A Dynamic Network with Individual Mobility for Designing Vaccination

Home , Individual mobility, Network theory

Transactions in GIS, 2010, 14(4): 533–545

Research Article

A Dynamic Network with Individual Mobility for Designing 1 Vaccination Strategies tgis_1201 533..546

Liang Mao Ling Bian Department of Geography Department of Geography University at Buffalo University at Buffalo State University of New York State University of New York

Abstract Vaccination is a primary means to control infectious diseases. Few studies on vaccination strategies have explicitly considered the mobility of individuals. This article aims to evaluate the efficacy of three vaccination strategies in a dynamic social network, in which individuals are mobile between and within communities. The three vaccination strategies are applied to this social network for evaluation, including a travel-based, a contact-based, and a random vaccination strategy. Simulation results show that the contact-based strategy, commonly seen in previous studies, is not always the most effective strategy in dynamic networks. This strategy is preferable for a population with a large number of intercommunity travelers, for instance in urban areas. On the other hand, the travel-based strategy, although directly accounting for individual mobility, is not necessarily the most effective in dynamic networks either. This strategy is recommended for a population with a small number of intercommunity travelers, such as rural areas. In addition, one advantage of the travel-based strategy over the other two is its efficacy in confining the spatial extent of affected areas. Results suggest that the intercommunity travel of individuals should be a major consideration for choosing proper vaccination strategies. By adding the spatial context into vaccination strategies, this research provides new insights into community-based planning for infectious disease control.

1 Introduction

Infectious diseases, such as severe acute respiratory syndrome (SARS) and recent H1N1 inﬂuenza, are some of the leading causes of death worldwide (Lopez and Murray 1998,

Address for correspondence: Liang Mao, Department of Geography, University of Florida, Gaines- ville, FL, 32611, USA. E-mail: liangmao@uﬂ.edu

Morens et al. 2004, Neumann et al. 2009). This category of diseases is characterized by transmission from one individual to another through a social network (Edmunds 1997, Potterat et al. 1999). To date, vaccination remains the primary means to control the disease transmission within a population. Because the supplies of vaccines are often limited, vaccination tends to target only a fraction of population and protect the remaining unvaccinated population (Fiore et al. 2007). Exploring effective vaccination strategies has been a paramount issue for researchers, health policy makers, and the general public (Longini and Halloran 2005, Weycker et al. 2005). The ultimate goal of vaccination strategies is to minimize the number of infected individuals and confine the affected areas (Riley and Ferguson 2006). Two crucial factors are often considered in the design of vaccination strategies. One is the social network, through which the disease is transmitted; the other is the targeted population, to which the vaccination is applied. Based on the understanding of these two factors, a variety of vaccination strategies have been proposed, and valuable guidance has been provided for implementation (Pastor-Satorras and Vespignani 2002, Rhodes and Anderson 1997, Zanette and Kuperman 2002). Current vaccination strategies are primarily based on static social networks. Indi- viduals in these social networks tend to be immobile and the contacts between them are fixed (Keeling and Eames 2005, Miramontes and Luque 2002). In reality, individuals move frequently on a daily basis, such as between homes and workplaces (Bian 2004). Their contacts form and break with location and time, resulting in a spatiotemporally dynamic social network. The dynamics of this social network may further affect the efficacy of vaccination strategies (MacPherson and Gushulak 2001, Sattenspiel and Dietz 1995). In the current literature, little is known about whether the current vaccination strategies would remain effective in a dynamic social network. Furthermore, in static social networks, the identification of the targeted population is mainly based on social relationships among individuals. The most typical strategy is to target individuals who have a great number of social contacts (Pastor-Satorras and Vespignani 2002). In a dynamic social network, how many contacts an individual has and who this individual has contact with, may change with location and time. In such a dynamic context, the characteristics of the targeted population may need to be reconsidered (Bian and Liebner 2007, Eubank et al. 2004). The objective of this research is to evaluate the efficacy of vaccination strategies in a set of dynamic social networks with mobile individuals. The objective is fulfilled by two steps. First, a set of dynamic social networks is established by varying the mobility of individuals within and between communities. The effects of individual mobility on the network structure are examined. Second, three vaccination strategies, including a travel- based, a contact-based, and a random vaccination strategy are applied to these dynamic social networks. The travel-based vaccination strategy is proposed, in particular, to correspond to individual mobility in the social network. The contact-based strategy is adopted from the current literature, and is widely studied in static social networks. The random strategy serves as a reference for comparison purposes. The efficacy of three vaccination strategies is evaluated by the percent of infection in the population, as well as the spatial extent of infection. The remainder of this article is organized into the following sections. The next section describes two sets of concepts: social networks and vaccination strategies, which are relevant to this research. The third section explains the design of dynamic social networks and three vaccination strategies. The fourth section that follows presents and

