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Open AcceResearchEpigrass: a tool to study disease spread in complex networksFlávio C Coelho*, Oswaldo G Cruz and Cláudia T Codeço

Address: Programa de Computação Científica, Fundação Oswaldo Cruz, Rio de Janeiro, Brasil

Email: Flávio C Coelho* - fccoelho@fiocruz.br; Oswaldo G Cruz - oswaldo@fiocruz.br; Cláudia T Codeço - codeco@fiocruz.br

* Corresponding author

AbstractBackground: The construction of complex spatial simulation models such as those used innetwork epidemiology, is a daunting task due to the large amount of data involved in theirparameterization. Such data, which frequently resides on large geo-referenced databases, has to beprocessed and assigned to the various components of the model. All this just to construct themodel, then it still has to be simulated and analyzed under different epidemiological scenarios. Thisworkflow can only be achieved efficiently by computational tools that can automate most, if not all,these time-consuming tasks. In this paper, we present a simulation software, Epigrass, aimed to helpdesigning and simulating network-epidemic models with any kind of node behavior.

Results: A Network epidemiological model representing the spread of a directly transmitteddisease through a bus-transportation network connecting mid-size cities in Brazil. Results showthat the topological context of the starting point of the epidemic is of great importance from bothcontrol and preventive perspectives.

Conclusion: Epigrass is shown to facilitate greatly the construction, simulation and analysis ofcomplex network models. The output of model results in standard GIS file formats facilitate thepost-processing and analysis of results by means of sophisticated GIS software.

BackgroundEpidemic models describe the spread of infectious dis-eases in populations. More and more, these models arebeing used for predicting, understanding and developingcontrol strategies. To be used in specific contexts, mode-ling approaches have shifted from "strategic models"(where a caricature of real processes is modeled in orderto emphasize first principles) to "tactical models"(detailed representations of real situations). Tactical mod-els are useful for cost-benefit and scenario analyses. Goodexamples are the foot-and-mouth epidemic models forUK, triggered by the need of a response to the 2001 epi-demic [1,2] and the simulation of pandemic flu in differ-

ent scenarios helping authorities to choose amongalternative intervention strategies [3,4].

In realistic epidemic models, a key issue to consider is therepresentation of the contact process through which a dis-ease is spread, and network models have arisen as goodcandidates [5]. This has led to the development of "net-work epidemic models". Network is a flexible conceptthat can be used to describe, for example, a collection ofindividuals linked by sexual partnerships [6], a collectionof families linked by sharing workplaces/schools [7], acollection of cities linked by air routes [8]. Any of thesescales may be relevant to the study and control of diseasespread [9].

Published: 26 February 2008

Source Code for Biology and Medicine 2008, 3:3 doi:10.1186/1751-0473-3-3

Received: 10 July 2007Accepted: 26 February 2008

This article is available from: http://www.scfbm.org/content/3/1/3

© 2008 Coelho et al; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Networks are made of nodes and their connections. Onemay classify network epidemic models according to nodebehavior. One example would be a classification based onthe states assumed by the nodes: networks with discrete-state nodes have nodes characterized by a discrete variablerepresenting its epidemiological status (for example, sus-ceptible, infected, recovered). The state of a node changesin response to the state of neighbor nodes, as defined bythe network topology and a set of transmission rules. Net-works with continuous-state nodes, on the other hand,have node' state described by a quantitative variable(number of susceptibles, density of infected individuals,for example), modelled as a function of the history of thenode and its neighbors. The importance of the concept ofneighborhood on any kind of network epidemic modelstems from its large overlap with the concept of transmis-sion. In network epidemic models, transmission eitherdefines or is defined/constrained by the neighborhoodstructure. In the latter case, a neighborhood structure isgiven a priori which will influence transmissibilitybetween nodes.

The construction of complex simulation models such asthose used in network epidemic models, is a daunting taskdue to the large amount of data involved in their parame-terization. Such data frequently resides on large geo-refer-enced databases. This data has to be processed andassigned to the various components of the model. All thisjust to construct the model, then it still has to be simu-lated, analyzed under different epidemiological scenarios.This workflow can only be achieved efficiently by compu-tational tools that can automate most if not all of thesetime-consuming tasks.

