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Modular and hierarchical structure of social contact networks
Physica A 392 (2013) 4619–4628
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Physica A journal homepage: www.elsevier.com/locate/physa
Modular and hierarchical structure of social contact networks Yuanzheng Ge a,∗ , Zhichao Song a , Xiaogang Qiu a , Hongbin Song b , Yong Wang b a
College of Information System and Management, National University of Defense Technology, 410073, Changsha, China
b
Institute of Disease Control and Prevention, Academy of Military Medical Sciences, 100013, Beijing, China
highlights • • • •
We model social contact networks with geographic and census information. We reconstruct multiple layers of a social contact network for a university. We reproduce the outbreak of influenza H1N1 as an evaluation. Local clustering of contact networks constrains the spread of diseases.
article
info
Article history: Received 6 June 2012 Received in revised form 23 May 2013 Available online 6 June 2013 Keywords: Social contact networks Modular structure Hierarchical structure Epidemic transmission Agent-based model
abstract Social contact networks exhibit overlapping qualities of communities, hierarchical structure and spatial-correlated nature. We propose a mixing pattern of modular and growing hierarchical structures to reconstruct social contact networks by using an individual’s geospatial distribution information in the real world. The hierarchical structure of social contact networks is defined based on the spatial distance between individuals, and edges among individuals are added in turn from the modular layer to the highest layer. It is a gradual process to construct the hierarchical structure: from the basic modular model up to the global network. The proposed model not only shows hierarchically increasing degree distribution and large clustering coefficients in communities, but also exhibits spatial clustering features of individual distributions. As an evaluation of the method, we reconstruct a hierarchical contact network based on the investigation data of a university. Transmission experiments of influenza H1N1 are carried out on the generated social contact networks, and results show that the constructed network is efficient to reproduce the dynamic process of an outbreak and evaluate interventions. The reproduced spread process exhibits that the spatial clustering of infection is accordant with the clustering of network topology. Moreover, the effect of individual topological character on the spread of influenza is analyzed, and the experiment results indicate that the spread is limited by individual daily contact patterns and local clustering topology rather than individual degree. © 2013 Elsevier B.V. All rights reserved.
1. Introduction Epidemics that break out in an environment with a large population size in a short period are normally transmitted by face-to-face contacts among human beings [1,2]. Social contact networks, the notable features of which are overlapping qualities of clustering communities, hierarchical structure and geospatial correlations [3,4], are applied to study the dynamic transmission of diseases, such as pandemic influenza and measles [5–8]. Differing topological analysis of ‘small-world’ [9] and ‘scale-free’ [10] networks, nodes in social contact networks are often provided with spatial information, like individual
∗
Corresponding author. Tel.: +86 0731 84574333. E-mail address: [email protected] (Y. Ge).
0378-4371/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physa.2013.05.051
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Y. Ge et al. / Physica A 392 (2013) 4619–4628 Table 1 Agent description. Attribute
Description
ID Location Community Contact list
A unique identifier for each individual. Geographic resident location, which is represented by the location identifier of a region, building or room. Social communities that individuals belong to, which are used to define the type of contact networks Ψ . For each type of social relation ϕ ∈ Ψ , a contact list contains all the other agents’ IDs that the agent might contact.
geospatial locations and spatial migration. Hence, a spatial-topology structure is required to model social contact networks, and it is also useful to predict the spread of epidemics and evaluate containment interventions [11,12]. In recent years, many researchers tried to reconstruct appropriate contact network models to study the impact of complex social collective behavior on disease transmission. However, how to figure out the contact pattern in a population remains a challenge. About human behavior, what we know is only a small part. So far, behavior investigation is the main method to capture and quantify individual social interactions. Survey questionnaires and wireless wearable devices are sent to participants to record their daily contact process [13–15]. The dynamic data, which are collected during individuals’ daily life, are useful to reconstruct social contact networks. However, statistic analysis of collected investigation data lacks the capability to build computational models for larger scale population and nonspecific scenes [16,17]. In addition to this, how to construct proper social contact networks based on various investigation data is another crucial problem [18]. The investigation data has to be transformed to computational models. As we know, one significant advantage of the computational model is that virtual scenarios can be repeated thousands of times at a low cost [19]. Hence, integrating individual-based computational models (especially the agent-based modeling method), spatial-topologies of social contact networks and behavior investigation data together is a promising solution to dealing with complex social problems [20–25]. In this paper, we propose a method to construct social contact networks with both spatial-topologies and overlapping communities by integrating modular models with a growing hierarchical structure. The modular model is used to describe the highly correlated local interactions, and individuals’ spatial distribution and their organization structure is used to specify the scale of sub-networks in each layer of the hierarchical network. We show that, by employing this method, it is possible to reproduce geospatial contact networks with the main properties of empirical social networks, including degree distribution and clustering coefficient. To evaluate the proposed method, a hierarchical social contact network of a university was constructed based on student geospatial distribution and organization date. Transmission experiments of influenza H1N1 were carried out in the constructed network, and the results were validated by the clinical data collected during the outbreak of influenza H1N1 in 2009. Further, the impact of individual topological character and the spatial clustering of transmission were studied and analyzed. 2. Methods The hierarchical structure of the contact network consists of multiple layers of networks, and each layer contains several independent networks, which are of the same type. The independent network is internally connected, and there is no edge between different independent networks. It likes a community, in which nodes have the same attributes. 2.1. Individual model In social contact networks, nodes are represented by agents, who are endowed with demographic attributes, social relationships, and geographic locations [26]. The model defined N agents who have no initial edges. For each agent (node), its initialized demographic attributes include agent ID, location, and communities that the agent belongs to. In order to formulize the multiple layers of contact networks, we defined a database table to store all individuals’ social relationships. Each individual has a contact list set, which can be defined as: L = (G, Ψ ) where G (short for Group) is a set of agents, and Ψ is a set of relations. For an agent a, one type of social relationship l = (g , ϕ) represents the fact that all the agents a′ ∈ g (g ⊂ G) have the social relation ϕ ∈ Ψ with agent a [27]. For a relation ϕ , the union of all the N agents’ groups (∪gi , i ∈ [1, N ]) is one layer of social networks. The layer number of a network is denoted by α . Table 1 lists the attributes of an agent and the corresponding descriptions. 2.2. Modular model The modular model is the first layer of the hierarchical network (α = 0), representing the most frequent contact relations, like students in the same dormitory and family members in a household [5]. Individuals in a modular model have a rather high probability to contact each other every day. Hence, a modular model is defined as a complete network, and the only parameter to control the network topology is the node number n. If there are many kinds of modular models in the whole
