Modelos Epidemiológicos

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Review Article

Spatial Approaches to ModelingDispersion of CommunicableDiseases A Review

Ling BianDepartment of GeographyUniversity at Buffalo

AbstractThe dispersion of communicable diseases in a population is intrinsically spatial. In

the last several decades, a range of spatial approaches has been devised to model

epidemiological processes; and they differ significantly from each other. A review of

spatially oriented epidemiological models is necessary to assess advances in spatial

approaches to modeling disease dispersion and to help identify those most appro-priate for specific research goals. The most notable difference in the design of these

spatially oriented models is the scale and mobility of the modeling unit. Using two

criteria, this review identifies six types of spatially oriented models. These include:

(1) population-based wave models, (2) sub-population models, (3) individual-based

cellular automata models, (4) mobile sub-population models, (5) individual-based

spatially implicit models, and (6) individual-based mobile models. Each model type

is evaluated in terms of its design principles, assumptions, and intended applications.

For the evaluation of design, four aspects of design principles are discussed: the

modeling unit, the interaction between the modeling units, the spatial process, and

the temporal process utilized in a design. Insights gained from this review can beuseful for devising much-needed spatially and temporally oriented strategies to

forecast, prevent, and control communicable diseases.

1 Introduction

Communicable diseases are transmitted from individual to individual. Efforts to develop

models to forecast epidemics of these diseases can be traced back to the 18th century

(Blower 2004). The last several decades have seen the most dynamic period of modeling

development. While the temporal dynamics of epidemics have always been a primary

Address for correspondence:Ling Bian, Department of Geography, University at Buffalo, Amherst,NY 14261, USA. E-mail: [email protected]

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focus of most models, attempts have also been made to model spatial dynamics in

epidemic processes, as the spread of communicable diseases through a population is

intrinsically both a spatial and temporal process. Geographers have led these attempts,

creating explicit conceptual frameworks to describe spatial processes. Increasingly,

researchers in other disciplines have, to varying degrees, incorporated spatial consider-ations into their models, especially with advances in Geographic Information Science

(GIScience). The spatial dynamics of epidemics can now be readily represented, although

some efforts are more explicit than others. A range of spatial approaches have been

devised and they differ significantly from each other, possibly due to their roots in a

diverse range of disciplines and a timespan of several decades when theories and methods

advanced rapidly. A review is necessary to evaluate the characteristics of these spatial

approaches, to help identify those most appropriate for specific research goals, and to

further our ability to forecast, prevent, and control these diseases. This review categorizes

and evaluates designs of spatial approaches incorporated in epidemiological models with

an emphasis on more recent developments.The most notable difference in the design of these spatially oriented models is the

scale of the modeling unit, which directly affects the spatial representation of many

aspects of model design and operation. Further, the mobility of these modeling units is a

salient difference that separates certain model designs from others, especially from the

perspective of spatial modeling. Using these two criteria, this review identifies six types

of models that incorporate spatial considerations. These include: (1) population-based

wave models, (2) sub-population models, (3) individual-based cellular automata models,

(4) mobile sub-population models, (5) individual-based spatially implicit models, and (6)

individual-based mobile models. These model types are first identified using these two

criteria and then by a chronology of their development, based on the time period when

discussion of the model type was most active in the literature.

Each model type is evaluated in terms of its design principles, assumptions, and

intended applications. For the evaluation of design, four aspects of design principles are

discussed: the modeling unit, the interaction between the modeling units, the spatial

process, and the temporal process utilized in a design. The assumptions and applications

are discussed with respect to these four principles. In addition, before the six types of

spatially oriented models are evaluated, classic non-spatial epidemiological models are

reviewed first. These classic models are the foundation of modern epidemiology, from

which many spatially oriented models are extended or derived. The review of classic

models is also organized by their design principles, assumptions, and applications.

The model design, instead of the model implementation, is the central issue of this

review because it is the conceptual design that guides model implementations. Often the

latter can be quite independent of a model design as implementation concerns change

according to the computing environment. In addition, the model design identifies para-

digmatic shifts in modeling concepts. The review is drawn from a variety of research

articles, ranging from those that are commentary or critique oriented, to design or

method oriented, to case study oriented. Spatial considerations are expressed in these

articles to various degrees, and at times not at all. This review extracts spatial consid-

erations from these articles whether they are explicitly described or implicitly alluded to.

Further, not all of the articles that have used a spatial design are included in this review,

as its goal is rather to distill primary spatial approaches from the literature than to offerstatistics from all relevant articles. The spatial approaches reviewed here focus on those

that address communicable diseases, i.e. the diseases transmitted between individuals

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through close contact or proximity, such as flu, measles, and foot and mouth disease,

among many others.

