Fuzzy cognitive maps and cellular automata: An evolutionary approach for social systems modelling

Applied Soft Computing 12 (2012) 3771–3784

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Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc

Fuzzy cognitive maps and cellular automata: An evolutionary approach for social systems modelling Vijay K. Mago a,∗ , Laurens Bakker a , Elpiniki I. Papageorgiou c , Azadeh Alimadad a,b , Peter Borwein a , Vahid Dabbaghian a a

MoCSSy Program, The IRMACS Centre, Simon Fraser University, Burnaby V5A 1S6, Canada Faculty of Health Sciences, Simon Fraser University, Burnaby V5A 1S6, Canada c Department of Informatics and Computer Technology, Technological Educational Institute of Lamia, Lamia, Greece b

a r t i c l e

i n f o

Article history: Received 15 July 2011 Received in revised form 17 February 2012 Accepted 21 February 2012 Available online 16 March 2012 Keywords: Fuzzy cognitive maps Cellular automata Evolutionary model Knowledge representation Injection drug users HIV

a b s t r a c t One of the first decisions to be made when modelling a phenomenon is that of scale: at which level is the phenomenon most appropriately modelled? For some phenomena the answer may seem too obvious to warrant even asking the question, but other phenomena cover the gamut, from high to low levels of abstraction. This paper explores how two modelling approaches that are ‘at home’ at opposite ends of the abstraction spectrum can be combined to yield an evolutionary modelling approach that is especially apt for phenomena that cover a wide range in this spectrum. We employ fuzzy cognitive maps (FCMs) to model the interplay between high-level concepts, and cellular automata (CA) to model the low-level interactions between individual actors. The combination of these models carries both beyond their respective limitations: the FCM concept is extended beyond the derivation of equilibrium outcomes from static initial conditions, to time-evolving systems where conditions may vary; CA are extended beyond the emergence of patterns from local interactions, to systems where global patterns have local repercussions. The applicability of the methodology is demonstrated by modelling the spread of human immunodeficiency virus (HIV) in an environment in which injection drug users share paraphernalia. © 2012 Elsevier B.V. All rights reserved.

1. Introduction Many systems in the real world are characterized as complex systems—non-linear, time-varying systems with feedback loops that make their behaviour hard to predict. Some of the complexity of such real-world systems lies in the different levels of abstraction (scales) at which mechanisms operate in them. Each level of abstraction is characterized with its own concepts or variables, and consequently has different interactions between these concepts. Thus, a model that is appropriate at one level may fall short at another to the point that even applying the same modelling technique at different levels may be inappropriate. This paper explores how modelling techniques that are ‘at home’ at different levels of abstraction can be coupled to reap the benefits of both techniques while cancelling some of their limitations. We propose a modelling technique that combines a macro-level approach with a micro-level approach to yield an especially apt evolutionary modelling technique, using the macro-level outcomes to parameterize the micro-level model, and feeding statistics from

∗ Corresponding author. E-mail address: [email protected] (V.K. Mago). 1568-4946/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.asoc.2012.02.020

the micro-level model back into the macro-level one to recalculate parameters. The macro-level model of choice is a fuzzy cognitive map (FCM), which aggregates the model domain into concepts and the global interactions between those concepts. The microlevel model of choice is a cellular automaton (CA; plural: cellular automata), which disaggregates the model domain into individual actors that interact locally. In our approach we allow for multiple CA and FCMs to interact (see Fig. 1). The applicability of the proposed technique is demonstrated by modelling and simulating human immunodeficiency virus (HIV) spread in an environment in which injection drug users (IDUs) share needles or paraphernalia, a key mode of HIV transmission among IDUs [47,67]. The complicating factor in this dynamic is that drug use and needle sharing are individual behaviours [41,42] that are nevertheless influenced by the behaviour of other individuals as well as law enforcement and health care agencies. An FCM is a natural choice for modelling the influence of such macrolevel actors (agencies) on behaviour, but it is unable to capture the interpersonal interactions in a heterogeneous population. A microlevel model like a CA, on the other hand, is especially apt to model interactions at the level of the individual, but often incorporates macro-level actors only as a single ‘environment’ variable [10,11], if at all. The combination of (multiple) FCMs and CA allows us not

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Fig. 2. Basic structure of FCM. Fig. 1. The structure of an evolutionary model coupling multiple FCMs and CA.

only to incorporate both the macro and micro levels, but also to model the feedback loop that exists between the micro level (e.g., the prevalence of needle sharing) and the macro-level (e.g., government policies), and vice versa. The remainder of this paper is organized as follows. Section 2 describes the basics of FCM and CA based modelling. Section 3 gives formal definitions for FCMs and CA, which are then used to further define the proposed FCM-CA based evolutionary approach. In Section 4, we apply the approach to our sample scenario of HIV spread when IDUs share needles. There again, we first introduce the FCMs and CA individually, before weaving them together. Finally, in Section 5, we report on the simulations we conducted on the resulting evolutionary system. 2. Fuzzy cognitive map and cellular automata Before defining our approach more thoroughly in Section 3 and combining FCMs and CA, we present FCM and CA modelling individually to give the reader some intuition of each of these models.

between two concepts Ci and Cj has a weight wij , which is proportional to the strength of the causal link from Ci to Cj . If wij > 0 then there is a postive causality between concepts, if wij < 0, the causality is negative, and if wij = 0 then there is no causality. Human knowledge and experience on the system determine the type and the number of nodes, as well as the weights (wij ) of the FCM. This knowledge in the form of fuzzy values, assigned by experts or mined from the literature, is transformed into numeric values, activation degrees Ai for each concept Ci and weights wij for the causal links between them. The goal of formalizing domain knowledge in this way is to infer the activation degrees of concepts that are at the end of causal chains. This is done by iteratively simulating causation until the FCM converges to a steady state. At each iteration, the value Ai of a concept is influenced by the values of concepts-nodes connected to it, and is updated according to Eq. (1) [62]:

⎛

(+1)

