Computer Science Department, Carlos III University of Madrid, Avda. de la Universidad 30, 28911, Leganés, Madrid, Spain
Abstract
Background
To understand how infectious agents disseminate throughout a population it is essential to capture the social model in a realistic manner. This paper presents a novel approach to modeling the propagation of the influenza virus throughout a realistic interconnection network based on actual individual interactions which we extract from online social networks. The advantage is that these networks can be extracted from existing sources which faithfully record interactions between people in their natural environment. We additionally allow modeling the characteristics of each individual as well as customizing his daily interaction patterns by making them timedependent. Our purpose is to understand how the infection spreads depending on the structure of the contact network and the individuals who introduce the infection in the population. This would help public health authorities to respond more efficiently to epidemics.
Results
We implement a scalable, fully distributed simulator and validate the epidemic model by comparing the simulation results against the data in the 20042005 New York State Department of Health Report (NYSDOH), with similar temporal distribution results for the number of infected individuals. We analyze the impact of different types of connection models on the virus propagation. Lastly, we analyze and compare the effects of adopting several different vaccination policies, some of them based on individual characteristics such as age while others targeting the superconnectors in the social model.
Conclusions
This paper presents an approach to modeling the propagation of the influenza virus via a realistic social model based on actual individual interactions extracted from online social networks. We implemented a scalable, fully distributed simulator and we analyzed both the dissemination of the infection and the effect of different vaccination policies on the progress of the epidemics. The epidemic values predicted by our simulator match real data from NYSDOH. Our results show that our simulator can be a useful tool in understanding the differences in the evolution of an epidemic within populations with different characteristics and can provide guidance with regard to which, and how many, individuals should be vaccinated to slow down the virus propagation and reduce the number of infections.
Background
In a world that is becoming more interconnected every day we find ourselves with increased frequency being in close vicinity to people that are outside our normal environment. To understand how infectious agents disseminate throughout a population it seems therefore essential to model the social model in a realistic manner. Monitoring the actual interactions between people is in general unrealistic, although it is plausible in time and spacerestricted environments. Largescale realistic population modeling is plagued with problems of being time and effort consuming; to add to this, individual contacts are normally either estimated or based on selfreported data. Lastly, while the insight gathered by experimenting with such a model could definitely be used for similar social environments, it remains to be understood what precisely determines this similarity. On the other hand, localscale modeling may be very precise but involves issues of consent and privacy as study participants usually need to agree to wearing some kind of a tracking device. It is unclear whether the local behavior of people that work in the same place, or attend the same event, can be extrapolated to global behavior.
Approach
Under these circumstances, we approach the problem from a novel angle: we approximate the actual social model by using contacts extracted from real social networks. The advantage is that these networks can be extracted from already existing sources and they faithfully record interactions between people in their natural environment. Our purpose is to understand how the infection spreads depending on the structure of the contact network and the individuals who introduce the infection in the population. This would help public health authorities to respond more efficiently to an epidemic since it would answer questions such as: How many people will be affected at any given time and how does the epidemic propagate? How many individuals will need hospitalization and treatment? How many individuals and which would need to be targeted to stop, or at least slow down, an epidemic? What would be an effective vaccination policy to implement? How long will the epidemics last with and without intervention? This work is a step towards successfully addressing these issues. More specifically, the purpose of the work we present in this paper is to accurately model the evolution of an epidemic in specific populations over a short to medium time span depending on the characteristics of the social model. Based on the dissemination patterns we observe, we study which vaccination policies are more successful than others in reducing the number of infected individuals and delaying the peak of infection. As part of this analysis, we need to asses to what extent social networks are a good approximation for facetoface contacts. Modeling the evolution of an epidemic involves modeling both the behavior of the specific infectious agent as well as the social structure of the population under study. In most existing approaches the population model is built based on using probability distributions to approximate the number of individual interactions. Some other approaches synthetically generate the interaction graphs
Related work
Interconnection networks
The majority of humantransmitted infectious diseases use physical contact as the main transmission mean. For this reason the dynamics of the propagation is tightly related to the structure and the characteristics of the network of connections between the individuals within a population
A different approach is followed by BioWar
Epidemic models
The typical mathematical model for simulating epidemics is the SIR model
Our contributions
The specific contributions of this work are the following:
• Population: We use real demographic data extracted from the U.S. Census to model group types with different characteristics. At the level of the individual, we allow modeling characteristics such as age, gender, and race.