2 Classic Models

Classic models refer to the population-based, temporally dynamic, mathematical epide-

miological models. They are non-spatial models whose primary focus is the temporal

dynamic of epidemic processes. These models use a population-based approach with a

population divided into a limited number of segments. The simplest division consists of

three mutually exclusive segments. One segment includes those individuals who are

susceptible to the disease (S). A second segment is comprised of those that are infectious

and can spread the disease to the susceptible individuals (I). The remaining segment refers

to those that are recovered from a previous infection (R) (Kermack and McKendrick

1927, Anderson and May 1991). The three population-segment models are commonlyknown as SIR models where the population segments are the modeling unit.

At any given time, a number of individuals are moved from the susceptible segment

into the infectious segment, and in the mean time, a number of individuals are moved

from the infectious into the recovered segment. Assuming that these changes are con-

tinuous, a differential equation set is typically used to express the dynamics:

dS dt SI = , (1)

dI dt SI gI = (2)

dR dt gI = (3)

where S, I, and R denote the susceptible, infectious, and recovered segments, respectively,

b is the infection coefficient and g is the recovery rate. The differential equation set

describes the temporal dynamics of an epidemic by estimating the size of the three

population segments through the course of an epidemic. The size of S, I, and R is available

or can be estimated from observed information. The parameters, such asbandg, are the

unknowns in the equation set. During the modeling, the observed number of new daily (or

other time periods) infection cases is plotted against time. The size of the infectious

population segment tends to rise after an epidemic begins and then declines after reaching

its peak. By adjusting the value of the parameters, the curve that is formulated by the

equation set is fitted to the observed number of infections through the course of an

epidemic. Values for the parameters are then derived for subsequent analysis to decipher

their implications in the epidemic process. As the population segment is the modeling unit,

the values of these parameters are usually estimated as an average over the entire segment.

SIR models are the simplest of a family of population-based models. More complex

models may take additional considerations into account. The three population segments

can be further divided into a greater number of smaller segments. The infectious segment,

for example, has been further divided into two population segments, those who are

exposed to an infection (E) and those who are infected and become infectious (I), whilethe S and R segments are kept unchanged. The four segments construct a SEIR model

that is also a basic form of this family of models. A variety of complexities have been

Spatial Modeling of Communicable Diseases 3



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introduced into the basic deterministic framework of SIR models. The basic modeling

framework, however, has remained the same throughout the history of modern epide-

miology (Anderson and May 1991). At the core of the design is the use of aggregated

population segments as the modeling unit and a focus on the temporal dynamic of

epidemics. This type of model does not consider spatial dynamics at all and employs afundamentally non-spatial modeling approach. There is also no explicit consideration of

the interactions between population segments.

In addition to their temporally continuous assumption, a series of other assumptions

are embedded in these models regarding individuals. Several such explicit and implied

assumptions are relevant to this discussion. First, all individuals within a population

segment are assumed to be identical. Second, all individuals interact with all other

individuals, the so-called homogeneous mixing assumption. Spatially oriented

assumptions are not explicitly discussed, but perhaps can be inferred. Individuals in a

homogeneously mixed population may also have a homogeneous spatial distribution,

and are immobile to allow the model to be executed (see the discussion of a wave modelin a later section).

The deterministic approach and a small number of parameters of these models allow

for a simple modeling process. These models can reasonably approximate the observed

dynamics of an epidemic and assess the collective state of a populations health. Classic

population models have been used for well over a century, and they have been the

foundation for modern epidemiology.

2.1 Population-Based Wave Models

In the 1980s, geographers proposed a spatial framework for epidemiological models that

explicitly considers the spatial dispersion of infectious diseases (Cliff and Ord 1981, Cliff

et al. 1986). A simple form of these spatial models is a three-ring wave model. The first

infection case occurs at the center of a space and spreads outwards in all directions in a

wave-like form. The infectious population segment is on the crest of the wave, the

susceptible segment is in front of the crest, and the recovered segment is behind the crest.

At a given time, the three population segments form a three-ring pattern. The second or

middle ring is the infectious segment, the outer ring is the susceptible segment, and the

recovered ring is at the center of the space around the starting point. The location of the

three population segments, or the three rings, changes dynamically as the infectious wave

spreads through the space (Cliff and Ord 1981, Cliff et al. 1986, Rhodes and Anderson

1997).