© 2010 Blackwell Publishing Ltd Transactions in GIS, 2010, 14(4) Vaccination on Dynamic Social Networks 535 discusses simulation results. The ﬁnal section concludes the article and articulates the implications of this research.

2 Social Networks and Vaccination Strategies

Conceptually, networks are composed of nodes and links. In the context of social networks, the nodes represent individuals and the links represent direct contacts among individuals (Scott 2000, Wasserman and Faust 1994). Through the links, diseases can spread from one individual to another in a population. The structure of social networks determines how diseases are transmitted between individuals, and subsequently the spatial and temporal heterogeneity in disease transmission (Koopman and Lynch 1999). The vaccination strategies are designed to control disease transmission, and thus their efficacy may also vary depending on the structure of social networks (May and Anderson 1984, McKenzie 2004). In network theory, the structure of social networks can be characterized by a set of parameters. One such parameter closely related to disease transmission is the “maximum connected component” (MCC). The MCC is defined as the largest subnetwork, in which all nodes are connected directly or indirectly by one or more links (Scott 2000). The total number of nodes in an MCC is referred to as the size of MCC, indicating the connectivity of a network. A larger size MCC indicates a higher degree of network connectivity, and implies a greater number of individuals that can be potentially infected during an epidemic (Holme 2004, Newman 2002). For a static network, the MCC remains con- stant because the contacts between individuals are fixed. With respect to a dynamic network, individuals may move and come into contact with different groups of individuals at different times and locations. The corresponding MCC may change as a result of individual mobility, and subsequently affect the efficacy of vaccination strategies. The effects of individual mobility patterns on the MCC and on the vaccination strategies are explored in this research. Vaccination means to immunize a fraction of the population and subsequently remove the targeted population from a social network. Current vaccination strategies identify the targeted population primarily based on social relationships among individuals, and can be characterized as contact-based strategies. Many studies have suggested that vaccination should be targeted to those individuals who are socially close to an infectious individual, such as family members or co-workers of this individual (Kiss et al. 2005, Rhodes and Anderson 1997). Other studies have reported that individuals with a large number of social contacts, or individuals having contacts across many social classes, should be a priority in vaccination (Pastor-Satorras and Vespignani 2002, Zanette and Kuperman 2002). In this research, the efficacy of a contact-based strategy is evaluated in dynamic social networks. In addition to the characteristics of the targeted population, vaccination strategies also consider the size of the targeted population, hereinafter called vaccination fraction. Vaccinating all individuals in a population (100% vaccination fraction) obviously achieves the highest efficacy, but it is often impractical due to the limited supply of vaccines (Emanuel and Wertheimer 2006, Longini and Halloran 2005). Various vaccination fractions have been studied with the combinations of different social networks and characteristics of targeted populations. These studies have reported that disease transmission can be controlled with a low vaccination fraction, if vaccines are applied to

(a) (b) Figure 1 A modeled social network with individual mobility. (a) The two tiers of population at day- (the upper tier) and night-time (the lower tier), respectively. Nodes are individuals. The lines between tiers link the same individual at different times and locations, representing the travel behavior of individuals. (b) The social network at the daytime tier. Each cluster of nodes represents a community. Lines represent the contacts among individuals within a community. The social network at the night-time tier has similar structure, but individuals’ spatial locations and social contacts are distinct

“proper” individuals in a social network (Burke et al. 2006, Keeling and Eames 2005, Longini et al. 2005). The identiﬁcation of these proper individuals is always an actively pursued topic in public health, which is addressed in this research.

3 Methodologies

The evaluation of vaccination strategies is conducted with two procedures. First, dynamic social networks are established by varying mobility of individuals within and between communities, referred to as intra- and intercommunity travel, respectively. The effects of mobility pattern (intra- versus intercommunity travel) on the network connectivity are investigated. Second, the three vaccination strategies are applied to the dynamic social networks. Their efﬁcacy is evaluated and compared under two typical mobility patterns.