Ai

=f

⎝A() +

FCM methodology is a symbolic representation for the description and modelling of complex systems at a high level. An FCM describes different aspects in the behaviour of a complex system in terms of concepts; each concept represents a characteristic of the system and these concepts interact with each other, showing the dynamics of the system. Kosko introduced FCM [31,32] as a signed directed graph for representing causal reasoning and computational inference processing, exploiting a symbolic representation for the description and modelling of a system. Concepts are utilized to represent different aspects of the system. The construction of an FCM usually requires the input of human experience and knowledge on the system under consideration (both qualitative and quantitative data) but can also be derived from data [59]. Thus, an FCM integrates the accumulated experience and knowledge concerning the underlying causal relationships among factors, characteristics, and components that constitute a system. An FCM consists of concept nodes, Ci , i = 1, . . ., n, where n is the total number of concepts. Each concept node Ci represents one of the key factors that play a role in the system, and it is characterized by a value Ai ∈ [0, 1] that is proportional to its activity or presence in the system. The concepts are connected by weighted arcs, which encode the relations among them. An example FCM with eight nodes and ten weighted arcs is illustrated in Fig. 2. Each connection

i

⎞

()

Aj

× wji ⎠

(1)

j= / i ()

2.1. Preliminaries of FCM based modelling

n 

()

where Ai and Aj are the activation degree of concepts Ci and Cj at iteration , wji is the weight of the interconnection from concept Cj to concept Ci and f is a threshold function that squeezes the result into the interval [0, 1]. The threshold function is used to reduce the unbounded weighted sum to a certain range, which allows for qualitative comparisons between concepts, at the cost of amenability to quantitative analysis. The most popular functions are continuous; however, some research works also utilize binary functions. The most commonly used ones are: binary, trivalent and sigmoidal. For a comparison of different threshold functions for FCMs, please refer to Tsadiras’s recent review paper [65]. Example FCM. To make the above more concrete, let us explore an example FCM for needle sharing among IDUs, illustrated in Fig. 3. It is distilled from one of the FCMs used in Section 4.1.

Fig. 3. An example FCM of factors that reduce needle sharing.

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Our example focuses particularly on why individuals may be deterred from sharing needles with fellow IDUs. Needle sharing among IDUs is a serious problem as it has a direct influence on the spread of serious diseases such as HIV. To prevent this practice, the government sanctions or promotes different organizations like Street Nurse Programmes and Drug User Organizations (DUOs). The more intent the government is on reducing needle sharing, the more support it will provide (positive arrows emanating from “Government Attention”) for such organizations. These organizations run a variety of programs to reduce needle sharing: education, supply of paraphernalia, health check-ups, etc. (the positive arrows pointing to “Deterrent from Needle Sharing”). On the other hand, the government is likely to also provide direct disincentives to share needles, complementing these ‘soft’ programs with harsh and stringent laws. This has the side effect of suppressing DUOs (the negative arrow), because drug use is further criminalized, indirectly decreasing the popularity and credibility of DUOs. This example shows how an FCM captures well the concepts and the different forces that play together to affect a particular issue, in this case needle sharing behaviour. There also lies its weakness: behaviour is fundamentally individual, not global. A low-level model like a CA can provide a solution.

2.2. Preliminaries of CA based modelling Cellular automata modelling is one of a family of modelling techniques that are based on the paradigm that simple rules that act at the micro level can produce complicated (computationally intractable [5,57]) patterns at the macro level [73]. CA were introduced by von Neumann [69] as an extension of Turing automata, to show that automata are capable of universal computation. They remained a relative curiosity (e.g., in the form of John Conway’s Game of Life [5]) until popularized by Stephen Wolfram, who showed that many real-world systems can be modelled using simple local rules [72]. CA modelling became widely used, first in the physical sciences, and later also outside of its comfort zone, in the social sciences. Its success there, for example in urban geography, is largely due to its conceptual simplicity [4,10]. The challenge lies in the fact that the ‘simple local rules’ are often unknown, and would need to be reverse engineered [24]. Conceptually, a CA consists of a collection of cells ci , i = 1, . . ., n representing n actors, embedded in a regular grid lattice. Any cell is in one of a number of states, and may change its state at each discrete time step. Whether or not it does and, if so, to which other state depends on its neighbours’ states. The transition function that maps a cell ci ’s neighbours’ states to a new state for that cell (ci ) is the core element of a CA. All cells usually change state synchronously, and homogeneously (using the same transition function for all cells). Asynchronous and heterogeneous CA exist [73], as well as similar models with a heterogeneous neighbourhood structure (referred

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infection susceptible

infected

death

Fig. 4. Automaton of a basic SI (susceptible-infected) model that takes into account death.

to as network models [19]), but these are not considered in the present work. Example CA. To make the above more concrete, let us explore an example CA model of disease transmission and spread. It has exemplar merit in its own right and forms the basis of one of the CA used in Section 4.2. In the most elementary form of disease spread, a population contains individuals of two types: those susceptible to the disease, and those infected by it. For simplicity, assume that infected individuals are also immediately infectious (i.e., they can transmit the disease to others as soon as they get infected), and that individuals do not recover, though they could die. The corresponding automaton is shown in Fig. 4. The spread of the disease is then governed by the structure of contacts in the population and the probability of disease transmission upon contact between a susceptible and an infected individual [1]. Let us take the neighbourhood shown in Fig. 5(a) (Fig. 5(b) and (c) are the other possible neighbourhood types), and use P to denote the probability of disease transmission upon contact. For example, in Fig. 6, cell c9 has two independent chances of getting infected: once by c10 and once by c14 . Its probability of infection is thus 1 − (1 − P)2 . In general, the probability that a Susceptible cell gets infected at a given time t is drawn from a binomial distribution and depends on the number of infected neighbours # infected (Ni , t − 1) (Ni are ci ’s contacts) at the previous time step (t − 1): Probability(ci gets infected at time t) = 1 − (1 − P)#infected (Ni ,t−1) (2) This is a clear example of the simple, local rules that make CA modelling a very fine-grained, low-level modelling technique. The lack of global control is also already visible: the probability of infection p may not remain constant, but change under global influences, e.g., a government-sanctioned campaign that reduces infective behaviour. A CA cannot model these changes, but an FCM can provide such input. 3. Evolutionary modelling approach to combine FCMs and CA In this section we present formally an evolutionary approach to modelling which combines FCMs and CA. Evolution is, by definition, an iterative process, so an evolutionary model needs to model

Fig. 5. Examples of CA neighbourhoods in two dimensions.

