A novel small-world model: Using social mirror identities for epidemic simulations

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SIMULATION

http://sim.sagepub.com/content/81/10/671

The online version of this article can be found at:

DOI: 10.1177/0037549705061519

2005 81: 671

SIMULATION

Chung-Yuan Huang, Chuen-Tsai Sun, Ji-Lung Hsieh, Yi-Ming Arthur Chen and Holin Lin

A Novel Small-World Model: Using Social Mirror Identities for Epidemic Simulations

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Using Social Mirror Identities

for Epidemic Simulations

Chung-Yuan Huang

Department of Computer and Information Science National Chiao Tung University and

Department of Computer Science and Information Engineering Yuanpei Institute of Science and Technology

306 Yuan Pei Road

Hsinchu 300, Taiwan, Republic of China gis89802@cis.nctu.edu.tw

Chuen-Tsai Sun Ji-Lung Hsieh

Department of Computer and Information Science National Chiao Tung University

1001 Ta Hsueh Road

Hsinchu 300, Taiwan, Republic of China

Yi-Ming Arthur Chen

Institute of Public Health National Yang-Ming University 155, Section 2, Li-Nong Street Taipei 112, Taiwan, Republic of China

Holin Lin

Department of Sociology National Taiwan University 1, Section 4, Roosevelt Road

Taipei 106, Taiwan, Republic of China

The authors propose a small-world network model that combines cellular automata with the social mirror identities of daily-contact networks for purposes of performing epidemiological simulations.The social mirror identity concept was established to integrate human long-distance movement and daily visits to fixed locations. After showing that the model is capable of displaying such small-world effects as low degree of separation and relatively high degree of clustering on a societal level, the authors offer proof of its ability to displayR0properties—considered central to all epidemiological studies. To

test their model, they simulated the 2003 severe acute respiratory syndrome (SARS) outbreak.

Keywords: Social mirror identity, small-world network model, multiagent system, cellular automata,

public health policy, network-based epidemic simulations

1. Introduction

Factors that influence the transmission dynamics of epi-demics include individual diversity and social networks

| | | | | | SIMULATION, Vol. 81, Issue 10, October 2005 671-699

constructed by interpersonal relationships and simple daily contact [1-6]. For instance, interactions among individu-als and contact routes both affect the outbreak of short-distance contagious diseases such as severe acute res-piratory syndrome (SARS) and enteroviruses [3, 5-11]. Due to the potential complexity of human interactions, researchers need a simulation model that can represent multiple social networks to analyze and control a wide range of potential transmission behaviors and epidemic characteristics.

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Furthermore, epidemic transmission speed and scope are closely related to daily human activities. Modern lifestyles are marked by strong habits with little day-to-day variety. For instance, the majority of adults in developed countries use the same transportation modes for short- and long-distance movement on a daily basis. The limited di-versity of transportation options to sites that are visited regularly (e.g., workplaces and schools) makes it easy for the rapid transmission of diseases within a town or city. Since it is hard to control the movement of individuals (e.g., method, timing, direction, and distance), researchers are repeatedly challenged by the task of simulating indi-vidual movement within a society—an issue referred to in the literature as the “mobile individual problem” [12-15]. Researchers who use small-world network models to investigate epidemics usually divide human contacts into short-distance (short-link) and long-distance (long-link) contact categories [1-6, 16-19]. While these models offer partial explanations for the mobile individual phenomenon, they fail to accurately express concurrent epidemic move-ment from one infectious agent to a group of susceptible people—for instance, coworkers, classmates, hospital em-ployees, or passengers taking the same bus. When apply-ing small-world network models to epidemics, indirect de-scriptions such as shortcuts and short/long or strong/weak links may not accurately reflect the repeated use of trans-portation tools for long-distance movement and for visiting multiple sites in one day. For this reason, epidemiologists, public health specialists, and health authorities cannot use most of the abstract small-world network models that have been proposed to test the efficacy of various public health policies and epidemic prevention strategies.

In this article, we propose a social mirror identity con-cept that accurately reflects human interaction (including long-distance movement and daily visits to fixed and/or multiple locations) in modern societies (Fig. 1). According to the social mirror identity concept, every visited location, every played role, and every performed activity is consid-ered a social mirror identity of the individual in question. A list of one’s social mirror identities might include fa-ther, husband, coworker, supervisor, subordinate, fellow passenger, store customer, or restaurant diner. Each role or activity at each location is considered a separate mir-ror identity. The mirmir-ror identity concept allows for a more complete and direct imitation of social phenomena and daily movement. In combination with cellular automata, we offer it as a solution to the mobile individual problem.

2. Related Epidemiological Models and Concepts

2.1 Compartmental Models

Many epidemiologists have used compartmental models to predict epidemic outbreak trends [20, 21]; the most ba-sic and well known is the SIR model created by William Kermack in 1927 (Fig. 2) [20]. During the 2002-2003 SARS outbreaks, many researchers used compartmental

models to estimate transmission dynamics and develop-mental tendencies [22-26] and to analyze super-spreader events (SSEs). However, those models were only capable of calculating change in the total number of infected indi-viduals per time step. During each simulation, differential equations were applied to calculate pivotal parameters, in-cluding the basic case reproduction number R0 [27, 28],

which is considered essential to the work of public health specialists and epidemiologists. To generate more accurate simulation results, some researchers divided each popula-tion into subgroups according to age, locapopula-tion of residence, infection rate, and other characteristics of interest to epi-demiologists [22-23, 26]. Regardless of characteristic or category, these simulation models ignore the fact that so-cial phenomena emerge from regular and frequent human interaction. In other words, compartmental models empha-size epidemic characteristics (e.g., transmission, mortality, and recovery rates) at the expense of population structure, social space, heterogeneity, localization, and interaction. Consequently, compartmental models are insufficient for analyzing public policy issues and epidemic prevention strategies.

The basic case reproduction number R0is an index

pa-rameter with an important reference value—the number of people infected by a patient prior to recovery or death. When R0is greater than 1, the number of infected patients

increases, and the transmission rate soars. An R0 of 1

in-dicates stability in the spread of the infection—in other words, each patient transmits the virus to one person on average. When R0is smaller than 1, a patient may or may

not transmit the virus, making the recovery rate higher than the infection rate. Accordingly, 1 is considered a plague threshold value; to prevent an epidemic from becoming a plague, R0must remain below the threshold value.

2.2 Simple Social Network Models

There are at least two ways of constructing a simple net-work model:

1. Lower dimensional lattices that represent social networks (Fig. 3a) [14, 29, 30]. Examples in-clude one-dimensional ring-shaped lattices and two-dimensional lattices with periodic boundary condi-tions (i.e., a doughnut-shaped surface). Since each node is connected to its adjacent nodes and the num-ber of connected nodes never changes, these models are sometimes referred to as regular network models. 2. Random networks that represent social networks (Fig. 3b) [31]. This type of network model and the compartmental model described above are equiva-lent in that both use statistics to represent many so-cial network characteristics. Random network mod-els are considered primitive means of representing complex, chaotic, and unpredictable societies.