The three-ring wave models project the three population segments described in

classic models into space, thus extending their temporally focused and non-spatial

model framework into the spatial dimension. Except for considerations specifically

intended for spatial modeling needs, the wave models inherit all the design principles

of classic models in terms of their use of a population segment as the modeling unit,

the exclusion of the interactions between the modeling units, and their temporal mod-

eling capability. All the assumptions of classic models regarding individuals are also

adopted verbatim, i.e. individuals are identical, homogeneously mixed, homogeneously

distributed, and immobile.

Diseases, as a phenomenon, move across space, but each individual who becomesinfected and subsequently transmits the disease remains immobile. The mobility of

diseases is represented as changes in the health states (susceptible, infectious, or recovered)

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of these immobile individuals much like the modeling of water waves, in which individual

water molecules vibrate vertically with zero horizontal motion, while the wave moves

forward. The motion of disease waves can be modeled in a similar fashion.

Population-based wave models mark the beginning of spatially explicit epidemiologi-

cal models. These models have a spatially oriented, explicit conceptual framework thatcould be implemented by the spatial analysis methods available at the time, for example,

the cellular automata method (Tobler 1970) and subsequently the micro-simulation

method (Amrhein and MacKinnon 1988). This seminal design has had a long lasting

influence on the spatial modeling approaches adopted in many generations of epidemio-

logical models.

In the past decade, population-based models have drawn increasing criticism. It is

argued that the homogeneity assumptions in these models inherently limit the usefulness

of population-based models in explaining the observed heterogeneity in disease trans-

mission, spatially as well as temporally (Holmes 1997, Koopman and Lynch 1999, Fuks

and Lawniczak 2001, Arita et al. 2003, Dye and Gay 2003, Francesconi et al. 2003,Meyers et al. 2003, Galvani 2004, Koopman 2004, Kretzschmar et al. 2004, Galvani and

May 2005, Watts et al. 2005). The strength of these models, however, remains their

ability to predict the health outcome at the population level. This is because the design

principles and assumptions associated with these models are intended for modeling at

this very level. One application for population-based wave models is for pandemics

where a disease sweeps through a large space like a wave, such as the 19181920 Spanish

flu that spread globally (Holmes 1997).

3 Sub-Population ModelsSub-population models divide a population into a substantial number of sub-

populations, where a sub-population is the basic modeling unit. The later 1990s saw a

surge in the development of this type of model. These models attempt to increase

heterogeneity in a population in order to produce more realistic results than those

produced by classic models. A population may be divided into a greater number of

smaller sub-populations using different criteria. Those models that divide a population

based on spatial considerations are called spatially structured models (Szymanski and

Caraco 1994, Lloyd 1995, Ferguson et al. 1997, Grenfell and Harwood 1997, Rhodes

and Anderson 1997, Torres-Sorando and Rodriguez 1997, Keeling 2000).

Most spatially structured models divide space into regular grid cells. Each cell

contains a sub-population; thus a sub-population is spatially registered. The sub-

population models are a straightforward derivative of classic models. Except for a finer

scaled modeling unit, the sub-population models use the same design principles as

population-based wave models to represent spatial and temporal processes, whilst

appearing to be developed independently of the population-based wave models. With an

increased number of modeling units, the interactions between them are added to these

models, although the mechanism of interaction is usually not explicitly explained.

Diseases pass between sub-populations through between-cell interactions and ultimately

move across space, while the cells remain immobile. In addition, sub-population models

inherit the assumptions of classic models at the sub-population level. Within a cell,individuals are assumed to be identical, homogeneously mixed, homogeneously distrib-

uted, and immobile.




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The breakdown of a population into finer-scaled modeling units and the interactions

between these units distinguish the sub-population models from the population-

based wave models, although the homogeneous mixing assumption and the associated

deterministic approach remain within a cell. This change adds spatial heterogeneity into

the estimation of a populations health and has had a major impact on many aspects ofepidemiological modeling. With certain limitations, the sub-population models bring the

modeling results to a more realistic level (Ferguson et al. 1997, Grenfell and Harwood

1997, Keeling 2000). Because of the intra-cell homogeneity assumption, sub-population

models are well suited for modeling disease dispersion between high-density, immobile

communities (Rhodes and Anderson 1997). Alternatively, these models are also well suited

for modeling disease transmission in livestock, such as localized breakouts of foot and

mouth disease among cattle and sheep on farms. Cattle and sheep are homogeneously

mixed sub-populations on individual farms, and the disease disperses through adjacent

farms and long jumps between them (Keeling et al. 2001, Doran and Laffan 2005).

Almost a decade later, especially after much discussion over the limitations of classicmodels, a variety of model designs have been proposed for sub-population models. One

such revision refines the transmission between cells by explicitly allowing exchange of

individuals or chance of interaction between cells (or other types of basic spatial units)

(Doran and Laffan 2005, Watts et al. 2005, Colizza et al. 2007, Mao and Bian 2010).