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network, the parameter is a set {ni }. Each modular has a high clustering coefficient C = 1. Under this definition, if all the modular models have n0 nodes, a contact network with N nodes is divided into N /n0 basic modular models. 2.3. Hierarchical structure The hierarchical network is generated based on the modular model, and it is a gradually growing process to connect nodes from modular models to a global hierarchal contact network. Based on the modular model, the next layer of the hierarchical network α = 1 is generated by two steps. Firstly, the number of modular models (ns ) that compose an independent network in layer α = 1 is fixed. We call the component networks (here, modular networks) ‘sub-networks’, which are used to construct the independent network in layer α = 1. Let (v, e) represent a modular network, and then all the fixed modular networks to construct a new independent network in layer α = 1 are denoted by {(vi , ei )}, where vi and ei are the node and edge set of a modular network i. Secondly, new edges are added between individuals in different modular networks to form a larger network. The newly generated network G is: G = (∪vi , ∪ei + e′ ),
i ∈ [1, ns ]
(1)
where e is the newly added edges. The next layer α = 2 and higher layers also follow the two steps: first to fix the number of sub-networks, and then to add new edges among these sub-networks. By this method, the higher layer networks are always generated based on the lower layer networks. Before adding new edges to form a new network, there is no overlapping node among the sub-networks. The sub-network itself is an independent network, which is also generated by its sub-networks, except the modular network. For instance, on campus, the dormitory contact network is the modular network, and the class contact network is the higher one. For any two dormitory networks Di and Dj , they are independent from each other when we consider them in the dormitory layer. However, in the class layer, if Di and Dj are determined to be the sub-networks of a class layer network, their nodes might have new connections in this layer. Here, one problem is how to fix the number of the sub-networks in a lower layer to compose an independent network in a higher layer. To solve this problem, we utilize both the spatial distance and population organizational structure to estimate how many layers should be abstracted, and how many sub-networks should be contained in a layer. Details will be shown in the case study later. The other problem is whether to add a new edge between a pair of individuals who are in different sub-networks. We define a probability distribution to calculate the connection probability, which is a function of layer α and an individual’s degree d: ′
p=λ
di Maxdi
+
dj
Maxdj
2
(2)
where λ is a decline coefficient specifying the decreasing degree of p, and λ = e−α . Maxdi is the maximum degree of links that individual i might have in layer α − 1, and di is its actual degree. According to Eq. (2), we can see that as the layer grows, the probability of adding new edges decreases. This means that the contact relations get sparser when the scale of a contact network grows larger (the higher layer), and individuals tend to contact with the ones who are spatially nearby. 3. Generating contact networks To validate the model described above, we applied the method to reconstruct the contact networks of one university in the light of enrolling data and individuals’ spatial distribution data. 3.1. Initialization In the university we investigated, there were about 13 000 susceptible individuals including students, teachers and other members of staff. Students were the main body of the susceptible individuals. In the experiment, the total number of students was initialized to 12 096, and all the students dwelt in 8 dormitory buildings. We also made a survey on the dwelling pattern of students in the dormitory buildings to get their organization structure and spatial distribution. The information includes: the number of students in one dormitory, the number of students in one class, and the class distribution in a dormitory building. We utilized this dwelling structure to define the hierarchy of the social contact networks on the whole campus. For the campus, 5 layers of contact networks were defined in turn: dormitory, class, dormitory floor, dormitory building, and the whole campus. Firstly, the dormitory contact network was defined as the modular model α = 0, which consists of 6 students. On the campus, all the student dormitories are of the same size. Based on this organization structure, the modular network is a complete network with 6 nodes. After this, all the agents were assigned a dormitory number. The next layer was the class contact network (α = 1), in which contacts are still highly intense but weaker than that in the dormitory layer. 21 dormitories compose a class, and new edges were added among individuals in the same class. The connection probability is calculated according to Eq. (2). Then in turn, dormitory floor (α = 2), dormitory building (α = 3) and the whole campus (α = 4) were generated one by one.
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a
b
Fig. 1. (a) Degree distributions of 4 layer contact networks, α = 1, 2, 3 and 4. (b) Clustering coefficients of 5 layer social contact networks, α = 0, 1, 2, 3 and 4.
3.2. Network properties From the dormitory network (α = 0) to the whole campus (α = 4), the degree distribution and clustering coefficients of generated contact networks were calculated for each layer, as shown in Fig. 1. The degree distribution shows single-scale character (Fig. 1(a)) [28]. As the layer grows, on the one hand, the degree distribution curves move right. It means that students have more contact objects when the spatial range expands. On the other hand, clustering coefficients decrease as an exponential function of layer α (Fig. 1(b)), which indicates that contact relations are clustering in local space and get sparse in a large spatial range. The average distance on the campus scale (layer α = 4) is 2.7915, showing obvious small-world character [29]. 4. Disease model To avoid the assumption of homogeneously mixed population [30], casual interactions among individuals are excluded. We assume that all the physical proximity interactions only happen between individuals who have social relationships. Contact behavior is limited by space and time, and spatial distance does affect contact frequencies [31,32]. Diseases transmitted by face-to-face contact like influenza can be described by two elements [33]: the number of contacts Nc and the actual transmission probability P for a contact between an infected individual and a susceptible one. 4.1. Contact frequency One individual has many acquaintances in the contact list, but it is impossible for the individual to contact all of them in a day. The actual number of individuals that one individual is able to contact in a day is limited [3]. To identify the effective contacts for influenza virus transmission, we only consider two contact patterns: conversation without physical contact and conversation with non-sexual physical contact. For an individual, the number of the two types of effective contacts Nc in a day follows a normal distribution around 26.05, and confidence interval (16.76, 35.34) is under confidence degree 95% [13]. Here, all these contacts are not uniformly assigned to each layer, but in proportion to the decline coefficient λ in Eq. (2). The basic principle is: the farther individuals are away from each other, the smaller probabilities they have of contact with each other. For a student a, his contact list is: L = {(g1 , dormitory), (g2 , class), (g3 , floor), (g4 , building), (g5 , campus)}. Student a might contact with a student a1 in the same dormitory (a1 ∈ g1 ) several times in a day, but might not contact with a student a2 on the same campus (a2 ∈ g5 , and a2 ̸∈ (g1 ∪ g2 ∪ g3 ∪ g4 )) several days. 4.2. Infection rate Transmission probability of a contact is defined as an alternative parameter pi , which fluctuates in terms of different diseases. To model infected individuals infectivity in different infectious periods, pi is multiplied by a coefficient pΓ (t ) to
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Fig. 2a. Newly infected individuals in hospital every day, which is a function of time t (day).
calculate the actual transmission probability P (Eq. (3)). pΓ (t ) is a function of infectious duration t (day), which is cumulated from the day that the individual is infected (Eq. (4)). Normally, individual infectivity reaches peak value one day after symptoms appear [34]. Here, we use Γ (4, 1) to generate the infectivity coefficient. P = pi pΓ (t ) pΓ (t ) =
−t 3
e t 6
(3)
,
t > 0.