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Figure 1. An example of the social mirror identity concept

Figure 2. General transfer diagram for the compartmental SIR model with susceptible populationS, infected populationI, and recovered populationR

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In either model, communities, cities, and countries can be defined as separate social networks; even our planet can represent one social network. One node represents one individual with status-determining attributes, for example, epidemiological progress, gender, age, or immunization. Connections between individuals are referred to as edges, with different edges representing different interpersonal relationships. Edges in AIDS simulations represent sexual relationships, while in SARS simulations, they represent close physical proximity. The states of all network nodes change simultaneously during each time step. The state of each individual node is determined by its original state, its neighbor’s state, and a set of interaction rules.

Some researchers have used two-dimensional cellular automata to explore local transmission mechanisms and epidemic characteristics [30, 32-37]. Cellular automata are considered specific and regular network models. They ex-hibit social properties such as population structure, local aggregation, social space, heterogeneity, and interaction— all of which are essential to understanding epidemiological and contagion issues (Fig. 4). They are useful for observing disease transmission during an epidemic, but they lack an important network property—small-world phenomenon— meaning that they generally fail to represent low degrees of separation among individuals [18]. Without this prop-erty, a social network model cannot accurately simulate real transmission dynamics or modern public health poli-cies associated with epidemic diseases.

2.3 Small-World Social Network Models 2.3.1 Triadic Closure

First proposed by Rapoport [38] in 1957, the triadic closure concept is based on the view of human beings as “birds of a feather.” Accordingly, employees in the same com-pany, classmates in the same school, and regular customers at a coffee shop have a much better chance of meeting each other and forming relationships than two strangers. In other words, relationships are formed because of what people have in common, not because of random probabil-ities. The triadic closure concept posits that two strangers with a common friend have a higher than average proba-bility of meeting each other and becoming friends them-selves. Triadic relationships are thus viewed as a funda-mental structural unit, complete with social rules governing connections among individuals. Connections established via multiple triads form large social networks. Whenever an epidemic outbreak occurs, healthy but susceptible lo-cals are most likely to become infected due to their triadic and/or polygonal closure relations with infectious patients.

2.3.2 Small-World Network Models

While working on his well-known letter delivery exper-iment in 1967, Milgram [39] proposed a concept called “six degrees of separation” to explain the phenomenon in which humans frequently interact with each other and form

groups, yet everybody in the world remains separated by only six other people. Milgram’s idea was verified in 1998 by Watts and Strogatz [16] (Fig. 3c). Their small-world network model (which contains the characteristics of high clustering and low degree of separation) was based on two concepts: (1) topological networks and structures are ubiq-uitous in the real world, and (2) they strongly influence so-cial issue dynamics and outcomes [5, 17-19, 40]. Because of their work, the capability of any social simulation model to portray high clustering and low degree of separation is now considered an important index for examining so-cial network models. Soso-cial individuals are characterized by long-distance movement, daily visits to fixed locations, multiple activity locations, and local clustering—meaning that the average distance between any two individuals is shortened. Geographic location and distance are therefore considered secondary causal factors in epidemic outbreaks.

2.3.3 The Small-World Phenomenon

Determining whether a social network model is indeed a small-world network model requires validation of a high clustering coefficient and a low degree of separation co-efficient. A clustering coefficient is used to evaluate the degree of connection between two neighboring nodes. In equation (1), graph G represents a social network, vi is a node in graph G, and kiis the vertex degree of node vi. The C(vi)clustering coefficient of node viis defined as the ratio of Ei(the number of edges that actually exist among the ki nodes) and ki× (ki – 1)/2. Accordingly, the C(G) clustering coefficient of the entire social network equals the average C(vi)value for all nodes.

C(vi)=

2× Ei ki× (ki− 1)

. (1)

The S(vi, vj)separation coefficient is used to evaluate the shortest distance between two random nodes, viand vj. The S(G)separation coefficient of the entire social network is the average length of the shortest distances between any two nodes. When the number of individuals in a society increases, the average separation coefficient between any two individuals increases logarithmically rather than pro-portionally [17].

3. The Proposed Model

As shown in Figure 5, our simulation model consists of two layers. The upper layer is a simplified multiagent sys-tem for simulating heterogeneous cohorts, and the lower layer contains two-dimensional n×n cellular automata that represent real-world activity spaces. Social mirror identi-ties are used to connect the two layers, thus establishing a small-world network model. By manipulating transmis-sion rules, disease parameters, and public health policies, the model can be used to simulate the transmission dy-namics of contagious diseases, verbal communication, and social issues.

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Figure 4. Cellular automata and state transition function

Figure 5. Cellular automata with the social mirror identity model (CASMIM)

3.1 Cellular Automata with Social Mirror Identities Model (CASMIM)

In the cellular automata with social mirror identities model (CASMIM), each individual is depicted as a single agent in the upper-layer multiagent system, and the places that an agent visits on a regular basis (e.g., homes, train stations, workplaces, and restaurants) are defined as that individ-ual’s social mirror identities. In typical cellular automata, lattices represent abstract agents. In our model, each

lower-layer cellular automata lattice represents a social mirror identity.

It is possible for multiple social mirror identities to be connected to the same agent. Each agent has many so-cial mirror identities representing fixed locations that are visited daily or very frequently. The number of social mir-ror identities connected to any single agent exhibits a nor-mal distribution. Snor-mall clusters formed by a mirror identity and its neighbors can represent family members, cowork-ers, fellow commutcowork-ers, health care workcowork-ers, relatives in

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hospitals, or diners in restaurants. The mirror identity con-cept uses simple social networks to preserve the properties of elements that interact with their neighbors within two-dimensional lattices and to reflect such activities as long-distance movement and daily visits to fixed locations.

In the example shown in Figure 1, Andy spends 1 hour every morning taking his wife Cindy to her job at a flower shop before driving to his insurance company office. Their son Bob takes a school bus to his elementary school. At least once a week, the three of them eat dinner at their favorite restaurant. After dinner, Andy often takes Cindy and Bob home before going with his friends Dick, Eric, and Frank to watch a baseball game. According to our proposed model, Andy, Bob, Cindy, Dick, Eric, and Frank are upper-layer agents, and Andy’s home and office and the restaurant and stadium are lower-layer mirror identities. Bob’s mir-ror identities are his home, school bus, classroom, and the restaurant. Cindy has only three mirror identities: home, the flower shop, and the restaurant. Andy’s automobile is considered an extension of their home node rather than a separate activity node since Andy rarely uses it to transport anyone outside of his family. Bob’s school bus is consid-ered a social mirror identity because he uses it 5 days per week and plays with many of the children who take the same bus.

