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Modelling, Analysis and Simulation of a Spatial Interaction Model
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IFAC-PapersOnLine 49-29 (2016) 221–225 Modelling, Modelling, Analysis Analysis and and Simulation Simulation of of aa Spatial Spatial Interaction Interaction Model Model Modelling, Analysis and Simulation of a Spatial Interaction Model Modelling, Analysis and Simulation of a Spatial Tamara Vobruba*. Andreas Körner**Interaction Model
Tamara Felix Vobruba*. Andreas Körner** Breitenecker*** Tamara Felix Vobruba*. Andreas Körner** Breitenecker*** Tamara Vobruba*. Andreas Körner** Felix Breitenecker*** Felix Breitenecker*** * Institute of Analysis and Scientific Computing Vienna University of Technology, Vienna, Austria * Institute of Analysis and Scientific Computing Vienna University of Technology, Vienna, Austria (e-mail: [email protected]). * Institute of Analysis and Scientific Computing Vienna University of Technology, Vienna, Austria (e-mail: [email protected]). * Institute Computing Vienna ** Instituteof ofAnalysis Analysisand andScientific Scientific Computing ViennaUniversity Universityof ofTechnology, Technology,Vienna, Vienna,Austria Austria (e-mail: [email protected]). ** Institute of Analysis and Scientific Computing Vienna University of Technology, Vienna, Austria (e-mail: [email protected]). (e-mail:[email protected]) ** Institute of Analysis and Scientific Computing Vienna University of Technology, Vienna, Austria (e-mail:[email protected]) ** Institute Computing *** Instituteof ofAnalysis Analysisand andScientific Scientific ComputingVienna ViennaUniversity Universityof ofTechnology, Technology,Vienna, Vienna,Austria Austria (e-mail:[email protected]) *** Institute of Analysis and Scientific Computing Vienna University of Technology, Vienna, Austria (e-mail:[email protected]) (e-mail: [email protected]) *** Institute of Analysis and Scientific Vienna University of Technology, Vienna, Austria (e-mail: Computing [email protected]) *** Institute of Analysis and Scientific Computing Vienna University of Technology, Vienna, Austria (e-mail: [email protected]) (e-mail: [email protected]) Abstract: This paper reviews the theoretical backgrounds and formal structure of Gravity Models for Abstract: This paperBehaviour, reviews the and formal of Gravity Models for Spatial Interaction as theoretical they buildbackgrounds the foundation for thestructure developed Migration Model. Abstract: This paperBehaviour, reviews the theoretical backgrounds and formal structure of Gravity Models for Spatial Interaction as they build the foundation for the developed Migration Model. Abstract: This reviews the theoretical backgrounds and offormal of examined. Gravity Models Furthermore the paper qualities and possibilities as well as the limits these structure models are Based for on Spatial Interaction Behaviour, as they build theas foundation for themodels developed MigrationBased Model. Furthermore the qualities possibilities as is well the limitstoofsimulate these are examined. on Spatial Interaction Behaviour, as they build the foundation for the Migration developed Migration into Model. this a Migration Model isand introduced, which implemented Movement and Furthermore the qualities and possibilities as is well as the limitstoofsimulate these models are examined. Based on this a Migration Model is introduced, which implemented Migration Movement into and Furthermore theThis qualities andused possibilities asmigration well as the limitstoofsimulate models examined. Based on within Europe. canisbe to analyse behaviour inthese general andare in terms of influencing this a Migration Model which is implemented Migration Movement into and within Europe. This canisbeintroduced, used to analyse behaviour in general and in terms of influencing this a Migration Model introduced, whichmigration is implemented to simulate Migration Movement into and parameters. within Europe. This can be used to analyse migration behaviour in general and in terms of influencing parameters. within Europe. This can be used to analyse migration behaviour in general and in terms of influencing parameters. © 2016, IFACQualitative (International FederationMathematical of Automatic Control) by Elsevier Ltd. AllMacro, rights reserved. Keywords: Simulation, Models,Hosting Probabilistic Models, Interaction parameters. Keywords: Qualitative Simulation, Mathematical Models, Probabilistic Models, Macro, Interaction Mechanism, Modelling, Simulation Keywords: Qualitative Simulation, Mathematical Models, Probabilistic Models, Macro, Interaction Mechanism, Modelling, Simulation Keywords: Qualitative Simulation, Mathematical Models, Probabilistic Models, Macro, Interaction Mechanism, Modelling, Simulation Mechanism, Modelling, Simulation models and this can be the used for decision support. 1. INTRODUCTION models and this can be the used for decision support. 