(4)
4.3. Root infected individuals To simulate the transmission of influenza H1N1 in the constructed social contact networks, infected individuals were imported as the root ones at the beginning. According to the case we studied, more than one individual were diagnosed as having flu at the beginning of the new semester. We applied two criteria to confirm whether a student was the root infected individual: students who showed symptoms in the first two days; at the same time, students who had no contact history with other infected students. The root infected students were located in 3 dormitory buildings, and there are respectively 6, 5, and 2 root students in them. In the simulation, all the infected students were supposed to be sent to the hospital for treatment for 1–2 days after being infected. Hence, students in hospital were removed from the contact networks. 5. Results and discussion We carried out three sets of experiments to evaluate the generated contact networks. One set reproduced the Langfang case under the same interventions as what were implemented during the outbreak. The other two sets analyzed the dynamic properties of the contact networks for influenza transmission. 5.1. Validating experiments During the outbreak in the Langfang case, three progressive non-pharmaceutical interventions were made to control the spread of the disease on the campus:
• The dormitory buildings were isolated on the 5th day, while students were allowed to move around within a building; • All the dormitories were isolated on the 6th day, while students were confined to their dormitories; • All the dormitory members of students diagnosed with infection were isolated on the 9th day. We implemented the three interventions by cutting off edges between layers of the generated contact networks. The experiments were repeated 1000 times under transmission probability pi = 0.042. Under this parameter, R0 of influenza H1N1 on the campus is 2.1. Fig. 2a shows the number of newly infected students every day (dots) who were sent to the hospital. The experiment results were compared with the clinical records (triangles). In the light of this chart, the accumulative number of infected individuals in the hospital increases rapidly during the first 8 days, and slows down when strict isolating interventions (the third intervention) were executed (Fig. 2b). It indicates that non-pharmaceutical interventions are efficient to prevent the
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Fig. 2b. Accumulative infected individuals in hospital. The dots represent the average result of 1000 experiments; the triangles represent the real clinical record.
Fig. 3. Percentages of accumulative infected individuals to total individuals in 8 dormitory buildings.
spread of influenza H1N1, especially when possible contacts among individuals are cut off. However, this measure disturbs people’s normal life and costs a lot economically. In Fig. 2a, the peak value of experiment results is higher than that of the real data. The reason is that when facing the outbreak, individuals consciously reduce interactive behavior and travels, and however, we did not take this factor into consideration in the disease model. The experiment results also exhibit a spatial clustering property in dormitory buildings. Fig. 3 shows the accumulative percentages of newly infected individuals to total individuals in a dormitory building. The tendency of results (dots) keeps accordant with the clinical data (squares). It indicates that the outbreak of influenza is only clustered in a few buildings. 5.2. Impacts of degree distribution Under the same transmission probability pi , we carried out transmission experiments of influenza H1N1 in the social contact networks without isolating interventions and allowing infected ones to go to hospital 1–2 days after being infected. In this set of experiments, one root infected individual was imported at the beginning. To identify the impact of individual degree on disease transmission, we stochastically pick one individual from all the individuals who have the same degree k as the root student. Transmission experiments were repeated 1000 times under each degree k. Fig. 4 shows the correlation between the degrees of root infected individuals and the average accumulative
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Fig. 4. Accumulative infected individuals under different degrees (k).
Fig. 5. Total infected individuals on the whole campus with one root infected individual for 10 000 repeated experiments.
number of infected individuals on the campus. It can be seen that there is no obvious mapping between degree distribution and influenza transmission results. The root individual degree has almost no impact on influenza transmission. The main reason is that the number of contacts that one individual might have in a day is limited, and this number follows a normal distribution around 26.5. The probability of contact number has no relation with individual degree distribution, and hence, the individual degree cannot directly impact individual daily contact number. In daily life, it is hard to confirm whether the individual who has more social relations can contact more acquaintances in a day. A fact is that proper contact is an activity of physical proximity, which takes time and covers distance. Since it is hard to identify the impact of individual degree, we stochastically selected one individual as the root infected one to study the transmission process in detail. The experiment is repeated 10 000 times under the same conditions. It can be seen that the results split into two distinct parts in Fig. 5, which indicate two possible means of transmission: Case 1: only several individuals are infected, and the number of newly infected individuals is no more than 100; Case 2: almost all the students on the campus are infected. In the first case, influenza is confined to a small range, and the percentage of the first case is 22.51%. Before infected students are sent to the hospital, they infected just a few students, and soon they are all in hospital. In the second case, influenza spreads across the campus, and this case takes 77.49%. It means that once influenza breaks, an outbreak on the campus will take place. If we assume the frequency as the probability, that a root individual will cause spread of influenza H1N1 on a large scale is a probable event.
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Fig. 6. (a) Average infected individuals of 10 experiments for each individual and x axis represents ID of the root infected individual in the experiment. (b) The number of dots in each belt in Fig. 6(a).