Each individual upper-layer agent has a set of attributes that demonstrate its epidemiological progress and social mobility status (Table 1, Fig. 6); all of the agent’s social mirror identities have access to these attributes. In addition, each social mirror identity has a group of private attributes that represent its current status, location, and special activ-ity locations—homes, hospitals, or dormitories (Table 2). Agents who possess individual social mirror identities have complete access to these attributes. In the Figure 1 exam-ple, Andy belongs to one group at home with Cindy and Bob, a second group at his office with his coworkers, a third group (also with Cindy and Bob) with other customers at their favorite restaurant, and a fourth group with his base-ball friends. Andy’s social mirror identities form a star-shaped topology, with Andy at the center and the mirror identities at the vertices.

According to our proposed model, the greater the num-ber of social mirror identities an agent has, the greater the agent’s influence. In epidemiological terms, the more so-cial mirror identities an agent has, the more likely the agent will become infected or transmit a disease to other agents. In cellular automata terms, the lattices surrounding a so-cial mirror identity represent neighbors, family members, classmates, colleagues, friends, hospital workers, passen-gers on the same bus, customers in the same restaurant, and so on. Andy’s lower-layer social mirror identity at the baseball stadium is adjacent to those of Dick, Eric, and Frank, and his lower-level social mirror identity at home is adjacent to those of Cindy and Bob.

Figure 6. Epidemiological and social mobility states

We decided to use the Moore neighborhood concept with a radius parameter of 1 (Fig. 7) in CASMIM because von Neumann neighborhoods lack triadic closure relation-ships between a lattice and its four neighboring lattices. In contrast, each lattice in a Moore neighborhood has triadic closure relationships with its eight neighboring lattices; this higher degree of local clustering matches Rapoport’s description of interactions in human societies. In the Figure 1 example, if Andy catches the flu from his friend Dick, he may infect his wife and son. According to triadic closure relationships, there is a high probability that Bob, Eric, and Frank will also become infected.

With a few important exceptions (e.g., AIDS), most epidemic simulation models assume that one time step is equivalent to one 24-hour period in the real world. We used that assumption when designing CASMIM. As shown in Figure 8, the statuses of upper-layer agents change si-multaneously with their lower-layer social mirror identity statuses during each time step. Each agent’s social mir-ror identity comes into contact with all of its surrounding neighbors’social mirror identities in random order per time step; contact order is not considered critical. The attributes of the social mirror identity, the agent, and other associated social mirror identities vary according to the attributes of the social mirror identities of neighboring agents, a set of

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Table 1. Agent attributes

Data Default

Attribute Type Description Value

ID Integer Unique serial number that identifies agent in CASMIM. 1∼ P

E Symbol RateF oreverI mmunedetermines proportion of agents classified asM(Immune) in the epidemiological progress attributeE(i.e., the population of permanently immune agents). All other agents are classified asS(Susceptible)—“not yet infected but prone to infection.”

Susceptible, Immune

Mobility Symbol Default value is “free”—no restrictions on interacting with the mirror identities of neighboring agents. When an agent is placed under home quarantine or hos-pital isolation, its Mobility status changes to Quarantined or Isolated, meaning that the agent is restricted to its rooted social mirror identity (home, hospital, or dormitory) and that the activities of all social mirror identities are temporarily suspended.

Free

Count Integer Records the number of an agent’s mirror identities; each agent has a minimum of 1 and a maximum ofM. The number of an agent’s mirror identities exhibits a normal distribution.

1∼ M

MirrorIdentity Set Data structure for containing mirror identities.

Age Symbol Agents are categorized as young (1 to 20), prime (21 to 60), and old (61 and above). Ages are randomly assigned according to RateY oung, RateP rime, and

RateOldparameters.

Young, Prime, Old

Super Boolean Denotes whether an agent is a super-spreader. If yes, set Super to “true”; if no, to “false.” The RateSuper parameter determines which agents are super-spreaders.

True, False

Immunity Permanent

Boolean Denotes whether an agent is permanently immune. If yes, set

ImmunityPer-manenty to “true”; if no, to “false.” The RateF oreverI mmunitydetermines which agents are permanently immune.

True, False

Day Integer Number of days for each of the three epidemiological progress states. If an infected agent has not yet recovered, Day is used to indicate the number of infected days. For recovered agents, Day is used to indicate the number of days since full recovery. If a recovered agent has temporary antibodies, Day is used to indicate the number of immune days.

RateCont act Real Rate of contact with other agents. For all agents, RateCont act values exhibit a normal distribution.

0_∼1

WearingMask Boolean Denotes whether an agent wears a mask. If yes, set WearingMask to “true”; if no, to “false.” Default value is “false.” When a mask-wearing pol-icy is enacted (for the general public or for health care workers), the

Pol-icyW earingMask.Parameter.RateP art icipat ionparameter is used to determine how many agents wear masks.

False

MaskType Real Average prevention grade of agent masks. The higher the number (closer to 1), the greater the efficacy.

0∼1

QuarantinedDay Integer Number of home quarantine days, with a range of 0 to

PolicyH omeQuarant ine.Parameter.DayQuarant ined.

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Table 2. Social mirror identity attributes

Root Boolean Each agent has one mirror identity whose Root = true; for all other mirror identities, Root = false. The rooted mirror identity is used to mimic special activity locations—for instance, homes, hospitals, and dormitories.

True, False

Suspend Boolean Default value is false for all mirror identities, denoting that they can move about without restriction. Except for rooted mirror identities, Suspend = true for all mirror identities of an agent in home quarantine or hospital isolation, repre-senting the idea that the agent cannot interact with other adjacent neighbors outside of its home or hospital until the end of the quarantine or recovery pe-riod. If the agent dies, Suspend = true for all mirror identities (including rooted mirror identity), representing the idea that the agent can no longer visit any other location.

False

Location (Integer, Integer)

The first number represents thex-axis coordinate and the second they-axis coordinate for the location of a mirror identity in the two-dimensional cellular automata. Each mirror identity is mapped to a single coordinate location; in other words, each coordinate location contains a single mirror identity of only one agent.

Neighbor Set Represents the coordinate locations of mirror identities of neighboring agents. We adopted the Moore neighborhood definition for our simulation model. Under this neighborhood structure, each mirror identity is defined as having eight neighbors.

interaction rules (to be described in section 4), simulation and epidemic disease parameters (Table 3), public health policy parameters (Table 4), and probabilistic causes (e.g., symptom detection rate).