1. INTRODUCTION Furthermore, this model is usedsupport. to simulate the models and this canmigration be the used for decision Furthermore, this model is used toSyria simulate the 1. INTRODUCTION andmovement this canmigration beinthe used for decision support. The Spatial Interaction Model is a macroscopic approach for models migration the summer 2015 from into and 1. INTRODUCTION The Spatial Interaction Model is a macroscopic approach for Furthermore, this migration model is used to simulate the migration movement in the summer 2015 from Syria into and describing any kind of spatial interaction behaviour between Furthermore, this migration modelrefugee is used toSyria simulate the Europe during called crisis, to into test this The Spatialany Interaction Model interaction is a macroscopic approach for within describing kind of spatial behaviour movement inthe theso summer 2015 from and within Europe during the so called refugee crisis, to test this The Spatial Interaction Model is a macroscopic approach for migration populations or regions. This Modelling approach hasbetween a wide migration movement in the summer 2015 from Syria into and kind of models in terms of usability in a specific scenario. describing any kind of spatial interactionapproach behaviour between populations or regions. ThistheModelling a wide called refugee crisis, to test this kind ofEurope models during in termsthe ofso usability in a specific scenario. describing any kind of spatial interaction behaviour between range of applications, simulationapproach of traffichas flows, the within within Europe during the so called refugee crisis, to test this populations or regions.from ThistheModelling has a wide range of applications, from simulation of traffic flows, the kind of models in terms of usability in a specific scenario. populations regions. This Modelling hasgoods a wide movement oforcommuters orthe migrants, theapproach trade with or kind of models in terms of usability in a specific scenario. 2. SPATIAL INTERACTION MODEL range of applications, from simulation of traffic flows, the movement of commuters orthe migrants, with goods 2. SPATIAL INTERACTION MODEL range of applications, from simulation ofSmith traffic flows, the transmission of messages. (see Senthe andtrade 2012, p. the 1)or movement of commuters or migrants, the trade with goods or the transmission of messages. (see Sen and Smith 2012, p. 2. SPATIAL INTERACTION MODELover space, movement of commuters or migrants, the trade with goods1)or Spatial interaction is a movement or transmission 2. SPATIAL INTERACTION MODELover space, the transmission of messages. (seetype Sen of andSpatial SmithInter2012,action p. 1) Spatial interaction is a movement or transmission The best known and widely used the transmission of messages. (seetype Sen of andSpatial SmithInter2012,action p. 1) which is resulting of a decision process involving different The best known and widely used Spatial interaction is a movement or transmission over space, which is resulting of a decision process involving different Models is the Gravity Model. The basic idea is the Spatial interaction is a movement transmission over space, influences. Interaction as physicalormovement is for example The best known and widelyModel. used type of Spatial Inter- isaction Models is the Gravity The basic idea the which is resulting of a decision process involving different influences. Interaction as physical movement is for example The best known widely used type of Spatial Interaction which description of theand Interaction between populations or regions is resulting of a decision process involving different migration movement, where nonphysical movement could be Models is ofthe Gravity Model. Thepopulations basic idea is the influences. description thereferring Interaction or regions Interactionwhere as physical movement is for could example migration movement, nonphysical movement be Models is the Gravity Model. The law basicofidea is the with a relation to between Newton’s gravity. In influences. Interaction as physical movement is for example the transmission of messages or the exchange of knowledge. description of thereferring Interaction between populations or regions with a relation to Gravity Newton’s law are of gravity. movement, where nonphysical movement could be the transmission of messages or the exchange of knowledge. description of the Interaction between populations or regions geography and demography Models used for Ina migration movement, where nonphysical movement could be with a relation referring to Gravity Newton’s law are of gravity. Ina migration geography demography Models used for of messages or theis exchange of knowledge. with a relation referring to ofNewton’s law of gravity. In the Thetransmission Spatial Interaction Model describing such spatial long time toand analyse the flow people, goods or capital. But the transmission of messages or theis exchange of knowledge. geography and demography Gravity Models are used for a The Spatial Interaction Model describing such spatial long time to analyse the flow of people, goods or capital. But geography and demography Gravity Modelsand are used for a The interaction with a relation, which is depending on different often there is a lack of mathematical theoretical Spatialwith Interaction Model isisdescribing such spatial long time to analyse the flow of people, goodsand or capital. But interaction a relation, which depending on different often there is a lack of mathematical theoretical Spatial Modelform, is describing spatial long time to to analyse the flowthe of people, goods or capital. But The attributes. In Interaction the most general this modelsuch is given by foundation understand backgrounds. (see Dennett often there is a lack of mathematical and theoretical interaction with a relation, which is depending on different In the general form, this model on is given by foundation backgrounds. (seetheoretical Dennett attributes. interaction with amost relation, which is depending different often there is understand athislack ofthe mathematical the following equation. 