Furthermore, to observe the probability of large-scale outbreak (Case 2), each individual on the campus was selected as the root student for transmission experiment. For each selected student, experiments were repeated 10 times. The average results are shown in Fig. 6(a), and the x axis is the student ID. It shows obvious separation (belts). The separation looks different from that in Fig. 5. However, they are the same in essence. Based on the analysis of transmission experiments for an individual above, we can see that there are only two possible conditions. Furthermore, the average number of students that a root individual infected is determined by the probability of Case 2. That also means that the more frequently Case 2 occurs, the larger the average number is. For instance, dots in Belt 8000 indicate the fact that Case 2 occurs 8 times in the 10 experiments for an individual. To study the distribution characters of the dots plotted in Fig. 6(a), we calculate the number of dots in each belt. The results are shown in Fig. 6(b), where the x axis is the frequency of dots, and the y axis corresponds to the belts. It can be seen that in most cases if no interventions are implemented, a root infected individual might cause an outbreak of influenza H1N1 on the campus. However, there is also some probability that the influenza is confined to a small range, and only a few students are infected. 5.3. Spatial clustering According to the definition of individual social relationship in Section 2.1, an individual is allowed to have multiple relationships at the same time. On the campus, each student has 5 types of relations. That means each student belongs to 5 independent networks. These networks respectively locate in 5 different layers of the hierarchical network. Since influenza H1N1 spreads from the root infected student, we picked out the 5 independent networks where the root student locates to analyze the transmission dynamics of different layers. The numbers of accumulative infected individuals in the 5 independent networks are calculated over time. Fig. 7 shows the accumulative infected ratio of the newly infected students to the total students in an independent network. For the 5 layers of the network, the accumulative ratio of layer α = 0 increases at the highest speed, which means that roommates are the highest risky group and all are infected in a short period (about 5 days). Once an infected individual is imported to a susceptible population, the influenza spreads rapidly in the lower layers (α = 0, 1), and then extends to the higher layers (α = 2, 3, 4). In Section 3, we showed the fact that the lower-layer networks have larger clustering coefficients (Fig. 1(b)), and here, the transmission experiments also exhibit that the outbreak has local clustering character. 6. Conclusion In this paper, we presented a novel approach to construct social contact networks based on modular and hierarchical structure with individuals’ spatial distribution information. It also provides an effective method to integrate individual geospatial locations with network topologies. This work can be offered to researchers interested in contagious infection, especially face-to-face transmitted diseases. Based on the constructed contact networks, the transmission experiments reproduced the outbreak of influenza H1N1 occurred in 2009. In the spread process, spatial-topology structure is found to play a significant role in an outbreak of a contagious disease. The local clustering of contact network constrains the spreading scale and speed of the disease. This effectiveness is crucial for controlling the spread of an epidemic disease and making non-pharmacological isolation interventions.
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Fig. 7. The accumulative infected ratio (R) of the newly infected students to the total students in an independent network.
In the constructed networks, individual degree exerts no obvious effect on the spread of influenza in the transmission process. Although individual social relations are heterogeneous from each other, effective contacts for all the individuals in a day are limited by space and time. One limitation of the experiment in the paper is that we did not consider the relation between individual degree and daily contact number. Hence, it is difficult to identify individual influence on the outbreak of an epidemic. As for future work, we will extend the method to larger-scale cases by integrating census data and national administrative hierarchical structure. More detailed investigation data is required to specify modular model and layers of the hierarchical structure. Secondly, individual models, especially the behavior algorithms, will be further studied to figure out what role an individual or a class of people plays in an outbreak. Thirdly, in the light of social contact networks, the threshold of an outbreak is another promising area. Since whether an outbreak occurs or not is a probability event, the threshold of an outbreak is useful to predict the spread of a disease and evaluate policies to be implemented. Acknowledgments The authors would like to thank National Nature and Science Foundation of China under Grant Nos. 9102403. We also acknowledge the invaluable comments and suggestions of anonymous reviewers, and the journal’s editors. References [1] C. Fraser, C.A. Donnelly, S. Cauchemez, et al., Pandemic potential of a strain of influenza A(H1N1): early findings, Science 324 (2009) 1557–1561. [2] V.J. Munster, E. de Wit, J.M.A. van den Brand, et al., Pathogenesis and transmission of swine-origin 2009 A(H1N1) influenza virus in ferrets, Science 325 (2009) 481–483. [3] J. Stehlé, N. Voirin, A. Barrat, et al., High-resolution measurements of face-to-face contact patterns in a primary school, PLoS One 6 (2011) e23176. [4] J. Mossong, N. Hens, M. Jit, et al., Social contacts and mixing patterns relevant to the spread of infectious diseases, PLoS Med. 5 (2008) e74. [5] L.M. Glass, R.J. Glass, Social contact networks for the spread of pandemic influenza in children and teenagers, BMC Public Health 8 (2008) 61. [6] L. Isella, J. Stehlé, A. Barrat, et al., What’s in a crowd? Analysis of face-to-face behavioral networks, J. Theoret. Biol. 271 (2011) 166–180. [7] K. Zhao, J. Stehlé, G. Bianconi, et al., Social network dynamics of face-to-face interactions, Phys. Rev. E 83 (2011) 056109. [8] D. Balcan, V. Colizza, B. Gonçalves, et al., Multiscale mobility networks and the spatial spreading of infectious diseases, Proc. Natl. Acad. Sci. 106 (2009) 21484–21489. [9] D.J. Watts, S.H. Strogatz, Collective dynamics of ‘small-world’ networks, Nature 393 (1998) 440–442. [10] R. Albert, A.-L. Barabási, Statistical mechanics of complex networks, Rev. Modern Phys. 74 (2002) 47–97. [11] S. Riley, Large-scale spatial-transmission models of infectious disease, Science 316 (2007) 1298–1301. [12] L. Hufnagel, D. Brockmann, T. Geisel, Forecast and control of epidemics in a globalized world, Proc. Natl. Acad. Sci. 101 (2009) 15124–15129. [13] W.J. Edmunds, G. Kafatos, J. Wallinga, Mixing patterns and the spread of close-contact infectious diseases, Emerg. Themes Epidemiol. 3 (2009) 10. [14] L. Isella, M. Romano, A. Barrat, et al., Close encounters in a pediatric ward: measuring face-to-face proximity and mixing patterns with wearable sensors, PLoS One 6 (2011) e17144. [15] M. Salathé, M. Kazandjieva, J.W. Lee, et al., A high-resolution human contact network for infectious disease transmission, Proc. Natl. Acad. Sci. 107 (2010) 22020–22025. [16] J.M. Epstein, Modelling to contain pandemics, Nature 460 (2009) 687. [17] D. Lazer, A. Pentland, L. Adamic, et al., Computational social science, Science 323 (2009) 721–723. [18] E. Ravasz, A.-L. Barabási, Hierarchical organization in complex networks, Phys. Rev. E 67 (2003) 026112. [19] K.M. Carley, D.B. Fridsma, E. Casman, et al., BioWar: scalable agent-based model of bioattacks, IEEE Trans. Syst. Man. Cybern. A 36 (2006) 252–265. [20] J. Parker, J.M. Epstein, A distributed platform for global-scale agent-based models of disease transmission, ACM Trans. Model. Comput. Simul. 22 (2011) 2. [21] F. Peruani, G.J. Sibona, Dynamics and steady states in excitable mobile agent systems, Phys. Rev. Lett. 100 (2008) 168103.