Disease transmission between cells is modeled using probability-driven or rule-based

approaches. Another revision is to add heterogeneity in individuals within a cell (or other

spatial units) according to the probability density function of certain characteristics of

sub-populations (e.g. population size, household size, and age structure). Individuals in

a cell are collectively represented according to these statistics (Ferguson et al. 2006).

Interactions between cells and the mobility of individuals are also represented by statis-

tical probabilities. These revisions, which mostly began around the middle of the last

decade, have kept a sub-population as the modeling unit. However, to various degrees

they have either incorporated stochastic interaction processes between sub-populations

or altered the homogeneity assumptions held in the original sub-population models.

4 Individual-based Cellular Automata Models

Individual-based cellular automata models (Holmes 1997) can be considered an exten-

sion of classic models. While also developed in the later part of the 1990s, cellular

automata models have departed further from classic models (including population-based

wave models) than sub-population models have. This model type also divides space into

cells, but a cell is intended to represent a discrete individual instead of an aggregation of

many identical individuals.

The design principles of these models are quite independent of classic models

(including population-based wave models). In addition to using an individual as the

modeling unit, the interactions between units are explicitly represented. A cell interacts

with a finite number of adjacent cells. The spatial and temporal spread of diseases is then

represented by localized transmissions that begin from an infectious cell and spreads to

adjacent susceptible cells. This type of model also facilitates long distance dispersions

as well by establishing new foci of transmission at locations that are a certain distanceaway from already infected cells (the so called leapfrogging in the cellular automata

literature).

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These design principles break several assumptions held by classic models (including

population-based wave models), especially the homogeneous mixing assumption. In

individual-based cellular automata models, individuals differ from one another and each

has an explicit spatial location. An individual interacts only with a limited number of

other individuals. Although the homogeneous distribution and immobility assumptionsare still held, the unique individual and finite local interaction assumptions make the

individual-based cellular automata models a considerable departure from classic models

(including population-based wave models). As a result, the estimated health outcomes

may differ (Holmes 1997).

For a spatial representation of disease transmission, the individual-based cellular

automata models adopt a typical rule-based cellular automata modeling approach. This

approach is fundamentally different from the deterministic approach used in the two

aforementioned types of models. Instead of using a single equation set to describe the

behavior of an entire population or a subpopulation, cellular automata models focus on

transmissions at a local level that collectively contribute to a dispersion pattern at thepopulation level. Because of the immobility assumption and the adjacent transmission

rule, these models are intended to model disease transmissions between immobile

individuals, such as plants (Holmes 1997).

From population-based wave models, sub-population models, to individual-based

cellular automata models, the modeling unit changes from a population segment to a

group of identical individuals and then a unique individual, respectively. Along with the

change in modeling unit is the finer scale of spatial representation. Despite this change,

the modeling units remain immobile in all three types of models. It is through a change

in the health state of individuals, rather than a change in their location (see discussion

of the wave models in Section 2.1), that diseases, as a phenomenon, move across

space.

Figure 1 shows a two-criteria space to identify the six model types. The vertical axis

represents the scale of modeling unit from a coarse scale at the bottom to a fine scale at

the top. The horizontal axis represents the mobility of these modeling units with increas-

ing mobility towards the right. Population-based wave models, sub-population models,

to individual-based cellular automata models, respectively, are placed from the bottom to

the top along the scale of modeling unit axis, to represent an increasingly finer modeling

unit. Along the mobility axis, all three types of models are placed at the left most position

because of the immobility of their modeling units.

5 Mobile Sub-Population Models

Mobile sub-population models are another derivative of classic models (Smallman-

Raynor and Cliff 2001). Most typical of this type are the transfer diffusion models

developed around the turn of the 21st century (e.g. Smallman-Raynor and Cliff 2001).

These models use a sub-population as the modeling unit, the same design as in sub-

population models, but these units are mobile. The interactions between sub-populations

are explicit and unique, bearing little resemblance to the interaction principle used in

sub-population models or population-based wave models.

These models identify one or more sub-populations as the source of infection. Withtime, these infected sub-populations may move to different locations. At new locations,

an infected sub-population may merge with other non-infected sub-populations or split




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itself into several smaller infected sub-populations before it subsequently moves, splits, or

merges again. In this way, diseases are transmitted across space through time. Because

these models tend to focus on where the merged or split units move to, the location of a

unit is always explicitly represented. Despite this distinct design, the mobile sub-

population models use identical assumptions as sub-population models and population-

based wave models at the sub-population level, except for the immobility assumption.