In this paper, we present a simulation software, Epigrass,aimed to help designing and simulating network-epi-demic models with any kind of node behavior. Withoutsuch a tool, implementing network epidemic models isnot a simple task, requiring a reasonably good knowledgeof programming. We expect that this software will stimu-late the use and development of networks models for epi-demiological purposes. The paper is organized asfollowing: first we describe the software and how it isorganized with a brief overview of its functionality. Thenwe demonstrate its use with an example. The examplesimulates the spread of a directly transmitted infectiousdisease in Brazil through its transportation network. Thevelocity of spread of new diseases in a network of suscep-tible populations depends on their spatial distribution,size, susceptibility and patterns of contact. In a spatialscale, climate and environment may also impact thedynamics of geographical spread as it introduces temporaland spatial heterogeneity. Understanding and predictingthe direction and velocity of an invasion wave is key foremergency preparedness.

EpigrassEpigrass is a platform for network epidemiological simu-lation and analysis. It enables researchers to perform com-prehensive spatio-temporal simulations incorporatingepidemiological data and models for disease transmissionand control in order to create complex scenario analyses.Epigrass is designed towards facilitating the constructionand simulation of large scale metapopulational models.Each component population of such a metapopulationalmodel is assumed to be connected through a contact net-work which determines migration flows between popula-tions. This connectivity model can be easily adapted torepresent any type of adjacency structure.

Epigrass is entirely written in the Python language, whichcontributes greatly to the flexibility of the whole systemdue to the dynamical nature of the language. The geo-ref-erenced networks over which epidemiological processestake place can be very straightforwardly represented in aobject-oriented framework. Consequently, the nodes andedges of the geographical networks are objects with theirown attributes and methods (figure 1).

Once the archetypal node and edge objects are definedwith appropriate attributes and methods, then a code rep-resentation of the real system can be constructed, wherenodes (representing people or localities) and contactroutes are instances of node and edge objects, respectively.The whole network is also an object with its own set ofattributes and methods. In fact, Epigrass also allows formultiple edge sets in order to represent multiple contactnetworks in a single model.

Architecture of an Epigrass simulation modelFigure 1Architecture of an Epigrass simulation model. A simu-lation object contains the whole model and all other objects representing the graph, sites and edges. Site object contaim model objects, which can be one of the built-in epidemiologi-cal models or a custom model written by the user.

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These features leads to a compact and hierarchical compu-tational model consisting of a network object containing avariable number of node and edge objects. It also does notpose limitations to encapsulation, potentially allowing fornetworks within networks, if desirable. This representationcan also be easily distributed over a computational grid orcluster, if the dependency structure of the whole modeldoes not prevent it (this feature is currently being imple-mented and will be available on a future release of Epi-grass). For the end-user, this hierarchical, object-orientedrepresentation is not an obstacle since it reflects the naturalstructure of the real system. Even after the model is con-verted into a code object, all of its component objectsremain accessible to one another, facilitating the exchangeof information between all levels of the model, a feature theuser can easily include in his/her custom models.

Nodes and edges are dynamical objects in the sense thatthey can be modified at runtime altering their behavior inresponse to user defined events. In Epigrass it is very easyto simulate any dynamical system embedded in a net-work. However, it was designed with epidemiologicalmodels in mind. This goal led to the inclusion of a collec-tion of built-in epidemic models which can be readilyused for the intra-node dynamics (SIR model family). Epi-grass users are not limited to basing their simulations onthe built-in models. User-defined models can be devel-oped in just a few lines of Python code. All simulations inEpigrass are done in discrete-time. However, custom mod-els may implement finer dynamics within each time step,by implementing ODE models at the nodes, for instance.

The Epigrass system is driven by a graphical user inter-face(GUI), which handles several input files required formodel definition and manages the simulation and outputgeneration (figure 2).

At the core of the system lies the simulator. It parses themodel specification files, contained in a text file (.epgfile), and builds the network from site and edge descrip-tion files (comma separated values text files, CSV). Thesimulator then builds a code representation of the entiremodel, simulates it, and stores the results in the databaseor in a couple of CSV files. This output will contain the fulltime series of the variables in the model. Additionally, amap layer (in shapefile and KML format) is also generatedwith summary statitics for the model (figure 3).