At this point, our simulation model is considered a small-world social network model with such simple social network attributes as population structure, area clustering, space, heterogeneity, localization, and interaction. It also has the social attributes of long-distance movement, daily visits to fixed locations, multiple activity nodes, and the small-world characteristic of low degree of separation—all of which are suitable for simulating epidemics, communi-cation networks, and other contagion problems. Moreover, one advantage of CASMIM is its use of the social mirror identity concept to reflect individual geographic mobility in special areas; this characteristic is particularly useful for analyzing public health policies.

3.2 Implementing CASMIM

Our simulation system (created with C++) consists of many functional modules, including CASMIM, an epidemiology module, a social mobility module (e.g., families, dormito-ries, and hospitals), and a public health policy module. We created a general-purpose and extendable software plat-form that is suitable for detailed numerical experimenta-tion and classroom demonstraexperimenta-tions of specific epidemic diseases and public health policy suites. The computational flowchart and system architecture for our proposed simula-tion system are shown in Figures 8, 9, and 10, respectively. To accommodate different requirements, we applied the visual component library (VCL) and event-driven

pro-gramming model that is part of the Borland C++ Builder to design the user interface and various input/output func-tions of the simulation system (Fig. 11). In addition to providing many specific statistical reports and charts on epidemic data, the simulation system offers two browser windows (micro-view and macro-view) to observe real-time epidemic disease infection situations in an agent so-ciety. After complied using the Borland C++ compiler and conversion into an executable application, the simulation system can be run on Windows with Dynamic Linked Library (DLL) files. Our simulation system is available at ftp://anonymous@140.126.75.253; for source code on particular contagious diseases or specific research require-ments, please contact the authors.

As shown in Figure 12a, the CASMIM construction pro-cess consisted of four steps, with each step using its respec-tive subprocedure to initialize the data structures and to establish relations among these data objects. The four steps were as follows: (1) call the initialize-multiagent-system subprocedure to initialize the upper-layer multiagent sys-tem, as shown in Figure 12b; (2) call the initialize-cellular-automata subprocedure to initialize the lower-layer cellular automata, as shown in Figure 12d; (3) call the distribute-mirror-identities-to-CA subprocedure to distribute all of an agent’s social mirror identities to cellular automata lattices so that the upper-layer multiagent system and lower-layer cellular automata connect with each other, thus establishing one-to-one mapping; and (4) call the set-rooted-mirror-identities-of-all-agents subprocedure to de-fine all of an agent’s rooted mirror identities (subproce-dure applications are discussed in section 4.2). For default

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Figure 8. Simulation flowchart

values and related explanations of agent and social mirror identity attributes, see Tables 1 and 2; for system param-eters used by agents, social mirror identities, and cellular automata during the initialization process, see Table 3.

The initialize-multiagent-system subprocedure repeats four additional steps until all agents are initialized, at which time it returns to the create-CASMIM procedure. The four steps are as follows: (1) selecting an un-initialized agent from the agent population, (2) giving the agent an ID, (3) initializing the agent’s attributes, and (4) calling the initialize-agent-mirror-identities subprocedure to initialize the data structure used by the agent to contain its social mirror identities (Fig. 12c).

The initialize-cellular-automata and initialize-multiagent-system execution processes are very similar.

According to the row-major layout, the attributes of each cellular automata lattice are initialized from top to bottom row and from left to right column. The initialize-agent-mirror-identities process is even more basic; it repeatedly executes two steps for every social mirror identity of an agent: it gives the identity a serial number and initializes its attributes (Table 2). Accessing the private attributes of a social mirror identity requires its serial number and the ID of the agent who possesses the identity.

In CASMIM, since each cellular automata lattice is con-nected to an agent’s social mirror identity, a procedure for lattices and social mirror identities to form one-to-one maps is required in order to provide access to each other’s private attributes. A coordinate attribute designated Loca-tion (x, y) is used by an agent’s social mirror identity to

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Figure 9. System flowchart

Figure 10. Simulation framework. Data on reported cases were collected from the World Health Organization (WHO) and national

health authorities. Input data were categorized as epidemic parameter (e.g., average incubation period, infection rate, distribution among age groups, mortality), imported case (e.g., time point, amount, imported during incubation or illness period), and activated public health policy (e.g., number of quarantine days, efforts to take body temperatures, restricting access to hospitals). Simulation output includes cellular automata states and various statistical charts.

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Table 3. Simulation system and epidemic disease parameters

PopulationAgent Set Stores total agent population in simulation system.

P Integer Total number of agents. 100,000

M Integer Upper limit of an agent’s mirror identities. 5

H Integer Height of two-dimensional lattice used in cellular automata. 500

W Integer Width of two-dimensional lattice used in cellular automata. 500

N Integer Total number of usable lattices (H× W) in cellular automata. 250,000

PeriodI ncubat ion Integer Average number of incubation days. 5

PeriodI nf ect ious Integer Average number of infectious days. 25

PeriodRecovered Integer Average number of recovered days. 7

PeriodI mmune Integer Temporarily immune to the disease.

RateSuper Real Percentage of super-spreaders in total population. 0.0001

RateY oung Real Percentage of young (0 to 20 years) agents in total population. 0.3

RateP rime Real Percentage of prime (21 to 60 years) agents in total population. 0.5

RateOld Real Percentage of old (60 years and above) agents in total population. 0.2

RateF oreverI mmunity Real Percentage of permanently immune agents in total population.

RateI nf ect ion Real Average infection rate. 0.045

RateDeat h Real Average death rate. 0.204

FrequencyCont act Real Number of contacts between an agent and its neighbors per time step. 4

Table 4. Public health policy parameters

Data

Policy Attribute Type Description

WearingMaskInGP Rate_RateP art icipat ion Real Policy participation rate.

P revent ion Real Infectious disease prevention rate.

WearningMaskInHW Rate_RateP art icipat ion Real Policy participation rate.

P revent ion Real Infectious disease prevention rate.

TemperatureMeasuring _RateRateDet ect ion Real Fever detection success rate.

P art icipat ion Real Measurement participation rate.

HomeQuarantine

Class Symbol A- and B-class quarantines.

DayQuarant ined Integer Number of home quarantine days.

RateP art icipat ion Real Policy participation rate.

RestrictedAccessToHospitals RateP art icipat ion Real Policy participation rate.

ReducedPublicContact RateP art icipat ion Real Policy participation rate.

record its position in the cellular automata. Two private attributes (MirrorIdentityNo and AgentID) are used by the cellular automata lattice to refer to the social mirror iden-tity and the agent who possesses it. Moreover, the private attributes Root and Suspend (Table 2) are used to model specific epidemic situations such as home quarantine, hos-pital isolation, the infectious condition of one’s family and neighborhood during a home quarantine, or the infectious condition of health care workers in a hospital where an agent is being held in isolation.