3.1 Dynamic Social Networks A dynamic social network is conceptualized as an individual-based two-tier network. Each individual in a population is a modeling unit, while the two tiers refer to the population at two time periods: day- and night-time, respectively. The establishment of a social network is based on four assumptions. First, individuals have explicit spatial locations at day- and night-time, respectively (Figure 1a). They are distributed in a number of communities, where they may live and work (Figure 1b). Second, individuals travel between locations on a daily basis (Figure 1a). Some individuals live and work in the same community, hereinafter referred to as intracommunity travelers. Other individuals live in one community and work in a different community, hereinafter referred to as intercommunity travelers.

Third, the intercommunity travelers are more likely to travel to a community near their home communities. This assumption is consistent with many observations, which demonstrates that human travel behavior decays with spatial distance (Brockmann et al. 2006, Clark and Unwin 1981, Rushton 1969). Fourth, individuals have contact with other individuals only in the same community, where the individuals live, work, or both (localized contacts). Within each community, a few individuals have a signiﬁcantly larger number of contacts than the remaining individuals, resulting in a “scale-free” network structure (Albert et al. 2000, Pastor-Satorras and Vespignani 2001). The localized contacts and the scale-free structure have been observed in many social networks that are responsible for disease transmission (Athanasiou and Yoshioka 1973, Liljeros et al. 2001, Riley et al. 2003, Wellman 1996). These four assumptions are incorporated in a computing model with a population of N = 5,000. This computing model is implemented by the following four steps. First, each individual in the population is assigned two sets of spatial locations, expressed as (x, y) coordinates, at day- and night-time, respectively. The 5,000 individuals are grouped into 50 communities using the K-mean clustering algorithm. Those individuals who are spatially close together are grouped into the same community. Second, a proportion of individuals ( p) are randomly selected to be the intercommunity travelers. The remaining proportion of individuals (1 - p) is the intracommunity travelers. Third, an intercommunity traveler from community i is assigned to travel to community j with a probability 22 that follows a distance-decay function, ()()11ddij∑ ij , where dij indicates the distance between communities i and j. Fourth, individuals within a community are linked by a preferential attachment algorithm, proposed by Albert et al. (2000), to implement a scale-free network structure. This algorithm links each new individual preferentially to individuals who already have a large number of links (Albert et al. 2000, Pastor-Satorras and Vespignani 2001). The average number of direct contacts that individuals have is set to eight, ensuring that every individual has at least two contacts in both day- and night-time. This number of direct contacts is deemed to be adequate relative to the size of the population, although eight may seem to be a little lower than real situations. In such a computing model, intercommunity travelers connect otherwise separate communities into a population-wide social network, because they have contacts in two different communities. Consequently, different mobility patterns, represented by the value of p (the proportion of intercommunity travelers in the population), may result in different degrees of network connectivity, represented by the size of MCC (see Section 2). In this research, the P-value is varied to evaluate this effect. In a preliminary study, the MCC is evaluated against a range of P-values, from 0 to 20% using a 1% increment, from 20 to 50% using a 5% increment, and from 50 to 100% using a 10% increment. Results indicate two types of MCC sizes, representing two distinct states of network connectivity. Two P-values are, then, selected to represent these two states of network connectivity (two predeﬁned networks) in the subsequent analysis. The details are described later in Section 4.1.

3.2 Design of Vaccination Strategies Three vaccination strategies, a travel-based, a contact-based, and a random vaccination strategy, are designed to vaccinate a fraction of the population via different identiﬁcation methods. The travel-based vaccination strategy identiﬁes individuals who travel between