• Contacts: We leverage data extracted from social networks to model the interaction patterns between individuals pertaining to the same social group. We allow customizing individual interaction behavior based on the day of the week and the time of day.
• Simulator: We implement a scalable, fully distributed simulator and we evaluate its performance on two platforms: a distributed memory multiprocessor cluster and a shared memory multicore processor.
• Results: We validate the results of the simulation against real data obtained from NYSDOH. We investigate the virus dissemination process and compare it with dissemination in networks which have exponential and normal contact distributions, as well as in a social model without timedependent interactions. We additionally study how infecting different type of individuals may affect the epidemic.
• Vaccination: We analyze and compare the impact of different vaccination policies on managing the virus dissemination process.
We first describe the modeling task and the simulation algorithm, followed by the analysis we undergo to understand the impact on the epidemics of the network structure and of the characteristics of the individuals that introduce the virus in the population. We then present and discuss the performance and simulation results of EpiGraph, including those for vaccination. We summarize the paper with the conclusions and some directions for future work.
Methods
The modeling task
This work focuses on understanding and predicting the effects of the flu virus propagation throughout specific populations over a short to medium time span. We specifically do not focus on extended time periods for which qualitatively different parameters may make a difference. In addition, in our model there is no entry into or departure from the population, except possibly through death from the disease. Neither are we considering the possibility that an individual may get reinfected once recovered, during the same epidemic. Generally diseases transmitted by viral agents confer immunity so the assumption is that if an infected individual recovers he will acquire immunity for a time period at least as extended as the simulation time for the infection. On the other hand we are modeling interaction features that may have a large impact in the case of a single epidemic outbreak but whose effects level out over time. Two such examples are the structure of the social model, as well as the connectivity characteristics of the specific individuals which introduce the virus in the population.
EpiGraph consists of two main components: (1) the social model for the population under study, including the patterns of contact between individuals within this population, and (2) the epidemic model, which captures the mechanism by which susceptible individuals get infected and go through the different stages of the infection. This model is specific to the infectious agent under study, in our case, to the influenza virus. We use the social model built as described in the following section as an input for the epidemic model.
Modeling the population
The social model is represented via an undirected connection graph and can capture heterogeneity features at the level of both the individual and each of his interactions. Each node models a single individual and may have specific characteristics such as gender, age and race. We use actual demographic information to instantiate the nodes. Each graph edge represents an interaction between two individuals; we use contact information from social networks to realistically approximate these connections. Connections are timedependent such that the graph captures the dynamic nature of interactions. In the current implementation two individuals interact based on the day and the time.
Individuals and groups
To most faithfully simulate the effects of an infectious agent spreading through a specific population we decided to use real instead of synthetic data. We use demographic information obtained from the Primary Metropolitan Statistical Area of Boston
These groups represent social structures such as companies, schools, or groups of stayhome parents and retired people that are interacting in education programs, hobby classes, kids' schools or any other kind of activities that make them come in contact. The second aspect which needs to be considered in the virus propagation is the individual characteristics of the members of this population. Severe illness and death regularly occur in elderly or otherwise unhealthy individuals. In most epidemics, 80% to 90% of deaths occur in persons over 65
Connections
Rather than assuming a distribution or generating synthetic interaction graphs, we use real information from social networks to model the social interaction patterns. The interaction network is built statically to reflect the existence of communication between individuals but abstracts away the timing for these interactions. To recover some of the dynamic nature of these interactions we introduce a time parameter depending on which an individual may interact with any number of other individuals following his own patterns. Each individual has contacts within his own group as well as with individuals from other groups. Let's take the example of a worker. He is going to interact frequently with people from the same work group during work hours, with friends during leisure hours, with random people when using public transportation, and with family during evening/night hours. We therefore model three kinds of interactions: (1) between individuals of the same group (intragroup connections), (2) between individuals of different groups (intergroup connections), and (3) between members of the same family. Each of these kinds of interactions is assigned to a specific daily time frame depending on the schedule for the main activity work, study, etc for leisure activities, and for family time. This makes the simulation more realistic, particularly over short time periods. In principle, it is possible to assign any timedependent interaction pattern separately for each individual.