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Contents lists available at SciVerse ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
Modular and hierarchical structure of social contact networks Yuanzheng Ge a,∗ , Zhichao Song a , Xiaogang Qiu a , Hongbin Song b , Yong Wang b a
College of Information System and Management, National University of Defense Technology, 410073, Changsha, China
b
Institute of Disease Control and Prevention, Academy of Military Medical Sciences, 100013, Beijing, China
highlights • • • •
We model social contact networks with geographic and census information. We reconstruct multiple layers of a social contact network for a university. We reproduce the outbreak of influenza H1N1 as an evaluation. Local clustering of contact networks constrains the spread of diseases.
article
info
Article history: Received 6 June 2012 Received in revised form 23 May 2013 Available online 6 June 2013 Keywords: Social contact networks Modular structure Hierarchical structure Epidemic transmission Agent-based model
abstract Social contact networks exhibit overlapping qualities of communities, hierarchical structure and spatial-correlated nature. We propose a mixing pattern of modular and growing hierarchical structures to reconstruct social contact networks by using an individual’s geospatial distribution information in the real world. The hierarchical structure of social contact networks is defined based on the spatial distance between individuals, and edges among individuals are added in turn from the modular layer to the highest layer. It is a gradual process to construct the hierarchical structure: from the basic modular model up to the global network. The proposed model not only shows hierarchically increasing degree distribution and large clustering coefficients in communities, but also exhibits spatial clustering features of individual distributions. As an evaluation of the method, we reconstruct a hierarchical contact network based on the investigation data of a university. Transmission experiments of influenza H1N1 are carried out on the generated social contact networks, and results show that the constructed network is efficient to reproduce the dynamic process of an outbreak and evaluate interventions. The reproduced spread process exhibits that the spatial clustering of infection is accordant with the clustering of network topology. Moreover, the effect of individual topological character on the spread of influenza is analyzed, and the experiment results indicate that the spread is limited by individual daily contact patterns and local clustering topology rather than individual degree. © 2013 Elsevier B.V. All rights reserved.
1. Introduction Epidemics that break out in an environment with a large population size in a short period are normally transmitted by face-to-face contacts among human beings [1,2]. Social contact networks, the notable features of which are overlapping qualities of clustering communities, hierarchical structure and geospatial correlations [3,4], are applied to study the dynamic transmission of diseases, such as pandemic influenza and measles [5–8]. Differing topological analysis of ‘small-world’ [9] and ‘scale-free’ [10] networks, nodes in social contact networks are often provided with spatial information, like individual
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Corresponding author. Tel.: +86 0731 84574333. E-mail address: [email protected] (Y. Ge).
0378-4371/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physa.2013.05.051
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Y. Ge et al. / Physica A 392 (2013) 4619–4628 Table 1 Agent description. Attribute
Description
ID Location Community Contact list
A unique identifier for each individual. Geographic resident location, which is represented by the location identifier of a region, building or room. Social communities that individuals belong to, which are used to define the type of contact networks Ψ . For each type of social relation ϕ ∈ Ψ , a contact list contains all the other agents’ IDs that the agent might contact.
geospatial locations and spatial migration. Hence, a spatial-topology structure is required to model social contact networks, and it is also useful to predict the spread of epidemics and evaluate containment interventions [11,12]. In recent years, many researchers tried to reconstruct appropriate contact network models to study the impact of complex social collective behavior on disease transmission. However, how to figure out the contact pattern in a population remains a challenge. About human behavior, what we know is only a small part. So far, behavior investigation is the main method to capture and quantify individual social interactions. Survey questionnaires and wireless wearable devices are sent to participants to record their daily contact process [13–15]. The dynamic data, which are collected during individuals’ daily life, are useful to reconstruct social contact networks. However, statistic analysis of collected investigation data lacks the capability to build computational models for larger scale population and nonspecific scenes [16,17]. In addition to this, how to construct proper social contact networks based on various investigation data is another crucial problem [18]. The investigation data has to be transformed to computational models. As we know, one significant advantage of the computational model is that virtual scenarios can be repeated thousands of times at a low cost [19]. Hence, integrating individual-based computational models (especially the agent-based modeling method), spatial-topologies of social contact networks and behavior investigation data together is a promising solution to dealing with complex social problems [20–25]. In this paper, we propose a method to construct social contact networks with both spatial-topologies and overlapping communities by integrating modular models with a growing hierarchical structure. The modular model is used to describe the highly correlated local interactions, and individuals’ spatial distribution and their organization structure is used to specify the scale of sub-networks in each layer of the hierarchical network. We show that, by employing this method, it is possible to reproduce geospatial contact networks with the main properties of empirical social networks, including degree distribution and clustering coefficient. To evaluate the proposed method, a hierarchical social contact network of a university was constructed based on student geospatial distribution and organization date. Transmission experiments of influenza H1N1 were carried out in the constructed network, and the results were validated by the clinical data collected during the outbreak of influenza H1N1 in 2009. Further, the impact of individual topological character and the spatial clustering of transmission were studied and analyzed. 2. Methods The hierarchical structure of the contact network consists of multiple layers of networks, and each layer contains several independent networks, which are of the same type. The independent network is internally connected, and there is no edge between different independent networks. It likes a community, in which nodes have the same attributes. 2.1. Individual model In social contact networks, nodes are represented by agents, who are endowed with demographic attributes, social relationships, and geographic locations [26]. The model defined N agents who have no initial edges. For each agent (node), its initialized demographic attributes include agent ID, location, and communities that the agent belongs to. In order to formulize the multiple layers of contact networks, we defined a database table to store all individuals’ social relationships. Each individual has a contact list set, which can be defined as: L = (G, Ψ ) where G (short for Group) is a set of agents, and Ψ is a set of relations. For an agent a, one type of social relationship l = (g , ϕ) represents the fact that all the agents a′ ∈ g (g ⊂ G) have the social relation ϕ ∈ Ψ with agent a [27]. For a relation ϕ , the union of all the N agents’ groups (∪gi , i ∈ [1, N ]) is one layer of social networks. The layer number of a network is denoted by α . Table 1 lists the attributes of an agent and the corresponding descriptions. 2.2. Modular model The modular model is the first layer of the hierarchical network (α = 0), representing the most frequent contact relations, like students in the same dormitory and family members in a household [5]. Individuals in a modular model have a rather high probability to contact each other every day. Hence, a modular model is defined as a complete network, and the only parameter to control the network topology is the node number n. If there are many kinds of modular models in the whole