The results of an Epigrass simulation can be visualized indifferent ways. A map with an animation of the resultingtimeseries is available directly through the GUI (figure 4).Other types of static visualizations can be generatedthrough GIS software from the shapefiles generated. TheKML file can also be viewed in Google Earth™ or GoogleMaps™ (figure 5).

Epigrass also includes a report generator module which iscontrolled through a parameter in the ".epg" file.

Epigrass is capable of generating PDF reports with sum-mary statistics from the simulation. This module requiresa LATEX installation to work. Reports are most useful forgeneral verification of expected model behavior and net-work structure. However, the LATEX source files generated

Workflow for a typical Epigrass simulationFigure 3Workflow for a typical Epigrass simulation. This dia-gram shows all inputs and outputs typical of an Epigrass simu-lation session.

Epigrass graphical user interfaceFigure 2Epigrass graphical user interface.

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by the module may serve as templates that the user canedit to generate a more complete document.

Building a model in Epigrass is very simple, especially ifthe user chooses to use one of the built-in models. Epi-grass includes 20 different epidemic models ready to beused (See manual for built-in models description).

To run a network epidemic model in Epigrass, the user isrequired to provide three separate text files (Optionally,also a shapefile with the map layer):

1. Node-specification file: This file can be edited on aspreadsheet and saved as a csv file. Each row is a node andthe columns are variables describing the node.

2. Edge-specification file: This is also a spreadsheet-likefile with an edge per row. Columns contain flow variables.

3. Model-specification file: Also referred to as the ".epg"file. This file specifies the epidemiological model to berun at the nodes, its parameters, flow model for the edges,and general parameters of the simulation.

The ".epg" file is normally modified from templatesincluded with Epigrass. Nodes and edges files on the otherhand, have to be built from scratch for every new network.Details of how to construct these files, as well as examples,

can be found in the documentation accompanying thesoftware, which is available at at the project's website [10]

MethodsExample modelIn the example application, the spread of a respiratory dis-ease through a network of cities connected by bus trans-portation routes is analyzed.

The epidemiological scenario is one of the invasion of anew influenza-like virus. One may want to simulate thespread of this disease through the country by the transpor-tation network to evaluate alternative intervention strate-gies (e.g. different vaccination strategies). In this problem,a network can be defined as a set of nodes and links wherenodes represent cities and links represents transportationroutes. Some examples of this kind of model are availablein the literature [8,11].

One possible objective of this model is to understand howthe spread of such a disease may be affected by the point-of-entry of the disease in the network. To that end, we maylook at variables such as the speed of the epidemic,number of cases after a fixed amount of time, the distribu-tion of cases in time and the path taken by the spread.

The example network was built from 76 of largest cities ofBrazil (>= 100 k habs). The bus routes between those cit-ies formed the connections between the nodes of the net-works. The number of edges in the network, derived from

Epigrass output visualized on Google-EarthFigure 5Epigrass output visualized on Google-Earth.

Epigrass animation outputFigure 4Epigrass animation output. Sites are color coded (from red to blue) according to infection times. Bright red is the seed site (on the NE).

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the bus routes, is 850. These bus routes are registered withthe National Agency of Terrestrial Transportation (ANTT)which provided the data used to parameterize the edges ofthe network.

The modelThe epidemiological model used consisted of a metapop-ulation system with a discrete-time SEIR model (Eq. 1).For each city, St is the number of susceptibles in the city attime t, Et is the number of infected but not yet infectiousindividuals, It is the number of infectious individuals res-ident in the locality, N is the population residing in thelocality (assumed constant throughout the simulation),and nt is the number of individuals visiting the locality, Θtis the number of visitors who are infectious. The parame-ters used were taken from Lipsitch et al. (2003) [12] torepresent a disease like SARS with an estimated basicreproduction number (R0) of 2.2 to 3.6 (Table 1).

To simulate the spread of infection between cities, weused the concept of a "forest fire" model [13]. An infectedindividual, traveling to another city, acts as a spark thatmay trigger an epidemic in the new locality. This approachis based on the assumption that individuals commutebetween localities and contribute temporarily to thenumber of infected in the new locality, but not to itsdemography. Implications of this approach are discussedin Grenfell et al (2001) [13].