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In some cases a descriptive character string may be substituted for one or both subscripts, e.g., ItS F or social p− . Rather than taking a somewhat random pick from the Greek alphabet for denoting parameters, we formalize their use in our modelling so that we can use them generically. Definition 1 (Parameter). A model x F or x CA has a (possibly empty) set of associated parameters x P = {x p1 , x p2 , . . .}. If these parameters are not set at the initialization of the model but may change over time, they are denoted x P(t) and x pi (t). 3.1. Models With this notation framework in hand, let us define each of the two component model types of our evolutionary model.

Fig. 6. Example situation for a disease spread CA. White cells are susceptible individuals, black cells are infected individuals.

time. Therefore, at least one of the component models needs to explicitly model the passing of time. Other models need to have congruent time scales, i.e., days and weeks, or weeks and weeks, but not weeks and months because no integer number of weeks fills up each month of the year. Congruent time scales are necessary to allow discretization of time, which in turn is necessary to simulate the model. They allow the models to interact at each discrete time step that they have in common. Indeed, the component models must interact, otherwise there would be no merit in joining them. Having set a time for this interaction (at common time steps), we need mechanisms for them to do so, which we will call ‘channels’. A channel transfers information between two models, and since the information is in a different form in each type of model, a channel needs to be formally defined for every pair of model types that may interact, and needs to be specified for each pair of models that do interact. In summary, to compose an evolutionary model one needs: • a collection of heterogeneous models, • a discretizable notion of time that works for each of these models, • a mechanism of information exchange between the models of the same type and of different types. In the following, we will proceed to define the models and channels involved in an evolutionary model combining FCMs and CA. Notation. Presenting the linkage between several models of two different types is challenging notation-wise. Therefore, we use a consistent notation style, which we introduce here generically before putting it to use in the subsequent definitions. For an arbitrary symbol , we may use subscripts both before and after the symbol, a superscript after it, and a variable in parentheses: () x i (t)

• The subscript before the symbol (x in the example) is used to distinguish between different FCM or CA (e.g., x F and y CA) or variables from FCMs or CA (e.g., x A and y C). • The subscript after the symbol (i in the example) is used to select or index elements if  is a matrix, vector or set (e.g., wij and Ai ). • The superscript after the symbol ( ∈ N in the example) is used for iterations of an algorithm that have no unit (hours, days, etc.). It is put in brackets to avoid confusion with the usual raising to some power. • The variable in brackets (t ∈ N in the example) usually denotes model time (e.g.,  i (t)).

Definition 2 (Fuzzy cognitive map). An FCM is a 4-tuple (C, W, A(1) , f) where 1. C = {C1 , C2 , . . ., Cn } is the set of n concepts nodes of the graph. 2. W : C × C → [− 1, 1] associates a strength wij between −1 and 1 with a pair of concepts (Ci , Cj ), denoting the weight of the directed edge from Ci to Cj (we take wii = 0). 3. A() ∈ [0, 1]n is the vector of activation degrees of concepts at iteration , so that A(1) contains the initial activation degrees. We use the shorthand A(∞) = lim A() →∞

(3)

to denote the activation degrees when the FCM has converged or reached its equilibrium point (see Algorithm 1). 4. f : R → [0, 1] is a threshold function used to ensure that all Ai remain in the interval [0, 1]. Definition 3 (Cellular automaton). A CA is a 4-tuple (C, N, S, ) where 1. C = {c1 , c2 , . . .} is the population, a set of cells or actors. 2. N : C → ℘(C), where ℘(C) denotes the power set of C, is a function that returns the neighbours of a cell ci (see Fig. 5). We use shorthand Ni for N(ci ). 3. S = {s1 , s2 , . . . } is a (possibly infinite) set of states that cells ci can be in at a specific time. We denote (ci , t) the state of ci at time t. We use shorthand  i (t) for (ci , t). The longer notation is maintained for use with sets of cells, e.g., (C, t) or (Ni , t). The return value of  is then necessarily a multiset. 4.  : S × S × · · · × S → S is a function that maps the states of the neighbours of a cell ci (at the previous time step) to a new state for ci . Note that we use capital Ci for FCM concepts, and lowercase ci for CA cells. 3.2. Channels With these two types of models, four channels are to be defined: two between models of the same type, and two between models of different type. Two channels are necessary for one heterogeneous pair of models, because a channel needs to be defined for transfer of information in each direction, i.e., from CA to FCM and from FCM to CA. The channels connect the cores of the models. In an FCM the set of concepts C is the core of the model, and in a CA the transition function  is the core of the model. Hence,

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Definition 6 (Channel from CA to FCM). A channel from a cellular automaton x CA to a fuzzy cognitive map y F is formed by using cell (1)

states (x C, t) from x CA to change activation degrees y Ai (t) of y F (1) y Ai (t)

= x CA→y F Fi ((x C, t − 1))

(5)

and letting y F reconverge. x CA→y F Fi is used to translate from the multiset of states returned by (x C, t − 1) (see Definition 3) to a number within the same range ([0, 1]) as the other FCM concepts [7]: x CA→y F

F : S × S × · · · × S → [0, 1]

(6)

Definition 7 (Channel from FCM to CA). A channel from a fuzzy cognitive map x F to a cellular automaton y CA is formed by using activation degrees x A(∞) from x F to set a parameter y pi (used in y ) in y CA: Fig. 7. Venn diagramme of the (co)domains of two transition functions between which a channel exists.

• a channel between two FCMs should connect concepts, • a channel between two CA should connect transition functions, • a channel from an FCM to a CA should connect concepts to a transition function input, and • a channel from a CA to an FCM should connect transition function output to a concept. Definition 4 (Channel between FCMs). Let there be two distinct fuzzy cognitive maps x F = (x C, x W, x A(1) , x f ) (1) (1) = and Remember that y F = (y C, y W, y A , y f ). yA (1)

(1)

(1)

(1)

[y A1 , y A2 , . . . , y Am , . . .] where y Am represents the mth element of this initial activation degree vector of y F. A transfer channel for updating is defined as: (2) x Ai

(1)

:= x Ai

(1)

← y Aj ,

∀k, 1 ≤ k ≤ |x C|

(4)

where |C| represents the size of C and ← represents the first iter(1) ation of FCM procedure. It means that y Am impacts the set of (1)

(1)

(1)

concepts, x A1 , x A2 , . . . , x A| C| in the first iteration. Later ( > 2), x F x and y F are independent and their inference algorithms (Algorithm 1) operates separately. Definition 5 (Channel between CA). Let x CA = (x C, x N, x S, x ) and = (y C, y N, y S, y ) be two linked CA. The linkage between them is established by sharing of (partial) states between the two CA.