• Intragroup connections: Which specific group an individual belongs to determines the actual number and patterns of interactions with other individuals from his own group. One of the main contributions of our work is that we model intragroup communications by scaling down real interaction graphs extracted from online Social Networks (SN) such as Enron and Facebook. The idea is to exploit the connectivity that exists in real business and leisure SNs and approximate facetoface contacts by a scaled version of virtual contacts. The graph extracted from the Enron email database consists of 70,578 nodes and 312,620 edges (corresponding to emails), while Facebook has 250,000 nodes and 3,239,137 edges (corresponding to postings). We use Enron's SN to model the worker and retired groups and Facebook's to create the school and stayhome groups. Note that the SNs are bigger than the generated groups. We scale each down by selecting as many random entries of the SN as group members, than connecting the nodes following the same patterns as those in the SN. The selection of random entries of the SN allows us to create different interconnection patterns for each group. This approach is more realistic than either synthetically generating the interaction graphs or using probability distributions to approximate the number of individual interactions.
• Intergroup connections: We create a number of intergroup contacts per individual based on the group characteristics which the individual belongs to. Mostly the intergroup contacts occur in the hours between finishing one's main daily activity such as work or study and going home in the evening, or during weekends. These reflect daily activities which occur in public places such as parks, public transportation, etc., where one generally interacts with unknown people or friends pertaining to a different group.
In addition to intra and intergroup contacts we also model a different type of social interaction: the contacts one has with members of his family. These may be pertaining to the same or to different groups and one has contacts with them from late night to morning, and during the weekends. We assign a different distribution for the type and duration of contacts of an individual during weekends.
Modeling the infectious agent
The epidemic model is based on the principles of the SIR model as it is described in
Figure
State diagram of the epidemic model
State diagram of the epidemic model. The set of states that an individual may be in during the infectious process, and the transitions that may be taken from each of the states. Captures the evolution of the infection within a host.
More recent work
• Presymptomatic infection: In this stage individuals are infectious but symptoms are not yet present, therefore no treatment can be administrated. Figure
• Primary stage of symptomatic infection: symptoms are present and a percentage of the individuals will seek medical care. This is the window of opportunity for initiating antiviral therapy. In general, antiviral drugs reduce both the period of infectiousness and the infectivity, but they may facilitate the emergence of drugresistant viral mutants. In this work we are not considering new viral strains. Figure
• Second stage of symptomatic infection: symptoms are present and a percent of the individuals will seek medical care. At this point viral therapy is no longer effective. Other types of treatment may be possible, as well as isolating the individual for instance via hospitalization such that he does not continue infecting susceptible individuals. Figure
The epidemic model for influenza has many parameters, some of the most important being the basic reproduction number
Vaccination
Our model allows vaccinating a subset of individuals either before the outbreak of the epidemics or at any other point during the outbreak. The lower half of Figure
In case of an epidemic the period of time between its onset and the time when a vaccine becomes available is usually problematic because of the lack of understanding of both the effects of the timing when the vaccine is administrated and the choice of who will receive the vaccine. These factors are not independent, and they have further implications not only in terms of the number of infected individuals and the speed of virus dissemination, but also for the gravity of the infection in different population groups. Our simulator allows analyzing the effects of implementing a vaccination program at different times throughout the dissemination of the infectious agent.
One of the advantages of our epidemic model is that it is possible to monitor the effect of interventions such as vaccination or hospitalization at an individual level. It is therefore possible to simulate various scenarios like vaccinating or isolating a specific collective, for instance the members of a specific company or school, or a given city area.
The simulation algorithm
Our simulation algorithm uses as inputs both the social model as well as the epidemic model. The simulation algorithm processes each connection of every individual to generate a probability with which the connection will serve for transmitting the infection. This probability depends on: (1) the connection type and current time: the connection types are intragroup, intergroup, and family, and each of them corresponds to a specific daily time slice; (2) the current states of the connected individuals in the epidemic model; (3) the personal characteristics of the individual subject to being infected.
To better understand the propagation characteristics for a connection graph based on social networks such as the one we are proposing, we also simulate propagation through two other types of graphs, both synthetically built based on probability distributions specifically exponential and normal distributions. In these cases there is no differentiation in groups of different group types. Later on in the paper we report on these simulations and we draw similarities and differences between the dissemination of the virus through these networks.