After initializing the upper-layer agent population and lower-layer cellular automata, the distribute-mirror-identities-to-CA subprocedure is called to establish one-to-one mapping between an agent’s social mirror identity and the cellular automata lattice. Through this procedure,

a small-world network model is created for simulations. The pseudo-code for the distribute-mirror-identities-to-CA subprocedure is

procedure distribute-mirror-identities-to-CA is

for index i from 1 to

System.Parameter.Pdo loop

AgentI ndex(i).AttributeLimit ←

random-by-normal-distribution

(1, System.Parameter.M)

for index y from 1 to

System.Parameter.Hdo loop for index x from 1 to

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Figure 11. Simulation system for contagious infections

Allocate:

ID← random(1, System.

Parameter.P ) // ID ∈ [1, System.

Parameter.P]

if AgentI ndex(I D).AttributeCount ≥

AgentI ndex(I D).AttributeLimit then

goto Allocate:

No ← AgentI ndex(r).AttributeCount←

AgentI ndex(r).AttributeCount + 1

call

connect-mirror-identity-with-cell(ID, No, x, y)

return

procedure

connect-mirror-identity-with-cell (parameter ID, No, x, y) is

AgentI ndex(I D).MirrorIdentityI ndex(N o).

AttributeLocat ion← coordinates(x, y)

EnvironmentCA.Cellx,y.AttributeAgent I D

← ID

EnvironmentCA.Cellx,y.

AttributeMirrorI dent ityN o← No

return

This subprocedure consists of two parts. In the first, a random number between 1 and System.Parameter.M (Ta-ble 3) is generated for each agent, and the number is as-signed as the upper limit of possible social mirror identities for an agent. The number of identities should exhibit a nor-mal distribution. In the second part, two methods are used to connect the cellular automata lattices with the agents’

social mirror identities. In the first method, each lattice is assigned a randomly chosen agent from top to bottom row and from left to right column, then each agent is connected to an available lattice with a social mirror identity that has yet to form its own lattice connection. If the number of connections between an agent’s social mirror identities and lattices has already reached its upper limit, the agent is released and another agent randomly chosen for the same procedure. In the second method, each agent’s social mirror identity is assigned to a randomly chosen cellular automata lattice. Determining which method to apply depends on a combination of simplicity and the particular requirements of the epidemic disease being examined.

The polymorphous Index(n) function used in the distribute-mirror-identities-to-CA subprocedure has two calling situations: AgentI ndex(n), where Index(n) refers to a certain agent with an ID of n, and AgentA .Mirror-IdentityI ndex(n), where Index(n) refers to a specific so-cial mirror identity of agent A with an serial num-ber of n. Index(n) has an inverse function desig-nated Trace(S), which also has two calling situations: AgentT race(S) (which returns the ID of a certain agent S) and AgentA.MirrorIdentityT race(S)(which returns the serial number of a specific social mirror identity S of agent A). According to these definitions, it is possible to deduce AgentA= Index(Trace(AgentA)). For examples of function Trace(S), see section 4.1.

After a social mirror identity and lattice are chosen, the distribute-mirror-identities-to-CA subprocedure calls

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Figure 12. Flowchart for initializing the cellular automata with the social mirror identity model (CASMIM): (a) create-CASMIM

procedure, (b) multiagent-system subprocedure, (c) agent-mirror-identities subprocedure, and (d) initialize-cellular-automata subprocedure.

the connect-mirror-identity-with-cell subprocedure to per-form a two-way reference. At this point, the serial number of the social mirror identity and the ID of the agent that possesses the identity are respectively recorded as the Mir-rorIdentityNo and AgentID of the cellular automata lattice. The lattice position is recorded in Location (x, y)—the coordinate attribute of the social mirror identity.

3.3 Small-World and Clustering Phenomena in CASMIM

We designed two sensitivity analysis experiments to de-termine whether our proposed model is (a) a small-world social network with the characteristics of high clustering and low degree of separation and (b) a robust simulation model in which small-world characteristics are not affected

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as long as four parameters (cellular automata height, cel-lular automata width, total agent population, and the upper limit of an agent’s mirror identities) are set within reason-able ranges. The first two parameters directly affect the set-tings of the third and fourth, and vice versa. Corresponding relationships among the four factors are shown in equation (2); the distribute-mirror-identities-to-CA procedure is de-scribed in section 3.2.

EnvironmentCA.AttributeH× EnvironmentCA.AttributeW

=

A∈PopulationAgent

|AgentA.AttributeCount|. (2)

Our first experiment focused on the relationship be-tween total agent population and degree of separation. We maintained a fixed average number of agent mirror identi-ties while changing the total agent population, beginning with 2000 and adding 2000 for each simulation up to a total of 200,000. Results are presented in Figure 13; the horizontal axis represents total agent population, and the vertical axis represents average degree of separation for the entire social network. The curves represent four exper-iments, with the average number of agent mirror identities set at 2, 4, 6, and 8. Each curve shows the average value for 20 runs.

According to our results, an increase in total agent pop-ulation was accompanied by a slow, logarithmic increase in average degree of separation for the entire social network. The average degree of separation remained sufficiently low to characterize our proposed model as a small-world social network model. In other words, the simulation model will always represent a small-world social network as long as all agents have an average of two or more mirror identi-ties, regardless of total agent population change. The lack of fluctuation in our model’s small-world characteristic is an indication of robustness for that parameter, even when the total agent population value changes.

Our second experiment focused on the relationship be-tween average number of agent mirror identities and degree of separation. We maintained a fixed population of 10,000 agents and manipulated the number of agent mirror iden-tities at a rate of 2n_{, with n = 0, 1, 2, 3, or 4. In Figure} 14, the horizontal axis represents the average number of agent mirror identities, and the vertical axis represents the average degree of separation for the entire social network. The results indicate that when the average number of mir-ror identities = 1 (i.e., each upper-layer agent has only one mirror identity in the lower-layer cellular automata, which is considered typical of cellular automata), the average de-gree of separation for the entire social network was very high. When the average number of agent mirror identities increased to 2 or more (with n≥ 1), the average degree of separation value decreased to 5.44, indicating the appear-ance of small-world characteristics. As the average number

of agent mirror identities increased to 4 (with n = 2), the curve in Figure 14 slowly decreased and stabilized. In other words, our proposed simulation model resembles a small-world social network as long as the average number of agent mirror identities exceeds 1. Our results indicate that the average number of agent mirror identities is a robust parameter; as long as it remains within a reasonable range (n≥ 1), small-world characteristics are not influenced by a change in value.

Figure 15 shows a normalized clustering coefficient curve and the separation coefficient curve (outcome) after normalizing the results of our second experiment. The clus-tering coefficient C(Agent) can be derived from equation (1), reduced according to the Moore neighborhood param-eters and the number of an agent’s social mirror identities, and expressed as equation (3) (where AgentA.AttributeCount represents the number of social mirror identities of agent A). The clustering coefficient C(PopulationAgent)of the en-tire social network is the average of C(Agent) for all agents.