y CA

xS

∩ yS = / ∅ For this to work, both CA must use the same set of cells C.

xC

= yC

In other words, the domains and codomains of x  and y  intersect (see Fig. 7). Channels between models of different types require a little more work to be done, because these models do not use the same currency, so to speak. The currency of FCMs are the activation degrees A() , and the currency of CA are the cell states (C, t). A channel between two models needs to convert the currency of the one into a form useful for the other model type. To maintain generality, we do not specify the conversion but instead denote it as a function, e.g., x CA→y F F. In this notation, the subscript prior to the F specifies in which channel it is employed (from x CA to y F in the example).

y pi (t)

= x F→y CA Fi (x A(∞) (t))

(7)

Here x F→y CA Fi serves the same purpose as x CA→y F F i in the previous definition: it translates between the two models: x F→y CA

F : [0, 1]|x C| → R

(8)

The careful reader may already have spotted the discrepancy between Eqs. (5) and (7) in Definitions 6 and 7, respectively: (1) y Ai (t)

y pi (t)

= x CA→y F F i ((x C, t − 1))

= x F→y CA Fi (x A(∞) (t))

Information from FCMs reaches CA instantly, whereas information from CA takes one time step to reach FCMs (hence the t − 1). In the context of anything other than a generic framework this discrepancy would not be worth mentioning, but here it warrants some explanation. The choice made above is an arbitrary one in the context of the framework, though it has to be made: information cannot be exchanged both ways instantly, because that would mean that both would have to be evaluated before the other, an impossibility. Therefore, the choice was made, and here it was made with the particular application of Section 4 in mind: there, the FCMs represent a process (government decision making) that occurs after data have been gathered, and so the explicit time lag was put in the channel from the CA to the FCMs. Although we chose to work with FCMs and CA, we contend that the reasoning and procedure described in this section is useful for a variety of combinations of heterogeneous models. 3.3. Process summary Aside from t’s and t − 1’s, the above definitions do not clarify the passing of time. In this short section we make the story that has already been told around the definitions explicit in algorithms. This section contains 3 algorithms: one for each model, and one for the evolutionary approach to link the other two algorithms. The combined evolutionary model dictates the passing of time, so the algorithms for FCM and CA describe the passing of a single unit of time. In a single unit of time, an FCM needs to converge to equilibrium. Algorithm 1 implements the limit of Eq. (3) (to A(∞) ) by iterating the distribution of activation degrees until the total change  is below a small threshold p . Algorithm 1 (Basic FCM inference algorithm).

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Input: initial activation degree vector A(1) ∈ [0, 1] ; edge weight matrix W ; convergence condition p Output: equilibrium activation degree vector A(∞) ←2 repeat f (A(−1) + A(−1) × W ) A() = () (−1) ← |Ai − Ai | i

 ←+1 until  < p

In a single unit of time, a CA updates the state of all cells simultaneously. Algorithm 2 implements this simultaneity by a strict distinction between the previous time step and the current. The states of a cell’s neighbours at the previous time step are gathered as input to the transition function, and the output is stored as the cell’s state at the current time step. Algorithm 2 (CA algorithm for a single time step from t − 1 to t). input: cell states (C, t − 1); parameters P(t) Output: new cell states (C, t) for all ci ∈ C do nhdState ← {j (t − 1)|cj ∈ N(ci )} i (t) ← (nhdState){subject to parameters in P(t)} end for

Time is driven forward by the evolutionary model algorithm (Algorithm 3), which incorporates and couples the two previous algorithms. Algorithm 3 (Evolutionary algorithm). Input: models x CA, . . . , y FCM, . . . and initial states (e.g., x (C, 1)) and activation degrees (e.g., y A(1) (1)); Initial parameters P(1); requested simulation time tend Output: final states (C, tend ) and activation degrees A(tend ) for t ← 2 to tend do Compute new initial activation degrees A(1) (t) from the CA states (C, t − 1) using Eq. (5) Invoke Algorithm 1 to calculate A(∞) (t) for each FCM Use the calculated activation degrees A(∞) (t) to set parameters in P(t) using Eq. (7) Invoke Algorithm 2 to advance time by one unit for each CA, from t − 1 to t end for

4. Application: modelling the spread of HIV among IDUs A number of studies try to investigate trends of HIV infection among IDUs using various statistical and epidemiological methods [27,68]. These studies attempt to analyze incidents rates, behavioural trends and the effectiveness of various programs run by governmental and non-governmental organizations to curb this social hazard. These studies are usually restricted to a few parameters or variables and also have to rely on either empirical data generated by organizations running such programs, or on a small set of data that is collected for that specific study only. These limitations provide an opportunity for complex systems modelling, which is able to take the information generated by the studies alluded to, and combine it to go beyond the boundaries of this previous research. The FCM-CA based evolutionary model presented in this paper is based on the notion that needle sharing among IDUs is not an abrupt incidence but a process that evolves over a period of time. The model was constructed specifically to investigate the interplay of environmental and social factors in the needle sharing-mediated spread of HIV among IDUs. Two FCMs at the macro level capture the overall positive and negative influences of an environment—factors that discourage and encourage needle-sharing, respectively. Two CA at the micro level simulate the effects that risk behaviours and HIV infection of individual IDUs have on other IDUs (social and actual contagion), in the environment defined by the FCMs. The prevalence of needle-sharing among IDUs then feeds back into the

Table 1 Main social factors (concepts) which influences the concept Inclination towards Sharing. ID

Social factors (concepts)

C1 C2 C3 C4 C5 C6 C7 C8 C9 C10

Unemployment Homelessness Economic stress Depression and risky behaviour Prostitution Social networking among IDUs Stringent laws Reduced supply of paraphernalia Incarceration and heightened policing Inclination towards sharing