EpiGraph uses sparse matrices to represent the contact graphs. This enables both optimized matrix operations and an efficient way to distribute and access the matrices in parallel. EpiGraph has been designed as a fully parallel application. It employs MPI
Simulation parameters
Parameter Name
Value
InfectiveBasicReproductionNumber
1.3730
LatentBasicReproductionNumber
0.6850
AsymptomaticBasicReproductionNumber
0.6850
InfectedTreatedBasicReproductionNumber
0.470
LatentTreatedBasicReproductionNumber
0.235
AsymptomaticTreatedBasicReproductionNumber
0.235
The basic reproduction numbers for a subset of the states in Figure 1. For a complete list of the parameters used by our simulator please refer to
Analyzing the impact of the network structure
It is wellknown that most human societies have superconnectors, people that act like hubs between the other members of the population and bear the weight of the connections in a social network. We naturally expect that the existence of these superconnectors will facilitate the spread of viruses and will make it harder to control the size of an epidemic. Is our social network such an aristocratic (rather than egalitarian) type of network? If we identify who the superconnectors are, what is the effect of vaccinating them (or isolating them from the network) for the dissemination of the virus? How can we reliably identify the superconnectors?
To start answering these questions we set up two experiments; the first is meant to analyze the network structure by comparing the dynamics of virus dissemination within our social networkbased network with that through other two networks which have exponential and normal probability distributions. The second experiment analyzes the effect on the epidemic of adopting different vaccination policies, some of them targeting the individuals having the largest number of connections.
Graph structure
Existing work such as
The simulation scenario for our social networkbased approach uses the demographic information of the city of Boston
Comparison of different network parameters
Contact Network
Average Contact Nr.
Social Network
45.088
9.649
119.068


Normal Distrib.
45.060
45.060
2050.854
26.250
1.000
Exp. Distrib.
45.016
45.016
2734.460
26.250
1.000
Comparison for several parameters of the social networkbased model, the normal distributionbased model, and the exponential distributionbased model. The parameters we are showing are: the average contact number, the connection degree, the average of the squared values of the number of contacts, the mean value at the peak of the probability distribution, and the standard deviation.
where
Figure
Number of connections in the social networkbased model
Number of connections in the social networkbased model. The histogram for the number of connections of all individuals modeled in the social networkbased model. The inset shows the distribution of the number of connections for the top 400 most connected individuals.
Number of connections in the exponential distributionbased model
Number of connections in the exponential distributionbased model. The histogram for the number of connections of all individuals modeled in the exponential distribution based model. The inset shows the distribution of the number of connections for the top 400 most connected individuals.
For the normal distribution most individuals have a number of connections close to the average and there are no superconnectors which may accelerate the propagation of the infection. The following section presents the results of simulating the virus propagation throughout these networks when the individuals that introduce the virus in the population are either average or highly connected.
Superspreaders
Depending on the properties of a connection graph it may be fundamental to understand not only the global behavior but also the individual behavior of the members of a population. Individual behavior may be a determining factor in the speed and extent of the infection propagation. In this context it is important to understand which are the individuals which spread the virus faster and further, and evaluate both the effects of infecting them, as well as vaccinating them with the purpose of containing an epidemic.
In an effort to better identify superspreaders in a given population we use the number of connections to define four group types: the individuals with high intergroup contacts, those with high intragroup contacts, those with highest numbers of overall contacts, and those with average number of overall contacts.
The simulation algorithm identifies these four population groups based on the number of connections. It can then evaluate the effects on the virus propagation of either infecting, or vaccinating, each of these different groups. The remainder of the paper presents the results of these simulations and evaluate different vaccination policies based on targeting some of these group types.
Results and discussion
The aim of this work is to understand the virus propagation process throughout a population both for prediction as well as for prevention purposes. A good, although difficult litmus test for the quality of the simulator is to compare its results with actual data. To prove the accuracy of the simulation results we compare them with the weekly data published by NYSDOH. We then analyze the virus propagation under different scenarios involving different types of interconnection networks and assuming that the virus is introduced in the population by groups of individuals with different characteristics. We also evaluate different vaccination policies meant to shorten and slow down the epidemic process.
Validation
Figure
Number of weekly newly infected for EpiGraph and NYSDOH
Number of weekly newly infected for EpiGraph and NYSDOH. In blue bars: the number of newly infected individuals per week as reported by NYSDOH. In red line: the predicted newly infected individuals in the greater Boston area as predicted by EpiGraph. The left yaxis represents the number of newly infected individuals as reported in NYSDOH. The right yaxis represents the number of newly infected individuals as predicted by EpiGraph.