C(Agent) =

3

(8surrounding Moore neighbors× Agent.AttributeCount)−1 . (3) In Figure 15, the horizontal axis represents the aver-age number of aver-agent mirror identities (increasing by 2n_, with n = 1, 2, 3, and 4), and the vertical axis range of 0 to 1 indicates a normalized unit. The normalized cluster-ing curve consists of small squares, and the average degree of separation curve consists of small triangles. The fig-ure also indicates that when the average number of agent mirror identities = 1, the degrees of separation and cluster-ing are both 1. When the average number of agent mirror identities exceeds 1 and increases gradually, (a) the de-gree of separation curve rapidly falls to between 0.1 and 0.01, and (b) the clustering curve decreases gradually and maintains a certain distance from the degree of separation curve. However, both curves support the assertion that our proposed model has the small-world social network char-acteristics of a high degree of clustering and a low degree of separation.

4. Modeling Contagious Epidemics and Setting Parameters

4.1 Modeling Epidemiological Features

When applying our proposed model to examine epidemic transmission dynamics, public health policies, and disease prevention strategies, epidemiologists need to categorize disease statuses according to specific epidemic character-istics, local conditions, and administrative requirements. We applied the state transfer concept of compartmental

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Figure 13. Effect of total agent population on average degree of separation

Figure 14. Effect of average number of agent mirror identities on average degree of separation

models and six disease statuses—S (Susceptible), E (In-cubated), I (Infectious), D (Deceased), R (Recovered), and M (Immune)—to represent an individual’s epidemi-ological progress state (Fig. 6, Table 1) and the behav-ioral and transformative results from interactions among individuals.

Before modeling the epidemiological features and pub-lic health popub-licies, we assumed that the epidemic was trans-mitted via close contact and exchanges of saliva. According to the transmission route and social network characteristics

used in our simulation, infections were further divided into contact and transmission stages, meaning that an agent’s mirror identity had to come into contact with the mirror identity of an adjacent neighbor for an infection to occur.

Based on the combination of Agent.Attribute.RateCont act and a random number c, each mirror identity of an agent determines whether it will interact individually with the mirror identities of its eight adjacent neighbors. If the value of the Suspend attribute for the mirror identity of agent A is false and the random number c is lower than

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Figure 15. Effect of average number of agent mirror identities on average degree of separation and average clustering coefficient

the contact rate, the mirror identity of agent A comes into contact with the mirror identity of its neighbor agent B. The Agent.Attribute.RateCont act depends on the enactment of a specific parameter—for instance, “reduced public con-tact.” In this section, we express these concepts using the following pseudo-code:

procedure contact is

for each A∈ PopulationAgent do loop

for each I ∈ AgentA.SetMirrorI dent ity

do loop

if AgentA.MirrorIdentityI.

AttributeSuspend = False then

for each J ∈ AgentA

.Mirror-IdentityI.SetN eighbor do loop

c← random(0, 1) // c∈ [0,1]

if c≤ AgentA.Attribute.RateCont act

then

infect(AgentA.MirrorIdentityI,

AgentT race(J ).MirrorIdentityJ)

return

procedure infect (parameter

AgentA.MirrorIdentityI,

AgentB.MirrorIdentityJ) is

if AgentA.AttributeE = I∧

AgentB.AttributeE = S then

n← Random(0, 1) // n∈ [0,1]

if n≤ System.Parameter.RateI nf ect ion

then

comment epidemiological state

changes from S to E

AgentB.AttributeE ← E // E

means incubated

AgentB.AttributeDay ← 1

return

Assume that agents A and B have adjacent mirror iden-tities; agent A is infected and contagious, and agent B is susceptible and prone to infection. When the two agents come into contact, a combination of infection rate (Sys-tem.Parameter.RateI nf ect ion)and a random number n de-termines whether or not agent B is infected by agent A. If n < the infection rate, agent B’s epidemiological state changes to E (Incubated), and the period attribute (Agent.AttributeDay)changes to 1 (denoting that symptoms have not yet appeared and that agent B cannot transmit the disease). The System.Parameter.RateI nf ect ionis determined by such factors as immunity rate—that is, whether agent A is a super-spreader, in home quarantine, in hospital iso-lation, and so on. Furthermore, agent A’s epidemiological state automatically changes from E to I (Infectious) once the incubation period (System.Parameter.PeriodI ncubat ion) is exceeded.

procedure

handle-epidemiological-progress-state (parameter AgentA) is

...

comment epidemiological state changes

from E to I

if AgentA.AttributeE = E then

if AgentA.AttributeDay > System.

Parameter.PeriodI ncubat ion then

AgentA.AttributeE ← I // I means

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...

from I to R or D

if AgentA.AttributeE = I then

if AgentA.AttributeDay >

System.Parameter.Period(I ncubat ion+Inf ectious)

then

d ← random(0, 1) // d ∈ [0,1]

if d ≤ System.Parameter.RateDeat h

then

AgentA.AttributeE ← D // D means

Deceased

AgentA.AttributeDay ← 0

else

AgentA.AttributeE ← R // R means

Recovered

...

from R to M

if AgentA.AttributeE = R then

System.Parameter.PeriodRecovered then

AgentA.AttributeE ← M // M means

Immune ...

from M to S

if AgentA.AttributeE = M∧ not

AgentA.AttributeI mmunityP ermanent then

System.Parameter.PeriodI mmune then

AgentA.AttributeE ← S // S means

Susceptible

... return

When agent A’s epidemiological state is I and it exceeds the System.Parameter.PeriodI nf ect ious infectious period, a combination of death rate (System.Parameter.RateDeat h) and a random number d determines whether the agent en-ters the D (Deceased) or R (Recovered) state. Death rates are influenced by such factors as age, whether or not the agent was placed under home quarantine during its incuba-tion and infective periods, whether it received treatment in hospital isolation, and its public activities during the illness period.

When agentA’s epidemiological state is R and it exceeds the System.Parameter.PeriodRecoveredrecovery period, it au-tomatically enters an M (Immune) state. In this state, the Agent.AttributeI mmunityP ermanent parameter determines whether agent A’s immunity is permanent or temporary— that is, whether complete recovery or renewed susceptibil-ity occurs following System.Parameter.PeriodI mmune.