FCMs where it influences the government to take action. As in the previous sections, we will first introduce the individual FCMs and CA, and then shed light on the interplay between FCMs and CA. 4.1. FCMs model structure of IDU In our example scenario, we are interested in the dynamics of two concepts, the Inclination towards Sharing (ItS) and Disinclination towards Sharing (DtS) which are supplied to the CA models as environment in which to simulate the IDU population at the micro-level. For this purpose, we constructed two FCMs: ItS FCM and DtS FCM. An FCM is usually constructed based on expert knowledge. The way this expert knowledge is gathered depends on the disciplinary context of the research. Often, a panel of domain experts provide this knowledge [29] as they are acquainted with the working of the system. In the context of our research, which is an epidemiological one, new theories and knowledge are often embodied in systematic reviews. In a systematic review, the literature is reviewed to identify, appraise, select and synthesize all high quality research evidence available on a particular topic. We followed this disciplinary tradition to construct our FCMs. We studied 32 research papers published after the year 2000 and 9 papers published during 1990 and 2000 to construct two separate FCM models as per Definition 2. The concepts extracted from these papers are summarized in Tables 1 and 2. The causal connections between these concepts, as identified by the papers, are shown in Fig. 8. One may view this as a single FCM with two distinct decision nodes or observable nodes. This approach has been used by Georgopoulos et al. [18] to diagnose the different types of speech language impairments and also by Dickerson and Kosko [15] to define behaviour in an undersea virtual world of dolphins, shark and fish. However, Fig. 8 shows that only two edges (from C3 to C20 Table 2 Main social factors (concepts) which influences the concept Disinclination towards Sharing. ID

Social factors (concepts)

C11 C12 C13 C14 C15 C16 C17 C18 C19 C20 C21 C22 C23 C24 C25

Drug User Organizations Paraphernalia supply by Street Nurse Programme De-marginalization by Street Nurse Programme Education by Street Nurse Programme Education by peer run safe injection site Paraphernalia supplies by peer run safe injection site Community based social counselors De-marginalization by community meetings Awareness due to health education Integrated families Integration by counseling/influencing Media Awareness through publicity Government support towards social systems Disinclination towards sharing

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Fig. 8. Two combined FCMs for modelling needle-sharing among IDUs.

and from C7 to C11 ) and one concept (C24 ) connect the two otherwise disjoint FCMs. Therefore, we define two separate FCMs: one for concepts that influence C10 , and one for concepts that influence C25 . The concept C24 cannot be in both or neither FCM under the formalism defined in Section 3, so it is taken to belong to the FCM it has most connections to. The edges that cross the boundary between the two FCMs fall under Definition 4. We extracted the information on the links between concepts from literature, i.e., the direction or flow of influence among concepts; type of influence i.e., positive or negative; and the weight on the edges. It is reasonably easy to find the first two aspects (shown in Fig. 8) but finding weights is a more intricate matter. One way to do this is to use statistical information, converting the standardized mean difference or correlation coefficient to an odds ratio, [20] but this methodology has some serious drawbacks as opposed to fuzzy logic concepts. For instance, sometimes a small fraction of change is very significant as compared to a large deviation e.g., a small change in human body temperature is very important as compared to seasonal temperature changes in environment. So, instead of relying on numeric values in literature, we searched for keywords that qualify the causal relationship. This approach is similar to asking human experts about their opinion. These written opinions were used in fuzzy IF-THEN rules to infer a defuzzified numeric weight which represents the cause and effect relationship (grade of causality) between each pair of concepts [44]. Generally, such rules operate on linquistic variables, say A, B and D, and assume the following form: IF value of concept Ci is A THEN value of concept Cj is B and thus the linguistic weight wij is D. In this case, we kept the value of A = 1. For B, we used the term set X(Influence). It comprises five linguistic variables: X(Influence) = {VeryLow(VL), Low(L), Medium(M), High(H), VeryHigh(VH)}, depicted in Fig. 9. The variable Influence, which is interpreted as a linguistic variable, takes values in the universe D = [− 1, 1], where the type of causality (positive or negative)

determines the sign, and the grade of causality determines the absolute value. We interpreted phrases from the reviewed papers that closely describe the influence of one concept on another, and matched them with a linguistic variable. The influences thus extracted from literature discern between different degrees of impact. An illustration: Let us consider the strength of influence between concept C6 (“Social networking among IDUs”) and concept C10 (“Inclination towards Sharing”). This edge describes the cause–effect relationship of social networking of IDUs on needle sharing. From Refs. [13,35,36], we extracted the most relevant texts as: Statement from [13]: “. . .needle sharing among injection drug users (IDUs) is associated profoundly with the social context of drug use. . .”. We interpreted this statement and inferred that this statement suggested linguistic weight to be VERY HIGH. Statement from [35]: “. . .drug partners have more influence on an individual’s drug-using pattern. . .”. The inferred linguistic weight is HIGH.

Fig. 9. Membership functions defined to capture the influence.

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Fig. 10. Crisp impact of Social networking among IDUs and risky behaviour on Inclination for Needle Sharing.

Statement from [36]: “. . .higher probability of sharing of needles. . .”. The inferred linguistic weight is HIGH. The way this information has been used in the IF-THEN rules of fuzzy logic is shown below: IF (Social networking among IDUs is ON) THEN (Inclination towards Sharing is VERY HIGH) with weight (1/3). The weight of this rule is determined to be 1/3 as this rule is supported by 1 reference out of 3. IF (Social networking among IDUs is ON) THEN (Inclination towards Sharing is HIGH) with weight (2/3). The weight here is 2/3 as this rule is supported by 2 references out of 3. Using the sum aggregation method, the centroid defuzzification method and the Mamdani inference mechanism, [32,62] a crisp weight value of 0.765 is calculated for this edge (see Fig. 10). The other weights in weight matrix W were inferred using the same approach. W was then used to determine the steady state of the FCMs, using Eq. (1). Tables A1 and B1 in the Appendix list the concepts, references and the defuzzified values on the edges for the two FCMs. Algorithm 4 describes the FCM reasoning algorithm to calculate the values of concepts C10 :ItS and C25 :DtS. To keep the values of concepts in the range [0, 1], we use a threshold function as described in item 4 of Definition 2. As mentioned in Section 2.1, the choice of threshold function is an important one. The authors in [7] conclude that sigmoid functions are more suitable for situations where representation of a degree of increase, a degree of decrease or stability of a concept is precisely required, so the sigmoid function f is selected for our approach: f (x) =

1 1 + e−p x

(9)

where p > 0 is a parameter that determines its steepness in the area around zero, and how quickly the function approaches the limiting values of 0 and 1 [7,65]. We used p = 0.5. Algorithm 4 (FCMs reasoning algorithm for calculating values of concepts ItS (C10 ) and DtS(C25 )).