Specifically, individuals within a sub-population are identical, homogeneously mixed,

and homogeneously distributed. Mobile sub-population models also inherit the same

deterministic approach used in classic models (including population-based wave models)

and sub-population models.

The mobility assumption brings to mobile sub-population models the flexibility to

model both spatial and temporal dynamics as observed in epidemic processes. Because of

the homogeneity assumption held in a sub-population, these models are most effective if

applied to mobile and high density populations, such as military or refugee camps.

Infected military units, for example, as mobile sub-populations, may be transferred to a

central location where they join other units. Subsequently some of the newly infected

units at the central location may be transferred to other locations to join other units, andconsequently diffuse a communicable disease across space (Smallman-Raynor and Cliff

2001).

Individual

Sub-

population

Modeling

Unit

Individual-based cellular

automata models

Individual-based spatially

implicit models

Individual-based mobile

models

Sub-population models Mobile sub-population

models

Population-based wave

models

Population

Mobility

Figure 1 Positions of the six model types in the scale-mobility space. The color scheme

represents health states: blue =susceptible, red =infectious, green =recovered

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6 Individual-Based Spatially Implicit Models

Individual-based and spatially implicit models present a significantly distinct modeling

framework from those of classic models, although both originated from the same research

community. The most typical example of this type is the discrete individual transmissionmodel developed toward the end of the 1990s (Ghani et al. 1997, Keeling 1999, Koopman

and Lynch 1999). The basic concepts underlying these models are two-fold. First,

individuals differ from one another and this simple fact should be the basic assumption for

epidemiological studies. Second, individualized interactions and infection play a vital role

in disease dispersion through a population (Koopman 2004).

The essential design of these models lies in the emphasis on unique individuals and the

interactions between them. The use of an individual as the modeling unit is similar to

individual-based cellular automata models, but is developed independently. Each indi-

vidual interacts with a limited number of other individuals in a social network, though not

necessarily those that are spatially adjacent. These individualized finite interactionsthrough social networks provide a mechanism for the transmission of disease through a

population across space and through time. The dynamics of an epidemic depend on the

structure of social networks. Commonly considered factors may include, for example, the

number of individuals with whom an individual interacts, the frequency and length of the

interaction, and the cross-connections within a group of individuals (Ghani et al. 1997).

The structure of social networks consequently determines how diseases propagate through

a population. Because the primary focus of these models is on the social connection, the

spatial dynamics in disease transmission is not always explicitly expressed.

In addition to the departure from the identical individuals and homogeneous mixing

assumptions, the homogeneous distribution and immobile individual assumptions are

not necessarily held for these models, either. The finite interactions not only break the

homogeneous mixing assumption, but also indirectly break the homogeneous distribu-

tion assumption, as individuals interact only at certain locations rather than all locations.

Further, these individuals may travel to different locations in order to interact with each

other. The explicitly expressed heterogeneity in individuals and in their interactions

makes it effective and necessary to use a stochastic approach in modeling the disease

transmission. In other words, the dispersion pattern at the population level may emerge

collectively from dynamic local processes.

Most of these models have been developed primarily for modeling sexually trans-

mitted diseases and usually involve a finite number of individuals (Kretzschmar and

Morris 1996, Ghani et al. 1997, Adams et al. 1998, van der Ploeg et al. 1998, Welch

et al. 1998, Koopman and Lynch 1999). Although spatially oriented considerations are

indirectly implied, this type of model holds a critical position in the evolution of the

spatial design of epidemiological models because of its unequivocal focus on unique

individuals and their individualized behaviors (interaction and infection). These models

also served as an important stepping-stone for the later development of their spatial

extensions, the individual-based mobile models.

7 Individual-based Mobile Models

The last type of model to be evaluated is individual-based mobile models that emerged

around the middle of the last decade (Bian 2004, Dibble and Fieldman 2004, Eubank




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et al. 2004, Huang et al. 2004, Dunham 2005, Ferguson et al. 2005, Longini et al. 2005,

Cooley et al. 2008, Lee et al. 2008, Yang and Atkinson 2008, Yang et al. 2008, Perez and

Dragicevic 2009, Xu and Sui 2009, Rakowski et al. 2010, Salath et al. 2010, Mao and

Bian 2011). This model type is a spatial extension of individual-based spatially implicit

models.These models account for the heterogeneity in all four design principles, i.e. in

individuals as the modeling unit, the interactions between them, the spatial process, and

the temporal process. For modeling disease transmission, these models tend to explore

the concept of time geography (Hgerstrand 1970, Pred 1977, Lenntorp 1978, Lytnen

1998, Kwan 1999, Miller 2005) and network theory (Watts and Strogatz 1998, Keeling