© 2010 Blackwell Publishing Ltd Transactions in GIS, 2010, 14(4) 538 L Mao and L Bian communities, and gives priority of vaccination to these individuals. The contact-based vaccination strategy prioritizes individuals who have a large number of direct contacts. The random vaccination strategy identifies a fraction of the population for vaccination based on random selection. Because vaccination strategies are sensitive to the size of the targeted population, as discussed in Section 2, a range of vaccination fractions f is considered. According to preliminary experiments, the value of f ranging from 1 to 10% using a 1% increment is most relevant to this research. The three vaccination strategies are designed to control an influenza epidemic in each of the two predefined networks. Influenza is chosen for the simulation because it is common and readily transmitted between individuals. Each individual is assigned with an infection status, susceptible, infectious, or recovered. An individual assigned susceptible status may change to the infectious status with a probability of 10%, once in contact with an infectious individual. Four days after being infected, an infectious individual changes to the recovered status, and is immune to infection for the remainder of the epidemic. One infectious individual is introduced into a susceptible population to ini- tialize the transmission. The simulation takes a bi-daily time step between day- and night-time over 150 days. Vaccination strategies are applied before the first infectious individual is introduced. The efficacy of these strategies is evaluated by the percent of the population that is infected and the spatial extent of infection. In total, for the two predefined networks, three strategies, and 10 vaccination fractions, 60 (2 ¥ 3 ¥ 10) combinations are simulated. For each combination, the efficacy of vaccination strategies is estimated by running 100 realizations, resulting in a total number of 6,000 realizations.

4 Simulation Results and Discussion

4.1 Effects of Mobility Pattern on Network Connectivity Figure 2 displays the effects of individual mobility pattern ( p) on the network connectivity with respect to the size of MCC. When p is zero, there are no intercommunity travelers existing in the social network, and the communities are isolated from each other. The MCC has the minimum size, which is approximately the size of a community (the community to which the ﬁrst infection is introduced). As p grows larger, a dramatic rise in the size of MCC is observed, and the size of MCC reaches the maximum at P = 8%. The added intercommunity travelers, as p increases to 8%, connect an increasing number of communities together, resulting in a rapid growth in the network connectivity. At the point of P = 8%, all individuals in the social network can be connected directly or indirectly. As p exceeds 8%, the size of MCC is stabilized at the maximum, but the connections between communities tend to be increasingly redundant due to the large number of intercommunity travelers. Two distinct states of network connectivity are emerging in Figure 2, as a result of alternative mobility patterns ( p). One is the “underconnected” state, occurring when the size of MCC is less than its maximum. In such a state, not all individuals in the social network are connected, and certain communities are still separated from other communities. The other is the “overconnected” state after all individuals are connected. All communities are connected with one another by at least one intercommunity traveler between each pair of them. Two P-values, P = 3% and P = 12%, are selected to represent

5000

4500 MCC < N MCC = N 4000

3500

3000

2500 Under- Over- connected connected Size of MCC 2000

1500

1000

500 010155 The % of intercommunity travelers in a population (p)

Figure 2 The size of the maximum connected component (MCC) as a function of p. The size of MCC indicates the degree of network connectivity, while the P-value represents the mobility pattern of individuals. The delineation between the under- and overconnected networks occurs at P = 8% typical under- and overconnected networks, respectively. These two deﬁned networks are used for evaluating the three vaccination strategies, in terms of the percent of infection and the spatial extent of infection, as described below.

4.2 Efﬁcacy of Vaccination Strategies on Percent of Infection Effective vaccination strategies are expected to produce a low percent of infection in the population. Figures 3a and b display the percent of infection resulting from the three vaccination strategies in the under- and overconnected networks, respectively. In the underconnected network (P = 3%), about 50% of the population is infected if no vaccination is applied (f = 0), representing a baseline situation of an inﬂuenza epidemic. As the vaccination fraction f increases, all of the three vaccination strategies can reduce the percent of infection, but to different degrees. The travel-based vaccination strategy outperforms the other two strategies. For a given value of f, the travel-based strategy produces the lowest percent of infection, followed by the contact-based strategy, while the random strategy produces the highest percent. It is also worthwhile mentioning that the performance of the travel- and contact-based strategies converge as the vaccination fraction exceeds 7%, both successfully suppressing the epidemic. In contrast with the underconnected network, the overconnected network (P = 12%) leads to about 80% of infection in the population without vaccination (f = 0). This baseline situation is much higher than the 50% in the underconnected network. The reason is that all individuals in the population can be connected through one or more links (the size of MCC = N), thus exposing it to greater potential infection. In this overconnected network, the contact-based vaccination strategy performs best, followed by the travel-based strategy, and then the random strategy. The latter two strategies perform similarly when the vaccination fraction is low. As the vaccination fraction

50 Random Vac. 80 45 Contact-based Vac. 70 40 Travel-based Vac. 35 60 30 50 25 40 20 30 15 % of Infections % of Infections 20 10 Random Vac. Contact-based Vac. 5 10 Travel-based Vac. 0 0 02468100246810 Vaccination Fraction % (f) Vaccination Fraction % (f) (a) (b)