Comparing the effect of the interconnection graphs
To estimate the impact of the structure of the interconnection network on the epidemic we simulate the virus propagation through the interconnection graphs introduced earlier in the paper by initially infecting a given percentage of the population that has specific individual characteristics. Specifically, we build four interconnection networks as follows: two which follow probability distributions normal and exponential, and two based on social networks, one as described in in the previous section and the other one flattened to reflect timeindependent connections. That is, every individual connects with all his contacts the whole 24 hours a day, regardless of group type (rather than only interacting during specific time slots). For each of these models, we select a percentage of the population to serve as the individuals who introduce the virus in the population; specifically we chose to infect 11 individuals. We simulate two different scenarios: in the first one we select the 11 individuals with the highest number of overall contacts; in the second one we select 11 individuals whose contact numbers are similar to the average contact number for the entire population. For the social networkbased graphs we model the greater Boston area; the average number of connections is 45. We maintain the same average number of connections for the other three graphs; the probabilitybased graphs nevertheless do not reflect either the social structures nor the timedependent interactions between individuals. Figure
Infecting individuals with maximum connection degree
Infecting individuals with maximum connection degree. Simulating the virus propagation through four different interconnection models when the virus is introduced in the population by 11 individuals with the highest number of overall contacts. The four models are the following: our social network SN (in black), SN flattened to have timeindependent connections (in blue), a normaldistribution model (in red), and an exponentialdistribution model (in green). The average number of connection is the same (45) for all the four networks.
Infecting individuals with average connection degree
Infecting individuals with average connection degree. Simulating the virus propagation through four different interconnection models when the virus is introduced in the population by 11 individuals whose contact numbers are similar to the average contact number for the entire population.
It is interesting to notice that the start time in the exponential distribution network is later than the one in the normal distribution network in the case of infecting individuals with average connection degree, but slightly earlier in the maximum connection case. This is due to the fact that the virus will start propagating faster in the exponential network if superconnectors introduce the virus in the population as they will have many more connections in the exponential than in the normal distribution case. Due to the fact that there aren't many of them, soon after the breakout the infection cannot sustain the same propagation speed. This is no longer the case if it is the average connection degree individuals which start the infection. In this case the exponential will lag behind because the normal distribution has more average connection individuals than the exponential one does. The number of infected individuals measured in millions for each of the four models is: 3.04 for the normal distribution, 2.57 for the exponential, 2.52 for the flattened social model, and 0.18 for the nonflattened social network.
Note that the nonflattened social model exhibits a much lower peak value (and a considerably later onset of the epidemic) than the other cases; we expect this to be mainly due to the fact that in the normal, exponential, and flattened models all individuals interact with all the individuals that they are in contact with at all times. This gives raise to many more infections than in the nonflattened case, where individuals connect with others only within a time slot of the day. The irregularities in the nonflattened graph are a result of simulating a more realistic and different behavior of individuals during weekends.
Interconnection patterns
In general we expect individuals that are highly connected to play an important role in the virus dissemination. Given a specific social model it is nevertheless not necessarily clear which kind of connections matter most. To get a better understanding we define several kinds of individual types depending on their interconnection patterns; we then infect a subset of individuals in these groups to compare the effects of the virus propagation. We are interested in the individuals with high intergroup, intragroup, and overall contacts, as well as those with a number of contacts similar to the population average. Given our internal representation, an efficient way to approximate the number of inter and intragroup contacts is to define a small window centered on the individual and count the connected individuals outside and within the window. As shown in Figure
Infecting individuals with different connection patterns within the social networkbased model
Infecting individuals with different connection patterns within the social networkbased model. Simulating the virus propagation through our social networkbased model when the virus is introduced in the population by individuals pertaining to four different types of groups: with maximum number of intergroup connections (in blue), with maximum number of intragroup connections (in red), with maximum number of overall connections (in green), and with number of connections similar to the population average (in black).
Somewhat less intuitive are the starting times corresponding to the maximum inter and intragroup connections, standing at days 64 and 59. The reason for this behavior is that, during weekdays (and for some individuals, Saturdays as well), one gets in contact with people outside his group (i.e. interconnections) only for 2 hours, compared to 8 hours for people inside his group (i.e. intraconnections). While family connections happen within a daily 14 hour slot, it may, or may not be the case that the family members are outside one's group. But more importantly, these connections are very few of the order of 2 or 3.