1999, Albert et al. 2000, Keeling and Eames 2005, Bian and Liebner 2007) in represent-

ing spatial and temporal dynamics. Individuals daily (or other time periods) travel

trajectories are represented as spatial-temporal lifelines. These lifelines intersect at dif-

ferent locations, such as homes, workplaces, and service places where individuals interact

with each other. The intersected lifelines form a social network. Diseases spread throughthis network by two mechanisms, the interactions between individuals at a location and

individuals travel between locations in order to interact with other individuals (Bian

2004, Miller 2005). Because the location and time of both the interactions and move-

ments between them are explicitly represented, the spatial dynamics of disease transmis-

sion can be explicitly modeled. These models can reveal a range of spatial and temporal

heterogeneities in disease transmission, depending on the structure of a social network

(Bian and Liebner 2007).

All assumptions underlying the individual-based mobile models are opposite to those

in classic models (including population-based wave models). Specifically, individuals

are unique, interact with only a finite number of other individuals, are heterogeneously

distributed, and mobile. This model type almost exclusively uses a stochastic modeling

approach that lets localized interactions and transmission to collectively contribute to a

dynamic spatial pattern at the population level.

Because the location and mobility of individuals are explicitly registered, this model

type is powerful for estimating the temporal speed and spatial extent of an epidemic. The

distinctive focus on unique individuals makes individual-based mobile models applicable

to modeling vulnerability to health threats at the individual level. The population scope

can range from hundreds of individuals in a workplace, hundreds of thousands in a

community, to millions in a region (Eubank et al. 2004, Cooley et al. 2008, Yang and

Atkinson 2008, Mao and Bian 2011, Stehl et al. 2011).

Still in their infancy, individual-based mobile models face many challenges. The

individual assumption, for example, requires a great amount of detailed information

about individuals that may not be available. Health status and other data at the indi-

vidual level are collected under restricted guidelines due to privacy concerns. To com-

pensate for the scarcity of such data, surrogate data have been used to estimate many

aspects of individual attributes and behaviors. Most recently, efforts have been made to

investigate individualized contact behavior at a high spatial and temporal resolution

using wireless devices and social media information (Crandall et al. 2010, Salath et al.

2010, Stehl et al. 2011). Findings from these studies are invaluable for the design of

individual-based mobile models. The availability of on-line information and methods to

harvest this information add much utility to this type of model. For example, byanalyzing individuals on-line health-seeking behavior, an epidemic can be tracked within

a shorter period of time than through conventional case reporting channels and at a

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reasonable level of accuracy (Ginsburg et al. 2009). Despite many challenges, individual-

based mobile models may hold great potential to eventually forecast spatial and temporal

dynamics of an epidemic in real time by incorporating this information.

The mobility assumption used in the last three types of models brings flexibility,

although to various degrees, to the modeling of heterogeneity and dynamics in diseasetransmission. Unlike the first three types of models in which the dispersion of diseases is

represented by the change in the health state of immobile modeling units, the last three

models explicitly change the location of modeling units. Unlike modeling the spread of

disease as passing waves, as represented in the first three types of models, the modeling

of the spread in the latter three types of models can be considered similar to that for the

motion of discrete objects. A sub-population or an individual can be readily represented

as an object. When the object moves, its parts (e.g. all individuals in a mobile sub-

population), its attributes (e.g. the health state), and its behaviors (e.g. interaction and

infection) move with it. This representation allows the modeling unit to move freely in

space, thus portraying the dynamic nature of infection sources, interactions, and diseasetransmission.

In Figure 1, mobile sub-population models hold an identical position as the sub-

population models along the modeling unit axis, as both types of models use a sub-

population as the modeling unit. Along the mobility axis, mobile sub-population models

are further to the right because their modeling unit is mobile. Individual-based spatially

implicit models are placed at the same position as individual-based cellular automata

models along the modeling unit axis, but further to the right along the mobility axis

because of the implied mobility of its modeling unit. Individual-based mobile models are

placed at the upper right most corner of the two-criteria space, because of its fine-scaled

modeling unit and its high degree of mobility. Table 1 compares the six types of models

in terms of their assumptions, design principles, and intended applications.

8 Modeling Unit, Mobility, and Stochasticity

The modeling units for the spatial dispersion of communicable diseases remained at the

population level from the early attempts in the 18th century to the middle 1990s. The

other two modeling units, sub-population and individual, came along in the decade that

followed. The evolution from population- to individual-based epidemiological models

reflects recent developments in several fields, such as computing sciences. Rapid

improvements in computing power can now support the modeling of a significantly

larger number of individuals than in the recent past. Related to this computing power is

the development of computing theories, such as object-orientation, that provide the

theoretical support for the individualized representation (Wegner 1990, Bian 2004).