25

Input: Initial activation degree vector A(1) ∈ [0, 1] ; edge weight matrix 576 W ∈ [−1, 1] ; convergence condition p (∞) Output: Inclination towards Sharing (A10 ); Disinclination towards (∞)

Sharing (A25 ) A(2) ← f (A(1) + A(1) × W ) w1,20 ← 0; w7,11 ← 0; w24,1 ← 0; w24,2 ← 0; w24,7 ← 0; {Remove the links between concepts so that they do not participate in next iteration} ←3 repeat f (A(−1) + A(−1) × W ) A() = () (−1) ← |Ai − Ai | i

 ←+1 until  < p w1,20 ← −1; w7,11 ← −1; w24,1 ← −1; w24,2 ← −1; w24,7 ← −1; {Restore the links so that the concepts participate during next FCM inference procedure, as per the feedback received from CA}

The only difference between a normal FCM reasoning algorithm and the way it is used here is that some (channel) edges are considered at the first iteration only and not during the remainder of the reasoning mechanism. This restriction, defined in Definition 4, helps us assume that these concepts have influence in the whole system but at the same time do not get updated at every iteration. 4.2. CA model structure Since the epidemic spread of HIV and the social spread of needle sharing behaviour are linked, the two CA (in our scenario) model two linked ‘epidemics’. The transmission of the HIV virus and the transmission of needle-sharing behaviour are both a form of contagion. However, the underlying mechanisms are different, so we use two distinct CA to capture them separately. Disease spread dynamics and adoption of needle sharing behaviour can be modelled in many ways, but we keep both concise and simple because the focus of this paper is on the interplay between the models. Using the formalism introduced in Definition 3, both CA (social CA and HIV CA) operate on the same set of cells (C) representing individuals, with a von Neumann neighbourhood (N; see Fig. 5(a)). The fact that they operate on the same set of cells forms the basis for a channel between the two, as per Definition 5.

V.K. Mago et al. / Applied Soft Computing 12 (2012) 3771–3784

larger than 1 in absolute value, a change for the better (+1; to nonsharing) or worse (−1; to sharing) is made

sharing neighbours



non-sharing

non-sharing or stayer neighbours death

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sharing

stayer

death

death

social i (t)

Fig. 11. The basic automaton of our Mover–Stayer model. Cells may start out in any one of the three states (incoming arrows above each state). A cell ci will by default stay in its state (loops below each state), unless moved to another state by its neighbours (arrows between non-sharing and sharing).

The states and the transition functions that operate on these differ between the two CA, although there is some cross-pollination in the sense of Definition 5.

=

sharing non-sharing social i (t − 1)

if counter i (t − 1) < −1 if counter i (t − 1) > 1 , otherwise

(11)

and the social counter is reset to 0. Stayers do not receive any influence, so they do not change their state: In addition, non-sharing and Stayer individuals die after 900 months (75 years) [7], and sharing individuals die after 600 months (50 years) [11]. This translates straightforwardly to labels on the state transitions of Fig. 11: counteri (t − 1) < − 1 and counteri (t − 1) > 1 for the state transitions and ages for death. To prevent artefacts of the deterministic death rate from affecting the simulation results, individuals’ ages are randomly assigned at the start of simulations.

4.2.1. Needle sharing behaviour: social CA We modelled the adoption of needle sharing behaviour using a Mover–Stayer model. Movers are those individuals that may change their behaviour depending on the influences they receive from their neighbours and environment, and Stayers do not change their behaviour, though they may influence others. In the context of our model, Movers are IDUs, and Stayers are service providers (nurses, social workers, etc.). The states (S) of this CA are: nonsharing, sharing and stayer, and the transition function () changes individual IDUs’ states based on neighbours’ influences (see Fig. 11). The model employs a novel approach to represent the effect of prolonged social relationships between members of a community where injection drug use is prevalent. In such an environment, an individual is more likely to share needles after lengthy relationships with IDUs who also share needles, and less likely to share needles after lengthy relationships with individuals (IDUs and service providers) who do not share needles. Thus, negative influences from sharing neighbours and positive influences from non-sharing and stayer neighbours accumulate in an individual’s social counter. For a given cell ci , let # + (Ni , t) count the number of cells in the neighbourhood of ci that exert a positive influence (stayers and non-sharing IDUs), and # − (Ni , t) counts the number of cells in the neighbourhood of ci that exert a negative influence (sharing IDUs). The value of the social counter of a cell ci at time step t is then:

4.2.2. HIV spread: HIV CA We modelled HIV spread by a susceptible-infected (SI) model that takes into account disease progression, so it is an extension of the example introduced in Section 2.2. The states (S) of this CA represent the number of time steps (months) an individual has been infected, and the transition function () probabilistically infects the neighbours of infected individuals according to a binomial distribution, as in Eq. 2. The transmission probabilities per risky contact (needle shared; 120 per month [50]) are different for three categories of infected individuals: in the incubation period (months 0–2) it is high (p = 0.05); during the latent period after this and the subsequent AIDS (months 3–83 and 84–96) it is low (p = 0.001); finally, infected individuals die (after 96 months; see Fig. 12) [16]. Since recovery from HIV infection is not possible, HIV progression is onedirectional (from susceptible to death). Sexual transmission is not included in the model, and it is assumed that viral transmission occurs only through needle sharing. These refinements of the basic SI model are still far from realistic, but lack of consistent information on the development of infectivity as the infection progresses prohibits a more realistic transition function. To prevent artefacts of the deterministic disease progression from affecting the simulation results, how long an infected individual has been infected at is randomly assigned at the start of simulations.

counter i (t) = counter i (t − 1) + social p+ × #+ (Ni , t) − social p−

4.3. Structure of evolutionary model

× #− (Ni , t)

(10)

where p+ and p− are model parameters that modulate the size of the positive and negative influences. When the social counter becomes

With the individual models now in place, the task left is to specify the channel functions F that convert the currencies of the two model types, and to show their relation to the overall story of this

σi (t) ≤ 96

susceptible

σi (t) > 83

σi (t) > 2

infection

P = 0.0

incubation P = 0.05

latent P = 0.001

AIDS P = 0.001

no infection

σ i (t) ≤ 2

σ i (t) ≤ 83

death

Fig. 12. Schematic view of the correspondence between disease progression (in stages) and the probability P of infecting another individual in a risky interaction (needle sharing).