Vaccination policies
Knowing whom to vaccinate and what is the time frame when this can be done to slow down an epidemic are questions that health officials are faced with in case of an outbreak. Currently, vaccination policies are more a matter of minimizing the impact of the virus on the individuals who seek treatment rather than an effort to curb the propagation. This does not reflect a lack of preoccupation but the fact that it isn't an easy problem to solve. In case of an outbreak there are seldom enough vaccines ready to administer to the majority of the population or even to the population that is most at risk. Our simulator can provide guidance about which individuals should be treated to slow down the propagation process and reduce the number of infections. Figure
The effect of different vaccination policies
The effect of different vaccination policies. Simulating the virus propagation through our social networkbased model when different vaccination policies are applied: no vaccination (in blue), vaccination of 28% of randomly chosen individuals (in green), vaccination of 28% of the population consisting of individuals with the highest number of overall connections (in red), vaccination of 10% of the population consisting of individuals with the highest number of overall connections (in black), and vaccination of the young and elderly individuals amounting to 23.44% of the population (in magenta).
• Vaccination of a 28% of randomly chosen individuals.
• Vaccination of school children and students, which were shown to be the main infection spreaders.
• Vaccination of elderly people, which have the greatest risk of contracting the virus.
• Vaccination of a 28% of the population representing individuals with the highest number of overall connections.
• Vaccination of a 10% of the population representing individuals with the highest number of overall connections.
Note that vaccinating young and elderly people curbs the propagation noticeably more by about a fifth than vaccinating 28% of the individuals at random does. The young and elderly make up 23.44% of the population. It is noteworthy to mention that vaccinating a mere 10% of the population by targeting the individuals with the highest number of overall connections reduces the infected numbers even more than the previous two cases; the start time of the epidemic in this case occurs slightly earlier. Lastly, by vaccinating 28% of the population consisting of individuals with the highest number of overall connections, the number of infected people is reduced to 27% of the case when vaccinating the young and elderly and 21% of the random vaccination of 28% of the population. More detailed simulations and analysis could be of help to health authorities in estimating the cost and feasibility of different vaccination policies relative to their effects in terms of the number of infected individuals and the starting time for an epidemic.
Performance
We developed EpiGraph as a scalable, fully parallel and distributed simulation tool. We ran our experiments on two platforms: an AMD Opteron 6168 cluster using 8 processor nodes and running at 800 MHz, and an Intel Xeon E5405 processor with 8 cores and running at 2 GHz. For the social networkbased graph which has 3,398,051 nodes and 150 million edges, the simulation algorithm runs in 2271 seconds on the cluster and 1429 seconds on the multicore processor. For the distributionbased models the running times can go up to a maximum of about 90 minutes.
Conclusions
This paper presents a novel approach to modeling the propagation of the flu virus via a realistic interconnection network based on actual individual interactions extracted from social networks. We have implemented a scalable, fully distributed simulator and we have analyzed both the dissemination of the infection and the effect of different vaccination policies on the progress of the epidemics. Some of these policies are based on characteristics of the individuals, such as age, while others rely on connection degree and type. The epidemic values predicted by our simulator match real data from NYSDOH.
Work in progress and future work
Work in progress involves studying the effects of using additional individual characteristics in understanding disease propagation throughout a population. We are also analyzing the characteristics of our social models such as clustering, node distance, and so on and investigating to what degree disease propagation and vaccination policies have a different effect for social networks with varying such characteristics. Lastly, we are investigating a deeper definition for superconnectors which involves more than one's direct neighbours, as well as an efficient technique to finding them. There are many ramifications of this work which lead to several directions for future investigation. We only mention a couple of them here. First we are interested in whether recording the actual position of each individual brings new insights to the social model. This provides a way to reconstruct interaction patterns with people inside and outside one's group. We are also interested in whether the duration of the individual contacts turns out to be relevant at a large scale and whether there is a connection between it and a notion of strong and weak connections which would reflect the degree to which a connection may serve as a channel for spreading the infectious agent between pairs of groups or individuals. Finally, it will be interesting to see how our approach scales to a nationwide simulation.
Abbreviations
NYSDOH: New York State Department of Health; SN: Social Network; SIR: SusceptibleInfectiousRecovered.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
GM, MCM and DES performed all coding and simulations. GM, MCM, DES, and JC conceived, designed the work, analyzed the data and wrote the paper. All authors read an approved the final manuscript.
Acknowledgements
This work has been partially funded by the Spanish Ministry of Science under the project TIN201016497.
This article has been published as part of