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network, the parameter is a set {ni }. Each modular has a high clustering coefficient C = 1. Under this definition, if all the modular models have n0 nodes, a contact network with N nodes is divided into N /n0 basic modular models. 2.3. Hierarchical structure The hierarchical network is generated based on the modular model, and it is a gradually growing process to connect nodes from modular models to a global hierarchal contact network. Based on the modular model, the next layer of the hierarchical network α = 1 is generated by two steps. Firstly, the number of modular models (ns ) that compose an independent network in layer α = 1 is fixed. We call the component networks (here, modular networks) ‘sub-networks’, which are used to construct the independent network in layer α = 1. Let (v, e) represent a modular network, and then all the fixed modular networks to construct a new independent network in layer α = 1 are denoted by {(vi , ei )}, where vi and ei are the node and edge set of a modular network i. Secondly, new edges are added between individuals in different modular networks to form a larger network. The newly generated network G is: G = (∪vi , ∪ei + e′ ),
i ∈ [1, ns ]
(1)
where e is the newly added edges. The next layer α = 2 and higher layers also follow the two steps: first to fix the number of sub-networks, and then to add new edges among these sub-networks. By this method, the higher layer networks are always generated based on the lower layer networks. Before adding new edges to form a new network, there is no overlapping node among the sub-networks. The sub-network itself is an independent network, which is also generated by its sub-networks, except the modular network. For instance, on campus, the dormitory contact network is the modular network, and the class contact network is the higher one. For any two dormitory networks Di and Dj , they are independent from each other when we consider them in the dormitory layer. However, in the class layer, if Di and Dj are determined to be the sub-networks of a class layer network, their nodes might have new connections in this layer. Here, one problem is how to fix the number of the sub-networks in a lower layer to compose an independent network in a higher layer. To solve this problem, we utilize both the spatial distance and population organizational structure to estimate how many layers should be abstracted, and how many sub-networks should be contained in a layer. Details will be shown in the case study later. The other problem is whether to add a new edge between a pair of individuals who are in different sub-networks. We define a probability distribution to calculate the connection probability, which is a function of layer α and an individual’s degree d: ′
p=λ
di Maxdi
+
dj
Maxdj
2
(2)
where λ is a decline coefficient specifying the decreasing degree of p, and λ = e−α . Maxdi is the maximum degree of links that individual i might have in layer α − 1, and di is its actual degree. According to Eq. (2), we can see that as the layer grows, the probability of adding new edges decreases. This means that the contact relations get sparser when the scale of a contact network grows larger (the higher layer), and individuals tend to contact with the ones who are spatially nearby. 3. Generating contact networks To validate the model described above, we applied the method to reconstruct the contact networks of one university in the light of enrolling data and individuals’ spatial distribution data. 3.1. Initialization In the university we investigated, there were about 13 000 susceptible individuals including students, teachers and other members of staff. Students were the main body of the susceptible individuals. In the experiment, the total number of students was initialized to 12 096, and all the students dwelt in 8 dormitory buildings. We also made a survey on the dwelling pattern of students in the dormitory buildings to get their organization structure and spatial distribution. The information includes: the number of students in one dormitory, the number of students in one class, and the class distribution in a dormitory building. We utilized this dwelling structure to define the hierarchy of the social contact networks on the whole campus. For the campus, 5 layers of contact networks were defined in turn: dormitory, class, dormitory floor, dormitory building, and the whole campus. Firstly, the dormitory contact network was defined as the modular model α = 0, which consists of 6 students. On the campus, all the student dormitories are of the same size. Based on this organization structure, the modular network is a complete network with 6 nodes. After this, all the agents were assigned a dormitory number. The next layer was the class contact network (α = 1), in which contacts are still highly intense but weaker than that in the dormitory layer. 21 dormitories compose a class, and new edges were added among individuals in the same class. The connection probability is calculated according to Eq. (2). Then in turn, dormitory floor (α = 2), dormitory building (α = 3) and the whole campus (α = 4) were generated one by one.
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a
b
Fig. 1. (a) Degree distributions of 4 layer contact networks, α = 1, 2, 3 and 4. (b) Clustering coefficients of 5 layer social contact networks, α = 0, 1, 2, 3 and 4.
3.2. Network properties From the dormitory network (α = 0) to the whole campus (α = 4), the degree distribution and clustering coefficients of generated contact networks were calculated for each layer, as shown in Fig. 1. The degree distribution shows single-scale character (Fig. 1(a)) [28]. As the layer grows, on the one hand, the degree distribution curves move right. It means that students have more contact objects when the spatial range expands. On the other hand, clustering coefficients decrease as an exponential function of layer α (Fig. 1(b)), which indicates that contact relations are clustering in local space and get sparse in a large spatial range. The average distance on the campus scale (layer α = 4) is 2.7915, showing obvious small-world character [29]. 4. Disease model To avoid the assumption of homogeneously mixed population [30], casual interactions among individuals are excluded. We assume that all the physical proximity interactions only happen between individuals who have social relationships. Contact behavior is limited by space and time, and spatial distance does affect contact frequencies [31,32]. Diseases transmitted by face-to-face contact like influenza can be described by two elements [33]: the number of contacts Nc and the actual transmission probability P for a contact between an infected individual and a susceptible one. 4.1. Contact frequency One individual has many acquaintances in the contact list, but it is impossible for the individual to contact all of them in a day. The actual number of individuals that one individual is able to contact in a day is limited [3]. To identify the effective contacts for influenza virus transmission, we only consider two contact patterns: conversation without physical contact and conversation with non-sexual physical contact. For an individual, the number of the two types of effective contacts Nc in a day follows a normal distribution around 26.05, and confidence interval (16.76, 35.34) is under confidence degree 95% [13]. Here, all these contacts are not uniformly assigned to each layer, but in proportion to the decline coefficient λ in Eq. (2). The basic principle is: the farther individuals are away from each other, the smaller probabilities they have of contact with each other. For a student a, his contact list is: L = {(g1 , dormitory), (g2 , class), (g3 , floor), (g4 , building), (g5 , campus)}. Student a might contact with a student a1 in the same dormitory (a1 ∈ g1 ) several times in a day, but might not contact with a student a2 on the same campus (a2 ∈ g5 , and a2 ̸∈ (g1 ∪ g2 ∪ g3 ∪ g4 )) several days. 4.2. Infection rate Transmission probability of a contact is defined as an alternative parameter pi , which fluctuates in terms of different diseases. To model infected individuals infectivity in different infectious periods, pi is multiplied by a coefficient pΓ (t ) to
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Fig. 2a. Newly infected individuals in hospital every day, which is a function of time t (day).
calculate the actual transmission probability P (Eq. (3)). pΓ (t ) is a function of infectious duration t (day), which is cumulated from the day that the individual is infected (Eq. (4)). Normally, individual infectivity reaches peak value one day after symptoms appear [34]. Here, we use Γ (4, 1) to generate the infectivity coefficient. P = pi pΓ (t ) pΓ (t ) =
−t 3
e t 6
(3)
,
t > 0.