4.2 Modeling Social Mobility, Families, and Hospitals Mirror identities have two private attributes: Root and Sus-pend (Table 2). As shown in the following pseudo-codes (for the set-rooted-mirror-identities-of-all-agents proce-dure), the Root attribute for most agents is true for one mirror identity but false for all others. In contrast, the Sus-pend attribute is false for all of an agent’s mirror identities. To facilitate our discussion, we will assume the presence of a rooted mirror identity—that is, one whose Root attribute is always true. Rooted mirror identities can be used to rep-resent one-of-a-kind units such as homes, dormitories, and hospitals.

procedure set-rooted-mirror-identities-of-all-agents is

do loop

AgentA.MirrorIdentityI.

AttributeSuspend← False

AttributeRoot ← False

n← random(1, AgentA.AttributeCount)

// n∈ [1,AgentA.AttributeCount]

AgentA.MirrorIdentityI ndex(n).

AttributeRoot ← True

AgentA.AttributeMobility← Free

return

If a health authority puts agent A under home quarantine (i.e., the mobility attribute of agent A is changed to Quar-antined), the Suspend attributes of all its mirror identities are marked as true with the exception of its rooted mirror identity. Accordingly, agent A cannot interact with other adjacent neighbors or move among various locations with the exception if it is home until the quarantine period ends. The lattice points surrounding agent A’s rooted social mir-ror identity are the mirmir-ror identities of the agent’s family members and/or cohabitants. Once the home quarantine is lifted, the Suspend attributes of all mirror identities except for the rooted mirror identity return to false, indicating a resumption of normal activities.

when event AgentA.AttributeMobility

= Quarantined do

do loop

AttributeRoot = False then

AttributeSuspend← True

= Free do

do loop

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The advantage of our proposed model is that it does not require fixed lattice points for representing hospitals. As-sume that agent B, with a confirmed epidemiological state of I , voluntarily enters isolation (i.e., its mobility attribute changes to Isolated). Similar to the preceding example, the Suspend attributes of all agent B mirror identities (ex-cept for its rooted mirror identity) are changed to true. This represents a scenario where agent B is receiving treatment in hospital isolation. The lattice points surrounding agent B’s rooted mirror identity represent medical staff, nurses, health care workers, and perhaps one or more family mem-bers who have special visitation privileges. If agent B re-covers, the Suspend attributes of the affected mirror iden-tities return to false, indicating a resumption of normal activities. If the agent dies, the Suspend attributes of agent B’s mirror identities (including its rooted mirror identity) are permanently changed to true.

= Isolated do

do loop

AttributeRoot = False then

= Free ∧ AgentA.AttributeE = R do

do loop

AttributeSuspend← False

= Free ∧ AgentA.AttributeE = Ddo

do loop

Table 5 presents the results of an intersection between epidemiological progress states and mobility states. The ta-ble allows users to address various potential combinations of situations that can occur during an epidemic outbreak. Health authorities can use this information to test various public health policies—for instance, decreasing or com-pletely eliminating the number of infectious patients who are allowed to leave their homes or hospitals (i.e., individ-uals with a disease status of E or I and a social activity status of Free, or with a disease status of S and a social activity status of Quarantined or Isolated).

4.3 Modeling Public Health Policies

4.3.1 Mask-Wearing Policy: General Public vs. Health Care Workers

The two parameters for a general public mask-wearing pol-icy are participation rate (the percentage of individuals in

the total population who actually wear masks) and preven-tion efficiency (the protecpreven-tion grade of the masks being used). Our simulation system uses the participation rate to select agents who abide by the policy. Agents with an S-status who wear masks are much less likely to become infected, depending on the prevention efficiency parameter. The same is true for I -status agents who wear masks be-fore and after their symptoms appear since the probability of the disease being spread to its neighbors will decrease, also depending on the prevention efficiency parameter.

The process for simulating a hospital employee mask-wearing policy is essentially the same. Once the policy is put into effect, agents who surround the rooted mirror identities of agents in hospital isolation either wear or do not wear masks based on the participation rate, and the probability of infection is also affected by the prevention efficiency parameter. Due to the high potential for infec-tion, health care workers are usually required or strongly encouraged to wear masks with high protection rates, mak-ing their participation rates very high. Since they tend to wear better quality masks, the prevention efficiency is also high.

when a mask-wearing policy in general public is enacted or changed do

if PolicyW earningMaskI nGP.Parameter.

RateP art icipat ion> 0 then

n← random(0, 1) // n∈ [0,1]

if n≤ PolicyW earingMaskI nGP.Parameter.

RateP art icipat ion then

AgentA.AttributeW earingMask← True

AgentA.AttributeMaskT ype←

PolicyW earingMaskI nGP.Parameter.

RateP revent ion

else

AgentA.AttributeW earingMask← False

when a mask-wearing policy in health worker is enacted or changed do

= Isolated do

for each N ∈ AgentA

.Mirror-IdentityRoot.

SetN eighbor do loop

c← random(0, 1) // c∈ [0,1]

if c≤ PolicyW earningMaskI nH W.Parameter.

RateP art icipat ion then

AgentT race(N ).AttributeW earingMask←

True

AgentT race(N ).AttributeMaskT ype←

PolicyW earingMaskI nH W.Parameter.

RateP revent ion

else

AgentT race(N ).AttributeW earingMask←

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Table 5. Intersection between epidemiological progress and social mobility states Epidemiological

Progress Mobility

State State Description

Susceptible Free Agent is healthy and free to move anywhere.

Susceptible Quarantined Agent is healthy but in quarantine since it may come into contact with an infectious agent.

Susceptible Isolated Agent is healthy but is mistakenly diagnosed as infected and therefore isolated by health care center.

Incubated Free Agent is infected and in an incubation period. It is free to move anywhere because it has not been properly diagnosed.

Incubated Quarantined Agent is infected and in an incubation period. It has yet to be examined. There is a possibility that one of its friends or family members has been diagnosed as infected; therefore, the agent is quarantined according to contact tracing and home quarantine policies.

Incubated Isolated Agent is infected and in an incubation period. After being examined, it is placed in hospital isolation.

Infectious Free Agent is infected and has symptoms, but it has yet to be examined or affected by a contact tracing policy. It can move anywhere and can easily infect other agents.

Infectious Quarantined Agent is infected and has symptoms, but it has yet to be examined.There is a possibility that one of its friends or family members has been diagnosed as infected; therefore, the agent is quarantined according to contact tracing and home quarantine policies.

Infectious Isolated Agent is infected and has symptoms. After being examined and diagnosed, it is placed in hospital isolation.

Deceased None Agent is dead.

Recovered Free Agent is recovered and free to move anywhere.

Recovered Quarantined Agent is recovered but is kept in quarantine because it has been in close contact with someone who has been diagnosed with the disease.

Recovered Isolated Agent is recovered but still in hospital isolation.

Immune Free Agent is immune and can move anywhere.