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evolutionary model. The story is that the FCMs provide an environment in which the CA play out, and the CA form the reality to which actors in the FCMs (notably the government) respond. Particularly, the environment provided by the FCMs, in the form of the activation degrees for Disinclination towards Sharing (DtS) and Inclination towards Sharing (ItS), affects the CA for sharing behaviour (social CA). Assuming that those strongly inclined to share would also more adamantly encourage others to share needles, and vice versa, then DtS A25 and ItS A10 can be mapped to social p+ and social p− , respectively (see also Definition 7): social p+ (t) social p− (t)

= DtS FCM→social CA F+ (DtS A(t)) = DtS A25 (t) = ItS FCM→social CA F− (ItS A(t)) = ItS A10 (t)

(12)

In this case the channel functions of Eq. (12) only extract the relevant value from the vectors of activation degrees provided. The channel feeding information back from the CA into the FCMs needs to perform more work. The government, in the form of the concept Government Support towards Social Systems, responds to the prevalence of needle sharing, and it does so after the fact, as alluded to in Section 3.2. The government may, for example, collect statistics on the prevalence of needle sharing among IDUs. Such statistics are usually bundled in a report, which takes some time to compile. Hence the choice to let the government, and consequently the FCMs, take information from the previous time step: (1) DtS A24 (t)

= social CA→DtS F F24 ((social C, t − 1)),

in which social CA→DtS F F24 should be a function of the prevalence of needle sharing in social CA. Remember that # − (Ni , t) counted the number of individuals with a negative influence (i.e., sharing IDUs) in Ni , the set of neighbours of cell ci . We use the same notation to count the number of sharing IDUs in the set of all cells for calculating the needle-sharing prevalence (the proportion of sharing IDUs) at time step t: needle-sharing prevalence =

#− (social C, t) |social C|

Table 3 Parameters for CA. Parameter

Value with reference

HIV transmission rate Initial stage (2 months) Clinical latency stage (84 months) AIDS stage (12 months) Needle-sharing rate Number of needles shared/month Life expectancy Infected Susceptible Stayer

[33]: p = 0.05 [33]: p = 0.001 [33]: p = 0.001 [50]: 120 [3]: 8 years [38]: 50 years [33]: 75 years

upper bounds may be imposed on a value x by scaling and shifting x: (pUb − pLb )x + pLb

(14)

In addition to having a lower and an upper bound, the government response, much like societal response, is often a critical mass one: it does not respond much to low prevalence, and then all of a sudden switches to high response past some critical prevalence level. Any further increase in prevalence beyond that does not increase the response much anymore due to saturation [54]. The hyperbolic tangent function is a viable candidate for capturing such a response: tanh(x) =

e2x − e−2x e2x + e−2x

(15)

The composition of Eqs. (13)–(15) gives the final channel function social CA→DtS FCM F24 : (1) DtS A24 (t)

= (pUb − pLb ) tanh

# (

− social C, t − 1) |social C|

+ pLb

(16)

5. Computer simulations (13)

The prevalence in this form already satisfies the requirement of Definition 6, that the outcome lie within the range [0, 1], but it is hardly realistic. The government does not respond linearly to issues in society. First of all, without needle sharing, and all else being equal, Government Support towards Social Systems is not at 0. Conversely, and for the same reason, Government Support towards Social Systems could hardly be at 1 (the government has other issues to worry about as well). So we assume a lower bound (pLb ) and Upper bound (pUb ) for government support. Generically, lower and

The evolutionary FCM-CA model of Section 4 was simulated using the MATLAB computing environment to illustrate the interplay of environmental and social factors in the needle sharingmediated spread of HIV among IDUs. To guide our exploration of the effect of the FCMs’ initial activation degrees on model outcomes, we formulated three hypothetical scenarios, of a poor country, a developing country and a developed country (see Table 4), each with different initial activation degrees and channel parameters. The parameters of the CA, summarized in Table 3, are kept at the same level for all simulations.

Table 4 Simulation results for the three sample scenarios. Model component

DtS FCM

Government support towards social systems Drug User Organizations Media Integrated families Community based social counsellors

ItS FCM

social CA→DtS FCM

Inputs

Unemployment Homelessness Stringent laws F24

Lower bound (pLb ) Upper bound (pUb )

Hypothetical case

Output

Poor country

Developing country

Developed country

0.1 0.001 0.25 0.85 0.01 13.8876

0.03 0.01 0.5 0.65 0.2 23.7753

0.6 0.75 0.8 0.2 0.5 61.5128

(∞) DtS A25 (1)

0.65 0.5 0.7 79.713

0.35 0.2 0.2 53.5539

0.1 0.08 0.25 15.7196

(∞) ItS A10 (1)

0.6 0.8 0.6

(1) DtS A24 (tend )

0.05 0.3 0.17

0.25 0.6 0.35

V.K. Mago et al. / Applied Soft Computing 12 (2012) 3771–3784

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Fig. 13. HIV infected population, needle sharing prevalence and Government support (poor countries). (a) 5000 time steps. (b) First 100 time steps.

5.1. Poor country

5.2. Developing country

In the first simulation our hypothetical environment provides a low degree of positive influence from government, drug organization, media, families, community and counsellors. In contrast the magnitude of negative influences such as unemployment, homelessness, and stringent laws is considered high. The specific initial activation degrees are listed in Table 4 for all three scenarios. Considering all positive and negative social and environmental influences that an individual can receive from the environment, the FCMs on their own would calculate inclination and disinclination towards sharing. Simulation results are shown in Fig. 13. In this scenario, the government’s capacity to intervene is limited (between pLb = 0.05 and pUb = 0.3) due to lack of resources. Initially, the prevalence of needle sharing and HIV rise sharply, an artefact of the model initialization: sharing and infected individuals are dispersed randomly within the population, giving them a large coverage (many non-sharing and/or susceptible neighbours to influence and/or infect). With time, two processes decrease the rate of new sharer recruitment and HIV infection: the government responds, affecting inclination and disinclination towards sharing, and clusters of sharing and/or infected individuals form which can only ‘grow’ at the edges because the neighbours of an individual in the center of a cluster are likely all already infected. After 2000 months the system stabilizes, and we see that the disease is endemic at a prevalence of around 37%. Further testing showed that in this particular scenario, government support would need to be raised to and sustained at 0.9 to eradicate HIV, likely an impossible task for the government of a poor country.