(4)
4.3. Root infected individuals To simulate the transmission of influenza H1N1 in the constructed social contact networks, infected individuals were imported as the root ones at the beginning. According to the case we studied, more than one individual were diagnosed as having flu at the beginning of the new semester. We applied two criteria to confirm whether a student was the root infected individual: students who showed symptoms in the first two days; at the same time, students who had no contact history with other infected students. The root infected students were located in 3 dormitory buildings, and there are respectively 6, 5, and 2 root students in them. In the simulation, all the infected students were supposed to be sent to the hospital for treatment for 1–2 days after being infected. Hence, students in hospital were removed from the contact networks. 5. Results and discussion We carried out three sets of experiments to evaluate the generated contact networks. One set reproduced the Langfang case under the same interventions as what were implemented during the outbreak. The other two sets analyzed the dynamic properties of the contact networks for influenza transmission. 5.1. Validating experiments During the outbreak in the Langfang case, three progressive non-pharmaceutical interventions were made to control the spread of the disease on the campus:
• The dormitory buildings were isolated on the 5th day, while students were allowed to move around within a building; • All the dormitories were isolated on the 6th day, while students were confined to their dormitories; • All the dormitory members of students diagnosed with infection were isolated on the 9th day. We implemented the three interventions by cutting off edges between layers of the generated contact networks. The experiments were repeated 1000 times under transmission probability pi = 0.042. Under this parameter, R0 of influenza H1N1 on the campus is 2.1. Fig. 2a shows the number of newly infected students every day (dots) who were sent to the hospital. The experiment results were compared with the clinical records (triangles). In the light of this chart, the accumulative number of infected individuals in the hospital increases rapidly during the first 8 days, and slows down when strict isolating interventions (the third intervention) were executed (Fig. 2b). It indicates that non-pharmaceutical interventions are efficient to prevent the
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Fig. 2b. Accumulative infected individuals in hospital. The dots represent the average result of 1000 experiments; the triangles represent the real clinical record.
Fig. 3. Percentages of accumulative infected individuals to total individuals in 8 dormitory buildings.
spread of influenza H1N1, especially when possible contacts among individuals are cut off. However, this measure disturbs people’s normal life and costs a lot economically. In Fig. 2a, the peak value of experiment results is higher than that of the real data. The reason is that when facing the outbreak, individuals consciously reduce interactive behavior and travels, and however, we did not take this factor into consideration in the disease model. The experiment results also exhibit a spatial clustering property in dormitory buildings. Fig. 3 shows the accumulative percentages of newly infected individuals to total individuals in a dormitory building. The tendency of results (dots) keeps accordant with the clinical data (squares). It indicates that the outbreak of influenza is only clustered in a few buildings. 5.2. Impacts of degree distribution Under the same transmission probability pi , we carried out transmission experiments of influenza H1N1 in the social contact networks without isolating interventions and allowing infected ones to go to hospital 1–2 days after being infected. In this set of experiments, one root infected individual was imported at the beginning. To identify the impact of individual degree on disease transmission, we stochastically pick one individual from all the individuals who have the same degree k as the root student. Transmission experiments were repeated 1000 times under each degree k. Fig. 4 shows the correlation between the degrees of root infected individuals and the average accumulative
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Fig. 4. Accumulative infected individuals under different degrees (k).
Fig. 5. Total infected individuals on the whole campus with one root infected individual for 10 000 repeated experiments.
number of infected individuals on the campus. It can be seen that there is no obvious mapping between degree distribution and influenza transmission results. The root individual degree has almost no impact on influenza transmission. The main reason is that the number of contacts that one individual might have in a day is limited, and this number follows a normal distribution around 26.5. The probability of contact number has no relation with individual degree distribution, and hence, the individual degree cannot directly impact individual daily contact number. In daily life, it is hard to confirm whether the individual who has more social relations can contact more acquaintances in a day. A fact is that proper contact is an activity of physical proximity, which takes time and covers distance. Since it is hard to identify the impact of individual degree, we stochastically selected one individual as the root infected one to study the transmission process in detail. The experiment is repeated 10 000 times under the same conditions. It can be seen that the results split into two distinct parts in Fig. 5, which indicate two possible means of transmission: Case 1: only several individuals are infected, and the number of newly infected individuals is no more than 100; Case 2: almost all the students on the campus are infected. In the first case, influenza is confined to a small range, and the percentage of the first case is 22.51%. Before infected students are sent to the hospital, they infected just a few students, and soon they are all in hospital. In the second case, influenza spreads across the campus, and this case takes 77.49%. It means that once influenza breaks, an outbreak on the campus will take place. If we assume the frequency as the probability, that a root individual will cause spread of influenza H1N1 on a large scale is a probable event.
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Fig. 6. (a) Average infected individuals of 10 experiments for each individual and x axis represents ID of the root infected individual in the experiment. (b) The number of dots in each belt in Fig. 6(a).