Further, renewed interest in network theory (Watts and Strogatz 1998, Albert et al.

2000) has supported the modeling of individualized interactions. Theoretical frame-

works and computation methods developed in GIScience have also allowed for the

representation of spatial dynamics of epidemics at an ever-increasing level of sophisti-

cation (Bian 2007). These advances have facilitated the ultimate break from the homo-

geneous mixing assumption used in population-based and sub-population-based models,

and the embracing of the heterogeneous interaction assumption seen in individual-based

models.The breakdown of a population into fine-scaled modeling units is necessary but not

sufficient to support the representation of mobility. Only when the modeling units are




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Table

1

Acomparisonofthesixmodelsintermsofassumptions,

designprinciples,andintend

edapplications

Population-Based

WaveModels

S

ub-Population

M

odels

Individual-Based

CellularAutomata

Models

Mobile

Sub-Population

Models

Individual-Based

SpatiallyImplicit

Models

Ind

ividual-Based

Mo

bileModels

Assumptions

Individuals

homogeneous

h

omogeneouswithin

asub-population

unique

homogeneouswithin

asub-population

unique

unique

Mixing

homogeneous

h

omogeneouswithin

asub-population

finiteinteraction

homogeneouswithin

asub-population

finiteinteraction

finiteinteraction

Distribution

implied

homogeneous

h

omogeneouswithin

asub-population

homogeneous

homogeneouswithin

asub-population

heterogeneous

heterogeneous

Mobility

impliedimmobilityimmobile

immobile

mobile

impliedmobile

mobile

DesignP

rinciples

Modelingunit

population

segment

s

ub-population

individual

sub-population

individual

individual

Interaction

betweenunits

none

implied

explicit

explicit

explicit

exp

licit

Spatialprocess

passingwave

a

djacentcell

transmission

adjacent

celland

leapfro

g

transm

ission

localandlong

distance

transmission

lifelineandsocial

network

lifelineandsocial

n

etwork

Temporalprocess

explicit

e

xplicit

explicit

explicit

explicit

exp

licit

Applications

Intended

popula

tion

anentire

population

h

igh-densityimmobile

populations

immobileindividuals

high-densitymobile

populations

connectedindividuals

allindividuals

Disease

pandemics

d

iseasesthataffect

livestock

diseases

thataffect

plants

diseasesthataffect

militarycamps

sexuallytransmitted

diseases

allthataffects

individuals

12 L Bian



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spatially registered and registered to multiple locations is it possible to represent the

dynamically located sources of disease transmission and, consequently, the heterogeneity

in disease dispersion. This level of mobility representation is achieved mostly during the

last decade following the rapid development of GIScience.

Associated with the fine scaled modeling units and their mobility is the increased useof stochastic modeling approaches (Koopman 2004). A fine-scaled modeling unit may

not be a sufficient factor to require stochastic approaches. The key is whether a modeling

unit is treated as a unique entity. It would be difficult to use an equation set to describe

unique modeling units and their heterogeneous attributes and behaviors, especially the

interactions between them. In comparison, stochastic approaches have considerable

advantages for modeling these heterogeneities and probability-driven complex infection

processes. For example, the original sub-population models are able to maintain the

deterministic approach at the sub-population level. Once a sub-population is treated as

a unique entity, the stochastic approaches are deployed. This is evidenced by the adoption

of stochastic approaches in revised versions of sub-population models (see Section 3).

9 Spatial and Temporal Representation and Implementation

Researchers from a range of disciplines have contributed to the development of the

spatial approaches incorporated in epidemiological models. These disciplines include

epidemiology, ecology, geography, computer science, mathematics and physics, among

others. Their varied disciplinary roots bring different perspectives to model design (as

discussed in earlier sections) and implementation considerations. Also, due to the diver-

sity of the articles included in this review (commentary-critique, design-method, and case

study), not all of them involve implementation considerations. Further, developed at

different periods over a time span of more than two decades, the implementation

considerations of these models are inevitably affected by theories and methods available

at the time. Proprietary software packages, in-house customized software packages, and

freeware have contributed to the implementation of these spatial approaches at any given

time. These packages have been originally developed either for epidemiological modeling

purposes that have incorporated spatial considerations or as GIS software that can be

used to support epidemiological modeling. Given these diverse issues, it might be impos-

sible to evaluate the implementation and operation concerns of the six types of models.

It is appropriate, however, to evaluate how space and time can be represented according

to the design principles of these models, using theories and methods available at the

present time.