4.3.2 Taking Body Temperature

Under a temperature measurement policy, the social mirror identities of individual agents decide individually whether or not to measure their body temperatures before coming into contact with their surrounding social mirror identities. Their decisions are made based on a combination of a par-ticipation rate parameter and a random number n. An n that is lower than the participation rate means that neighboring agents are following the practice of measuring the tem-peratures of all agents that want to contact them. Success thereby depends on the detection rate parameter—a com-bination of participation rate and thermometer accuracy. During the 2003 SARS epidemic, most countries accepted the World Health Organization (WHO) recommendation to enforce this policy, but execution was considered ex-pensive in terms of manpower and social costs. It was rel-atively easy for infected individuals to avoid having their body temperatures taken.

4.3.3 Reducing Public Contact

Some researchers have recently studied reduced public contact as a means of controlling the spread of disease [7, 10]. In our simulation, the infection process was affected by a combination of contact and infection rate. Reducing public contact decreased the contact frequency of a tar-geted group of agents. The combination of participation

rate and a random number n determined whether the mir-ror identities of two agents interacted. An n that exceeded the participation rate indicated that an agent avoided con-tact with the mirror identities of its neighboring agent.

4.3.4 A- and B-Class Home Quarantines

According to an A-class home quarantine policy, when-ever an infected agent is identified, all agents surrounding the infected agent’s mirror identities must decide whether they accept home quarantine, based on the participation rate parameter. If they do, their mobility attribute changes from Free to Quarantined. As in the hospital isolation ex-ample, all of the mirror identities of agents that decide to enter home quarantine become inactive until the separa-tion period is complete. This requirement does not apply to rooted mirror identities. This policy requires consider-able manpower and social costs to execute.

Although similar in most respects to the A-class policy, a B-class policy affects a larger number of agents. For in-stance, if one mirror identity of agent C is adjacent to a particular mirror identity of agent D (i.e., if agents C and D are a cohabiting couple), this represents one degree of separation. If one mirror identity of agent D is adjacent to a particular mirror identity of agent E (perhaps coworkers in the same office), this represents two degrees of separation between agents E and C. Under a B-class policy, both D and E would be required to enter home quarantine.

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4.3.5 Hospital Access

During the 2002-2003 SARS epidemic, Singaporean and Taiwanese health authorities imposed strict rules concern-ing hospital visitations [10]. To simulate this “hospital ac-cess control” policy, we assumed that agent A showed symptoms and was admitted to a hospital for treatment in isolation. If the rooted mirror identities of agents A and B are adjacent, this indicates that agent B is on the hospital staff, a nurse, a health care worker, or a very close rela-tive. If agent C’s nonrooted mirror identities are adjacent to agent A’s rooted mirror identity, it indicates that agent C is a distant relative, friend, classmate, or coworker. Under a strict visitation policy, agent B is allowed to come into contact with agent A, but agent C is not.

4.4 Basic Reproductive NumberR0with

Correspond-ing Parameters in CASMIM

To present a reasonable and precise picture of the transmis-sion dynamics of an epidemic, we adjusted certain param-eters according to the most recently available information on contact rate, transmission rate, number of contacts, and average transmission period. In addition to predicting and estimating overall disease trends, we also applied the ba-sic case reproduction number R0to estimate all values for

the parameters just described to increase the precision and reliability of the simulation process and outcome.

According to Anderson and May [27] and Becker [28], R0can be expressed as equation (4), where c represents the

number of times an infectious person comes into contact with an uninfected person,β is the probability of transmit-ting the infection to each contact, and D is the length of time a person remains infectious.

R0= c × β × D. (4)

According to the characteristics of our proposed model, equation (4) can be amended as equation (5). β and D are the same in both equations, withβ replacing the in-fection rate RI nf ect ion (System.Parameter.RateI nf ect ion)and D replacing the average infected period Pinf ect ious (Sys-tem.Parameter.PeriodI nf ect ious).

R0= (avg. of social mirror identity × no. of neighbors

× RCont act× TCont act)× RI nf ect ion× PI nf ect ious. (5) As shown in equation (5), element c in equation (4) can be broken down, with “avg. of social mirror identity” representing the average number of agent mirror identi-ties, “no. of neighbors” the number of neighbors for each mirror identity (which under the Moore neighborhood structure equals 8), RCont act (Agent.Attribute.RateCont act) the contact rate of an agent’s mirror identity and the mirror identities of its neighbors, and TCont act (Sys-tem.Parameter.FrequencyCont act) the average number of contacts of an agent’s mirror identity with the mirror iden-tities of any other neighbor during a time step.

Since the average numbers of mirror identities for an agent and its neighbors are constants, they do not require updating, even when other disease transmission parame-ters change. Thus, only four parameparame-ters in equation (5) are associated with epidemics: contact rate (RCont act), number of contacts (TCont act), transmission rate (RI nf ect ion), and av-erage infected period (PI nf ect ious). All of these require ad-justment according to the latest information released by health authorities.

5. Simulating SARS with CASMIM

5.1 Comparing Simulation Results with Actual Cases

After initializing our model and setting up system and epi-demic disease parameters (Table 3) according to informa-tion distributed by WHO and the U.S. Centers for Disease Control and Prevention (CDC) [7, 10, 41-48], we simulated the transmission dynamics of SARS in different areas and compared the effectiveness of various public health poli-cies and disease prevention strategies. We used the simula-tion definisimula-tions and parameters identified in secsimula-tion 4 and assumed that one time step = 1 day in the real world.

Since SARS originated in China’s Guangdong province, we viewed the SARS viruses in all other coun-tries as being imported and used the number of imported cases announced by local health authorities to determine transmission source information—for example, number of infectious people entering a country, the time step during which they entered, and whether they entered as incubated or infected individuals (Tables 6-9). We incorporated pub-lic health popub-licies at certain time steps according to actual announcements made by local health authorities and ad-justed our simulation environment, epidemic, and public health policy parameters according to data from the CDC [42, 44, 45, 47] and Sebastian and Hoffmann [10].

5.1.1 Statistical Analyses for Epidemic Simulation We used five statistical tests to examine the reliability and validity of time-series data generated by the simulation system (Table 10): a chi-square test for homogeneity of proportions, a correlation coefficient (CC, equation (6)), coefficient of efficiency (CE, equation (7)), mean square error (MSE, equation (8)), and mean absolute error (MAE, equation (9)). {Xt| t = 1 . . . n ∧ t ∈ ℵ} represents time-series data for the number of individuals who were actually infected each day. {Yt|t = 1 . . . n∧t ∈ ℵ} represents time-series data for the numbers of infected individuals each day generated by our simulation system. In both data sets, t represents the time step (ranging from 1 to a maximum value of n), Xt represents the number of actual infected individuals at time step t , Yt represents the number of in-fected individuals generated by the simulation system at time step t, X represents the mean number of actual in-fected individuals, and Y represents the mean number of infected individuals in the simulation.