Figure 3 The efficacy of three vaccination strategies as a function of f in two networks, P = 3% in (a) and P = 12% in (b). The Y axis represents the percent of infection, and the X axis represents the vaccination fraction f. The three curves represent the three vaccination strategies: the random (circle), contact-based (square) and travel-based (triangle) vaccination strategies, respectively reaches a relatively high level (above 7%), the travel-based strategy begins to take effect. The random strategy remains the least effective. Its efficacy remains low even after 10% of the population has been vaccinated. The three vaccination strategies exhibit different efficacy under two different networks. In the underconnected network (P = 3%), the travel-based vaccination strategy is the most effective, probably because of the following reasons. There are only a small number of individuals traveling between communities, and thus this strategy can easily break the entire network into isolated communities. The disease transmission is confined within individual communities, infecting only a small number of individuals. The contact-based strategy is less effective because it only vaccinates those who have a great number of contacts, but not necessarily the intercommunity travelers. The network connectivity remains relatively high as communities are kept connected, until a high vaccination fraction (7%) is reached, when all intercommunity travelers are removed from the network. Turning to the overconnected network (P = 12%), the contact-based vaccination strategy performs the best, probably as the result of the following reasons. There exist a large number of intercommunity travelers in the social network, keeping all communities redundantly connected. The travel-based strategy cannot break the social network into isolated communities as effectively as it does in the underconnected network, and thus the disease can be transmitted freely between communities. On the other hand, because of the scale-free structure of the network, the contact-based strategy can quickly remove those highly connected individuals, and increase the social distance between remaining individuals. The disease transmission is more difficult, because the infection from one individual needs to pass more individuals to reach another. The random strategy is insensitive to both individual mobility and number of contacts, making it the least effective strategy regardless of how the networks are connected. This is consistent with the established results, which show that the random

(a) Travel-based (b) Contact-based (c) Random Figure 4 The spatial patterns of infection in the underconnected network produced by three vaccination strategies. The black nodes represent infected individuals and gray nodes represent susceptible individuals vaccination strategy only works when the vaccination fraction is above 80% (Cohen et al. 2003, Pastor-Satorras and Vespignani 2001). Although the contact-based strategy is widely recommended, it may not always be the most effective in dynamic networks. This strategy can be replaced by the travel-based strategy in networks that are underconnected because of a few intercommunity travelers. On the other hand, even though the travel-based strategy is designed directly to correspond to individual mobility, it is not necessarily the most effective in dynamic networks either. This strategy would lose its efﬁcacy if applied to networks that are overconnected due to many intercommunity travelers.

4.3 Efficacy of Vaccination Strategies on Spatial Extent of Infection From a spatial perspective, effective vaccination strategies are expected to confine the spatial extent of affected areas. Figures 4 and 5 show the spatial patterns of infection in the under- and overconnected networks, respectively. The location at which the infection occurs is mapped for the three vaccination strategies. To make the illustration legible, only 2,500 individuals are simulated and displayed. In the underconnected network, the travel-based vaccination strategy performs the best, followed by the contact-based strategy, and then the random strategy. The travel-based strategy produces a small spatial cluster of infection, in addition to the smallest number of infection cases, as described previously (Figure 4a). Both the contact-based and random strategies (Figures 4b and c) cause a wide spread of infection, affecting many communities in the population. This is because both strategies do not consider the distinction between intra- and intercommunity travel of individuals. Between the two, the contact-based strategy is deemed more effective because it produces a smaller number of infection cases. With respect to the overconnected network, the travel-based strategy remains the most effective strategy from a spatial perspective. Although it produces more infection cases than the contact-based strategy, the cases are highly clustered in certain communities (Figure 5a). The infection can only happen locally, because individuals travel between communities following the distance-decay function. The contact-based strategy is less effective in confining the spatial spread of diseases, but it can reduce the number

(a) Travel-based (b) Contact-based (c) Random Figure 5 The spatial patterns of infection in the overconnected network produced by three vaccination strategies. The black nodes represent infected individuals and gray nodes represent susceptible individuals of infection cases to the greatest extent (Figure 5b). The random strategy is the least effective, because it fails to control both the number of infection cases and the spatial extent (Figure 5c).

