertaking testing and to calculate the range of epidemic sizes consistent with different outcomes. The analyses suggested that the optimal five-year age group to test is 25-29 years, although a random sample of appendix tissue from all age groups is nearly as informative. A random sample of tonsil tissue from all age groups is less informative, but the information content is improved if sampling is restricted to tissues removed from those over ten years of age. Based on the assumption that the test is able to detect infection in the last 75% of the incubation period, zero detected infections in an initial random sample of 1000 tissues would suggest that the epidemic will be less than 870,000 cases. If infections are detected, then the model prediction suggests that both relatively small epidemics (800+ cases if one is detected or 8300+ if two are detected) and larger epidemics (21,000+ cases if three or more are detected) are possible. It was concluded that testing will be most informative if undertaken using appendix tissues or tonsil tissues removed from those over ten years of age. Large epidemics can only be excluded if a small number of infections are detected and the test is able to detect infection early in the incubation period.