The number of individuals arriving in a city (nt) is basedon annual total number of passengers arriving trough allbus routes leading to that city as provided by the ANTT(Brazilian National Agency for Terrestrial Transportation).The annual number of passengers is used to derive anaverage daily number of passengers simply by dividing itby 365.

Stochasticity is introduced in the model at two points: thenumber of new cases is draw from a Poisson distribution

with intensity and the number of infected indi-

viduals visiting i is modelled as binomial process:

where n is the total number of passengers arriving from agiven neighboring city; Ik, t and Nk are the current numberof infectious individuals and the total population size ofcity k, respectively. δ is the delay associated with the dura-tion of each bus trip. The delay δ was calculated as thenumber of days (rounded down) that a bus, traveling atan average speed of 60 km/h, would take to complete agiven trip. The lengths in kilometers of all bus routes werealso obtained from the ANTT.

Vaccination campaigns in specific (or all) cities can be eas-ily attained in Epigrass, with individual coverages for eachcampaign on each city. We use this feature to explore Vac-cination scenarios in this model (figures 6 and 7).

The files with this model's definition(the sites, edges and".epg" files) are available as part of the Additional files 1,2 and 3 for this article.

AnalysisTo determine the importance of the point of entry in theoutcome of the epidemic, the model was run 500 times,randomizing the point of entry of the virus. The seedingsite was chosen with a probability proportional to thelog10 of their population size. These replicates were runusing Epigrass' built-in support for repeated runs with theoption of randomizing seeding site. For every simulation,statistics about each site such as the time it got infectedand time series of incidence were saved.

The time required for the epidemic to infect 50% of thecities was chosen as a global index to network susceptibil-ity to invasion. To compare the relative exposure of citiesto disease invasion, we also calculated the inverse of time

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Table 1: Parameters used in the models and their meaning. Parameter n and θ have their values derived stochatically during the simularion, therefore their values are not given here.

Symbol

Meaning. Value

β contact rate (t-1) [1.4, 2.27]1

θ number of infectious visitorsα mixing parameter 1n number of visitorsN population (S + E + I + R) city populationr fraction of I recovering from

infection (day-1)0.2

e fraction of E becoming infectious (day-1)

0.2

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elapsed from the beginning of the epidemic until the cityregistered its first indigenous case as a local measure ofexposure.

Except for population size, all other epidemiologicalparameters were the same for all cities, that is, diseasetransmissibility and recovery rate. Some positional fea-tures of each node were also derived: Centrality, which is isa measure derived from the average distance of a given siteto every other site in the network; Betweeness, which is thenumber of times a node figures in the the shortest pathbetween any other pair of nodes; and Degree, which is thenumber of edges connected to a node.

In order to analyze the path of the epidemic spread, wealso recorded which cities provided the infectious caseswhich were responsible for the infection of each othercity. If more than one source of infection exists, Epigrassselects the city which contributed with the largest number

Cost in vaccines applied vs. benefit in cases avoided, for a simulated epidemic starting at the highest degree city (São Paulo)Figure 6Cost in vaccines applied vs. benefit in cases avoided, for a simulated epidemic starting at the highest degree city (São Paulo).

Cost in vaccines applied vs. benefit in cases avoided, for a simulated epidemic starting at a relatively low degree city(Salvador)Figure 7Cost in vaccines applied vs. benefit in cases avoided, for a simulated epidemic starting at a relatively low degree city(Salvador).

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of infectious individuals at that time-step, as the mostlikely infector. At the end of the simulation Epigrass gen-erates a file with the dispersion tree in graphML format,which can be read by a variety of graph plotting programsto generate the graphic seen on figure 8.

Results and discussionScalability of epigrassThe computational cost of running a single time step in anepigrass model, is mainly determined by the cost of calcu-lating the epidemiological models on each site(node).Therefore, time required to run models based on larger

networks should scale linearly with the size of the net-work (order of the graph), for simulations of the sameduration. The model presented here, took 2.6 seconds fora 100 days run, on a 2.1 GHz cpu. A somewhat largermodel with 343 sites and 8735 edges took 28 seconds fora 100 days simulation. Very large networks may be limitedby the ammount of RAM available. The authors are work-ing on adapting Epigrass to distribute processing amongmultiple cpus(in SMP systems), or multiple computers ina cluster system. The memory demands can also beaddressed by keeping the simulation objects on an object-oriented database during the simulation. Steps in thisdirection are also being taken by the development team.