5 Conclusions and Implications

This article focuses on evaluating vaccination strategies in a social network with mobile individuals. Dynamic social networks are established by explicitly representing individuals that take intra- and intercommunity travel. The simulation results show that the mobility pattern (intra- versus intercommunity travel) has profound effects on the network connectivity, and consequently on the efficacy of vaccination strategies. The contact-based strategy is more effective in the overconnected networks, while the travel-based strategy performs the best in the underconnected networks. The results suggest that understanding the travel of individuals between and within communities is crucial for choosing proper vaccination strategies. The incorporation of community-level travel data into health surveillance can provide valuable support for public health decisionmaking. In reality, the under- and over-connected networks may correspond to two types of societies, where the spatial and social behaviors of individuals differ. The modeled communities in a social network can be defined by considering different socio-economic and cultural characteristics of geographic units (Knox and Marston 2004, McMillan 1986, Morgan and Moss 1965). The underconnected networks can correspond to rural areas, where small villages or towns are sparsely distributed and far apart from one another. These villages or towns can be considered as communities in vaccination strategies. Only a small number of individuals may travel between communities regularly each day, and the connections between communities are relatively weak. Under this circumstance, the travel-based vaccination strategy would be more preferable. On the other hand, the overconnected networks can represent urban areas, where communities may be defined as districts for administration, taxation, education, or other purposes. A large number of individuals travel within and between communities on a daily basis, and

© 2010 Blackwell Publishing Ltd Transactions in GIS, 2010, 14(4) Vaccination on Dynamic Social Networks 543 the connections between communities are robust. In such a situation of high network connectivity, the contact-based vaccination strategy would be more preferable. Practically, an optimal vaccination strategy should assess the trade-offs between efficacy and implementation (Euler et al. 2005, Longini and Halloran 2005). The random vaccination strategy is easy to implement, because no information is required about the population. Anyone can receive vaccination by going to a designated station, regardless of demographic, social, and spatial characteristics. This strategy, however, takes effect only when a large fraction of population is vaccinated, leading to high costs of vaccines. The contact-based vaccination strategy has been extensively studied before, but it is difficult to implement for two reasons. First, identifying individuals with a large number of contacts requires information about every individual in a social network, which is often infeasible to acquire (Gomez-Gardenes et al. 2006, Holme 2004). Second, in health surveillance, the contacts of an individual are difficult to define clearly. The information about “who has contact with whom” is highly dependent on subjective assessment (Cohen et al. 2003). Compared to the previous two strategies, the travel-based vaccination strategy might be a more practical choice. First, the travel-based strategy could confine the spatial spread of the disease, minimizing the affected areas. Other control measures applied with vaccination, such as antiviral drug use, household quarantine, and school/workplace closure, only need to be targeted to certain communities, rather than the entire population. The socio-economic burdens of the epidemics can be greatly reduced. Second, the information about intra- and inter-community travelers is widely available in the existing census data and travel survey reports. The patterns of individual mobility can be easily estimated, based on which, proper vaccination strategies can be chosen. Lastly, the travel-based strategy would be easier to implement. Vaccination stations can be established at transportation hubs, such as bus terminals and subway stations, where intercommunity travelers occur frequently. This research is a pilot study of vaccination strategies in a hypothetical population. The principles identified in this research, i.e. the intricate effects of social network and spatial mobility on disease control, can be extended to a realistic population for public health analysis and policy implementation. More sophisticated models can be established based on census, workplace, land use, and behavioral survey data. As the social, spatial, and behavioral aspects of individuals are incorporated, comprehensive evaluations of vaccination strategies can be performed, but the modeling complexity is increased as well. In summary, the transmission of diseases is a mixed product of two mutually dependent components: the social and spatial relationships between individuals (Bian 2004, Cliff and Haggett 2004, Keeling 1999). Ignoring either component may prevent us from exploring effective vaccination strategies. Much attention has been paid to the social component, while the spatial component has often been under-studied. This research integrates both components simultaneously into vaccination strategies through the explicit consideration of individual mobility. The results provide new insights into community-based planning for controlling the impending influenza pandemic and other emerging infectious diseases.

Endnote

1 An earlier version of this manuscript was presented at the 2009 Summer Assembly of the University Consortium of Geographic Information Science and awarded the Transactions in GIS Best Student Paper Prize.

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