Furthermore, to observe the probability of large-scale outbreak (Case 2), each individual on the campus was selected as the root student for transmission experiment. For each selected student, experiments were repeated 10 times. The average results are shown in Fig. 6(a), and the x axis is the student ID. It shows obvious separation (belts). The separation looks different from that in Fig. 5. However, they are the same in essence. Based on the analysis of transmission experiments for an individual above, we can see that there are only two possible conditions. Furthermore, the average number of students that a root individual infected is determined by the probability of Case 2. That also means that the more frequently Case 2 occurs, the larger the average number is. For instance, dots in Belt 8000 indicate the fact that Case 2 occurs 8 times in the 10 experiments for an individual. To study the distribution characters of the dots plotted in Fig. 6(a), we calculate the number of dots in each belt. The results are shown in Fig. 6(b), where the x axis is the frequency of dots, and the y axis corresponds to the belts. It can be seen that in most cases if no interventions are implemented, a root infected individual might cause an outbreak of influenza H1N1 on the campus. However, there is also some probability that the influenza is confined to a small range, and only a few students are infected. 5.3. Spatial clustering According to the definition of individual social relationship in Section 2.1, an individual is allowed to have multiple relationships at the same time. On the campus, each student has 5 types of relations. That means each student belongs to 5 independent networks. These networks respectively locate in 5 different layers of the hierarchical network. Since influenza H1N1 spreads from the root infected student, we picked out the 5 independent networks where the root student locates to analyze the transmission dynamics of different layers. The numbers of accumulative infected individuals in the 5 independent networks are calculated over time. Fig. 7 shows the accumulative infected ratio of the newly infected students to the total students in an independent network. For the 5 layers of the network, the accumulative ratio of layer α = 0 increases at the highest speed, which means that roommates are the highest risky group and all are infected in a short period (about 5 days). Once an infected individual is imported to a susceptible population, the influenza spreads rapidly in the lower layers (α = 0, 1), and then extends to the higher layers (α = 2, 3, 4). In Section 3, we showed the fact that the lower-layer networks have larger clustering coefficients (Fig. 1(b)), and here, the transmission experiments also exhibit that the outbreak has local clustering character. 6. Conclusion In this paper, we presented a novel approach to construct social contact networks based on modular and hierarchical structure with individuals’ spatial distribution information. It also provides an effective method to integrate individual geospatial locations with network topologies. This work can be offered to researchers interested in contagious infection, especially face-to-face transmitted diseases. Based on the constructed contact networks, the transmission experiments reproduced the outbreak of influenza H1N1 occurred in 2009. In the spread process, spatial-topology structure is found to play a significant role in an outbreak of a contagious disease. The local clustering of contact network constrains the spreading scale and speed of the disease. This effectiveness is crucial for controlling the spread of an epidemic disease and making non-pharmacological isolation interventions.
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Fig. 7. The accumulative infected ratio (R) of the newly infected students to the total students in an independent network.
In the constructed networks, individual degree exerts no obvious effect on the spread of influenza in the transmission process. Although individual social relations are heterogeneous from each other, effective contacts for all the individuals in a day are limited by space and time. One limitation of the experiment in the paper is that we did not consider the relation between individual degree and daily contact number. Hence, it is difficult to identify individual influence on the outbreak of an epidemic. As for future work, we will extend the method to larger-scale cases by integrating census data and national administrative hierarchical structure. More detailed investigation data is required to specify modular model and layers of the hierarchical structure. Secondly, individual models, especially the behavior algorithms, will be further studied to figure out what role an individual or a class of people plays in an outbreak. Thirdly, in the light of social contact networks, the threshold of an outbreak is another promising area. Since whether an outbreak occurs or not is a probability event, the threshold of an outbreak is useful to predict the spread of a disease and evaluate policies to be implemented. Acknowledgments The authors would like to thank National Nature and Science Foundation of China under Grant Nos. 9102403. We also acknowledge the invaluable comments and suggestions of anonymous reviewers, and the journal’s editors. References [1] C. Fraser, C.A. Donnelly, S. Cauchemez, et al., Pandemic potential of a strain of influenza A(H1N1): early findings, Science 324 (2009) 1557–1561. [2] V.J. Munster, E. de Wit, J.M.A. van den Brand, et al., Pathogenesis and transmission of swine-origin 2009 A(H1N1) influenza virus in ferrets, Science 325 (2009) 481–483. [3] J. Stehlé, N. Voirin, A. Barrat, et al., High-resolution measurements of face-to-face contact patterns in a primary school, PLoS One 6 (2011) e23176. [4] J. Mossong, N. Hens, M. Jit, et al., Social contacts and mixing patterns relevant to the spread of infectious diseases, PLoS Med. 5 (2008) e74. [5] L.M. Glass, R.J. Glass, Social contact networks for the spread of pandemic influenza in children and teenagers, BMC Public Health 8 (2008) 61. [6] L. Isella, J. Stehlé, A. Barrat, et al., What’s in a crowd? Analysis of face-to-face behavioral networks, J. Theoret. Biol. 271 (2011) 166–180. [7] K. Zhao, J. Stehlé, G. Bianconi, et al., Social network dynamics of face-to-face interactions, Phys. Rev. E 83 (2011) 056109. [8] D. Balcan, V. Colizza, B. Gonçalves, et al., Multiscale mobility networks and the spatial spreading of infectious diseases, Proc. Natl. Acad. Sci. 106 (2009) 21484–21489. [9] D.J. Watts, S.H. Strogatz, Collective dynamics of ‘small-world’ networks, Nature 393 (1998) 440–442. [10] R. Albert, A.-L. Barabási, Statistical mechanics of complex networks, Rev. Modern Phys. 74 (2002) 47–97. [11] S. Riley, Large-scale spatial-transmission models of infectious disease, Science 316 (2007) 1298–1301. [12] L. Hufnagel, D. Brockmann, T. Geisel, Forecast and control of epidemics in a globalized world, Proc. Natl. Acad. Sci. 101 (2009) 15124–15129. [13] W.J. Edmunds, G. Kafatos, J. Wallinga, Mixing patterns and the spread of close-contact infectious diseases, Emerg. Themes Epidemiol. 3 (2009) 10. [14] L. Isella, M. Romano, A. Barrat, et al., Close encounters in a pediatric ward: measuring face-to-face proximity and mixing patterns with wearable sensors, PLoS One 6 (2011) e17144. [15] M. Salathé, M. Kazandjieva, J.W. Lee, et al., A high-resolution human contact network for infectious disease transmission, Proc. Natl. Acad. Sci. 107 (2010) 22020–22025. [16] J.M. Epstein, Modelling to contain pandemics, Nature 460 (2009) 687. [17] D. Lazer, A. Pentland, L. Adamic, et al., Computational social science, Science 323 (2009) 721–723. [18] E. Ravasz, A.-L. Barabási, Hierarchical organization in complex networks, Phys. Rev. E 67 (2003) 026112. [19] K.M. Carley, D.B. Fridsma, E. Casman, et al., BioWar: scalable agent-based model of bioattacks, IEEE Trans. Syst. Man. Cybern. A 36 (2006) 252–265. [20] J. Parker, J.M. Epstein, A distributed platform for global-scale agent-based models of disease transmission, ACM Trans. Model. Comput. Simul. 22 (2011) 2. [21] F. Peruani, G.J. Sibona, Dynamics and steady states in excitable mobile agent systems, Phys. Rev. Lett. 100 (2008) 168103.
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