Epidemiological analysesThe model presented here served maily the purpose ofillustrating the capabilities of Epigrass for simulating andanalyzing reasonably complex epidemic scenarios. Itshould not be taken as a careful and complete analysis ofa real epidemic. Despite that, some features of the simu-lated epidemic are worth discussing. For example: Thespread speed of the epidemic, measured as the time takento infect 50% of the cities, was found to be influenced bythe centrality and degree of the entry node (figures 9 and10). The dispersion tree corresponding to the epidemic, isgreatly influenced by the degree of the point of entry of

Spread of the epidemic starting at the city of Salvador, a city with relatively small degree (that is, small number of neigh-bors)Figure 8Spread of the epidemic starting at the city of Salva-dor, a city with relatively small degree (that is, small number of neighbors). The number next to the boxes indicated the day when each city developed its first indige-nous case.

Effect of degree(a) and betweeness(b) of entry node to the speed of the epidemicFigure 9Effect of degree(a) and betweeness(b) of entry node to the speed of the epidemic.

Effect of betweeness of entry node on the speed of the epi-demicFigure 10Effect of betweeness of entry node on the speed of the epidemic.

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the disease in the network. Figure 8 shows the tree for thedispersion from the city of Salvador.

Vaccination strategies must take into consideration net-work topology. Figures 6 and 7 show cost benefit plots forthree vaccination strategies investigated: Uniform vaccina-tion, top-3 degree sites only and top-10 degree sites only. Vac-cination of higher order sites offer cost/benefit advantagesonly in scenarios where the disease enter the networkthrough one of these sites.

ConclusionEpigrass facilitates greatly the simulation and analysis ofcomplex network models. The output of model results instandard GIS file formats facilitates the post-processingand analysis of results by means of sophisticated GIS soft-ware. The non-trivial task of specifying the network overwhich the model will be run, is left to the user. But epi-grass allows this structure to be provided as a simple listof sites and edges on text files, which can easily becontructed by the user using a spreadsheet, with no needfor special software tools.

Besides invasion, network epidemiological models canalso be used to understand patterns of geographicalspread of endemic diseases [14-17]. Many infectious dis-eases can only be maintained in a endemic state in citieswith population size above a threshold, or under appro-priate environmental conditions(climate, availability of areservoir, vectors, etc). The variables and the magnitudesassociated with endemicity threshold depends on the nat-ural history of the disease [18]. Theses magnitudes mayvary from place to place as it depends on the contact struc-ture of the individuals. Predicting which cities are sourcesfor the endemicity and understanding the path of recur-rent traveling waves may help us to design optimal sur-veillance and control strategies.

Authors' contributionsFCC contributed with the software development, modeldefinition and analysis as well as general manuscript con-ception and writing. CTC contributed with model defini-tion and implementation, as well as with writing themanuscript. OGC, contributed with data analysis andwriting the manuscript. All authors have read andapproved the final version of the manuscript.

Additional material

AcknowledgementsThe Authors would like to thank the Brazilian Research Council (CNPq) for financial support to the authors.

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Additional file 1

Click here for file[http://www.biomedcentral.com/content/supplementary/1751-0473-3-3-S1.csv]

Additional file 2

Click here for file[http://www.biomedcentral.com/content/supplementary/1751-0473-3-3-S2.csv]

Additional file 3

Click here for file[http://www.biomedcentral.com/content/supplementary/1751-0473-3-3-S3.epg]

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16. Raffy M, Tran A: On the dynamics of flying insects populationscontrolled by large scale information. Theor Popul Biol 2005,68(291-104 [http://dx.doi.org/10.1016/j.tpb.2005.03.005].

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18. Keeling MJ, Grenfell BT: Disease extinction and communitysize: modeling the persistence of measles. Science 1997,275(5296):65-67.

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