In the second scenario we assumed our hypothetical environment to be more similar to the developing countries where individuals will receive more positive and fewer negative influences from the environment compared to the first scenario. The results of our model show that in this environment IDUs are less likely to share their needles, as should be expected. Simulation results are shown in Fig. 14. The government’s capacity to intervene in a developing country is larger, and it also tends to use at least some of this capacity by default, so government support ranges from pLb = 0.25 to pUb = 0.6. Consequently, the prevalence of HIV stabilizes at a lower level, around 30%. This is largely due to higher government support reducing the needle-sharing rate. The situation looks similar to that of the previous scenario, but further analysis shows dissimilarities as well: the level of government support required to eradicate HIV is now at 0.63, because other societal factors contribute more. The disease is still endemic, but less extra effort would be required to change that. 5.3. Developed country In our last simulation we considered an ideal environment, closer to that of developed countries, in which the environment provides the highest degree of positive and lowest degree of negative influence compared to the other two scenarios. As we can see in Table 4, the inclination towards sharing decreasesand disinclination toward sharing increases. Simulation results are shown in Fig. 15.

Fig. 14. HIV infected population, needle sharing prevalence and Government support (developing countries). (a) 5000 time steps. (b) First 100 time steps.

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Fig. 15. HIV infected population, needle sharing prevalence and Government support (developed countries). (a) 5000 time steps. (b) First 100 time steps.

Developed countries have the most resources available compared to the previous two scenarios, and the range of government support reflects this: we set it to lie between pLb = 0.6 and pUb = 0.8. The rapid decrease of needle sharing and HIV seen in Fig. 15 should not be ascribed to this high level of government support. In simulations in which government support is kept near zero, HIV is still eradicated eventually, and also needle sharing disappears. In conjunction with the previous two scenarios, a parallel can be drawn with the reality that a government’s ability to effect change is both limited and supported by the environment it acts in. If in a poor situation, a government on its own can hardly make a decisive impact; in a near-perfect situation, the problem will almost take care of itself. Most real-world scenarios are found between those two extremes. 6. Conclusion

a low level, where many actors interact and generate emergent system behaviour. At that low level lies the strength of CA modelling. CA, in turn, cannot capture high-level heterogeneous forces playing out on a system-wide scale. The marriage of these two modelling techniques is therefore an appropriate one. We have developed an extensible formalism for joining these models that could readily be used to incorporate also other types of models. We have subsequently applied the proposed modelling technique to the spread of HIV among IDUs, taking into account high-level socio-economic factors as well as low-level social influence. The FCM-CA model provides policy makers with an overall picture of the whole system that surrounds the IDUs, particularly as it pertains to needle sharing, a main mode of HIV transmission in this population. Conflict of interest

This paper has presented the fusion of FCMs and CA into a more comprehensive modelling technique, evolutionary modelling. It was proposed to address the scale limitations of both models while capitalizing on their strengths. FCM modelling is well-suited for a high-level view of a phenomenon, at the level of global actors. On its own, it is not suitable for investigating how phenomena play out at

The authors declared no conflict of interest. Appendix A. Supporting literature for disinclination Table A1.

Table A1 Description of the concepts and the numeric values on the edges of FCMs for Disinclination for Needle Sharing. Concept 1

Street Nurse Programme: education Street Nurse Programme: demarginalization Street Nurse Programme: paraphernalia supplies Peer run safe injection sites: education Peer run safe injection sites: paraphernalia supplies Publicity: awareness Community meetings: demarginalization Counselling and influencing: bondage/integrate Health education: awareness

Concept 2

Linguistic terms or numeric values used in references

Defuzzified numeric value

Reference 1 (linguistic weight)

Reference 2 (linguistic weight)

Reference 3 (linguistic weight)

Disinclination for Needle Sharing Disinclination for Needle Sharing Disinclination for Needle Sharing Disinclination for Needle Sharing Disinclination for Needle Sharing

[25]:(HIGH)

[74]:(LOW)

[70]: (VERY HIGH)

0.525

[25]: (MEDIUM)

[56]: (HIGH)

[58]: (VERY HIGH)

0.648

[25]: (HIGH)

[9]: (HIGH)

[34]: (VERY HIGH)

0.765

[45]: (HIGH)

[17]: (HIGH)

[34]: (VERY HIGH)

0.765

[23]: (HIGH)

[58]: (VERY HIGH)

[29]: (MEDIUM)

0.648

Disinclination for Needle Sharing Disinclination for Needle Sharing Disinclination for Needle Sharing

[46]: (HIGH)

[43]: (VERY HIGH)

[8]: (LOW)

0.525

[22]: (HIGH)

[45]: (HIGH)

[6]: (MEDIUM)

0.659

[39]: (VERY LOW)

[34]: (HIGH)

[60]: (HIGH)

0.594

Disinclination for Needle Sharing

[40]: (MEDIUM)

[14]: (HIGH)

[49]: (MEDIUM)

0.591

V.K. Mago et al. / Applied Soft Computing 12 (2012) 3771–3784

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Table B1 Description of the concepts and the numeric values on the edges of FCMs for Inclination for Needle Sharing. Concept 1

Economic stress Economic stress Prostitution Social networking among IDUs: risky behaviour Depression: risky behaviour Reduced supply of paraphernalia Heightened policing: incarceration

Concept 2

Prostitution Depression: risky behaviour Inclination for Needle Sharing Inclination for Needle Sharing Inclination for Needle Sharing Inclination for Needle Sharing Inclination for Needle Sharing

Linguistic terms or numeric values used in references

Defuzzified numeric value

Reference 1 (linguistic weight)

Reference 2 (linguistic weight)

Reference 3 (linguistic weight)

[75]: (MEDIUM) [12]: (HIGH)

[53]: (HIGH) [21]: (HIGH)

[2]: (MEDIUM) [48]: (VERY HIGH)

0.591 0.765

[26]: (HIGH)

[64]: (VERY HIGH)

[66]: (HIGH)

0.765

[13]: (VERY HIGH)

[36]: (HIGH)

[35]: (HIGH)

0.765

[28]: (HIGH)

[61]: (HIGH)

[37]: (HIGH)

0.75

[30]: (HIGH)

[63]: (VERY HIGH)

[51]: (LOW)

0.525

[55]: (HIGH)

[52]: (HIGH)

[71]: (HIGH)

0.75

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