The field and object views are perhaps one of the most important theoretical

developments in GIScience (Goodchild 1992, Cova and Goodchild 2002, Bian 2007).

The inclusive object-field dichotomy offers a profound framework to guide the repre-

sentation of spatial phenomena, whether they are discrete object-like or continuous

field-like entities. The two types of GIS data models, raster and vector, can be used to

represent both object-like and field-like phenomena, depending on the intended purpose,

scale, and convention (Couclelis 1992, Bian 2007).

With regard to the six types of models discussed above, raster data models are most

effective to support population-based wave models, sub-population models, andindividual-based cellular automata models. This is because their spatial arrangement is in

the form of regular grid cells, and each cell is treated as a basic modeling unit. Closely




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associated with raster data models is the cellular automata modeling approach. This

approach changes the attribute of a cell based on the current state of the cell and its

surrounding cells according to a set of pre-determined rules (Couclelis 1989, Clarke et al.

1997). It is most powerful when used to model spatial transmission under the influence

of adjacent cells in these three types of models. For modeling long distance dispersion,however, raster data models and the cellular automata approach are not always as

effective as the vector data model, although leapfrogging has been incorporated in the

cellular automata approach. The immobility of the cells limits the ability of this type of

model in representing free motion (Tischendorf 1997, Bian 2003), especially if attempt-

ing to represent interactions in the form of a network.

Vector data models are most effective for mobile sub-population models, individual-

based spatially implicit models, and individual-based mobile models because of the

mobility of their modeling units. Vector data models allow for the representation of free

motion in space, thus they are effective in representing mobile individuals and mobile

sub-populations, and ultimately the spatial dynamics of an epidemic. In comparison toraster data models, vector data models are more complex and require sophisticated

spatial data management capabilities. Proprietary software packages can normally meet

this need but they lack the ability to support complex spatial operations. Additional

programming tools are often required, typically in-house software packages or freeware,

in order to integrate with GIS packages in a working model.

The agent-based spatial modeling approach has been increasingly popular in epide-

miological modeling. In addition to possessing attributes, the agents may have many

behaviors, such as executing actions, interacting with other agents, perceiving its envi-

ronment, and acting in response to both other agents and their environment (Jennings

and Wooldridge 1996, OSullivan and Haklay 2000, Brown and Xie 2006, Sengupta and

Sieber 2007, Tang and Bennett 2010). Agent-based modeling is applicable to nearly all of

the six types of models reviewed above. An agent can be an individual in the three

individual-based models, or a sub-population in the two sub-population models. It is

more advantageous, however, if agents represent unique modeling units, be it unique

individuals or unique sub-populations.

Temporal representation has always been explicitly expressed since the beginning of

epidemiological model development, yet the representation of temporal change has

remained simple. Time is normally represented as time steps with regular intervals,

depending on the intended temporal resolution of a model, although an increasingly finer

temporal resolution is used in more recent models.

10 Conclusions

A review of spatial approaches incorporated in epidemiological models helps assess the

current state of our knowledge of, and ability to model, the spatial dynamics of epidem-

ics. In less than three decades, approaches to modeling the dispersion of communicable

diseases have progressed from population-based wave models at one corner of the

two-criteria space to individual-based mobile models at the opposite corner (Figure 1).

During this time, the schools of thought, design principles, and computation methods

behind these approaches have evolved and diversified. It may be true that all models arewrong, but some are useful (Box and Draper 1987). Each model type has its own role in

contributing to the development of health policies (McKenzie 2004).

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Insights gained from these evaluations can be valuable to devise much-needed

spatially and temporally oriented strategies to control and prevent communicable dis-

eases. The implementation of pre-infection strategies (e.g. vaccinations) and post-

infection strategies (e.g. quarantine) can be better located and timed towards those

individuals and communities considered most vulnerable. Well targeted strategies mayreduce the adverse economic and social impacts often brought by mass vaccination and

quarantine.

The six types of models are categorized according to the scale and mobility of

modeling units in order to identify and evaluate major approaches to modeling spatial

dynamics in epidemic processes. Epidemiological models that incorporate spatial

approaches are certainly not limited to these six types. For example, it is possible to use

deterministic approaches in an individual-based mobile model (Reluga et al. 2011). The

measure of scale and mobility can be continuous as well and a model may be placed

anywhere in the two-criteria space. The revised versions of sub-population models are

examples that may not fit into any of six distinct types, as their design principles andassumptions differ from one another. This will remain true as models continue to evolve

and diversify.

Acknowledgements

The valuable comments and suggestions provided by three anonymous reviewers aregratefully acknowledged.

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