2012, p. 2) toSo work aims to review and the In the most general form, this model is given by foundation toSounderstand the backgrounds. (seetheoretical Dennett attributes. the following equation. 2012, p. 2) this work aims to review the foundation the backgrounds. (see a Dennett backgroundstoofunderstand Gravity Models and to develop formal attributes. In the most general form, this model is given by following equation. (1) 2012, p. 2) So this work aims and to review the theoretical backgrounds of Gravity Models to develop a formal the I i , j = f ( Ai , R j , C i , j ) the following equation. 2012, p. 2) So this work aims to review the theoretical model description. (1) I i , j = f ( Ai , R j , C i , j ) backgrounds of Gravity Models and to develop a formal model description. (1) backgrounds of Gravity Models and to develop a formal I ii ,, jj = f ( Aii , R jj , C ii ,, jj ) (1) model description. I i , j = f ( Ai , R j , C i , j ) Based on this formalism a Migration Model is developed, model description. Based onaims this formalism a Migration Modelmovements is developed, I i , j is a real number and is representing the interaction which to describe international of Here Based onaims this formalism a Migration Modelmovements is developed, which tostructure describe international of Here I i , j is a real number and is representing the interaction Based onaims this a Migration Modelmovements developed, migration. The formalism of such a model, asiswell as the Here I i , jthe is apopulations real number is representing interaction orand regions i and j. It isthe described by which to describe international of between Here I ii ,, jjthe is apopulations real number is representing interaction migration. The ofare such a model, as well as the orand regions i attributes and j. It isthe described by which aims tostructure describe international movements of between of this the opportunities and the limits examined. As part in i, in j and a function of attributes i migration. The structure ofare such a model, as well as the the between j in j and the of populations orA regions i attributes and j. It is R described by the limits examined. As part of this opportunities and in i, a function attributes A R migration. The structure of such aand model, as well as the between the populations or regions i model is methodologically analysed characterised. i and j. It is described by j opportunities and the limits are examined. As part of this the in i, attributes in j and a function of attributes between i and j. separation attributes model is methodologically and characterised. C i , j Aii in i, attributes R jj in j and opportunities and the limitsanalysed are examined. As part of this the separation a functionattributes of attributes between i and j. A R C i, j i j model is methodologically analysedcan andhave characterised. International migration movements a wide range of separation attributes C i , j between i and j. model is methodologically analysedcan andhave characterised. International migration movements a wide range of i , j between i and j. separation attributes C The interaction is depending on a decision process which is impacts and may influence political decisions and from i, j International migration movements can have aand wide range of The interaction depending on aconditions. decision process whichthe is impacts may influence political decisions from happening on theisbasis of certain For example International migration movements can haverelations. a wide range national and integration politics to international In of The interaction isbasis depending on aconditions. decision process whichthe is impacts and may influence political decisions and from happening on the of certain For example national integration politics to international relations. In The interaction isbasis depending aconditions. decision whichthe is impacts may influence political decisions and from decision ofona the commuter aoncertain trafficprocess route could be terms of and analysing and understanding this movements happening of for certain For example integration politics to international relations. In national decision of a commuter for a certain traffic route could be terms of analysing and understanding this movements happening on the basis of certain conditions. For example the national integration politics to international relations. In the influenced by costs this route, the availability of public modelling and simulation is of great importance. Beside decision ofbya the commuter aroute, certain traffic route of could be terms of analysing and understanding this movements influenced costs offor this the availability public modelling and simulation is ofitself greatalso importance. Beside the decision a the commuter for certain traffic route could be terms of analysing and understanding this transport of and distance to awork. This influencing factors simulation of the movements themovements investigation of influenced by the costs of this route, the availability of public modelling and simulation is of great importance. Beside the transport and the distance to work. This influencing factors simulation of the movements the investigation of influenced by the of this route, the availability of public modelling and simulation is ofitself greatalso importance. Beside the are represented bycosts the attributes , and . Formally all influencing factors can be performed. Furthermore it is A R C j and simulation of the movements itself alsoFurthermore the investigation transport and the to work. This influencing factors are represented by distance the attributes all influencing factors can be performed. it is of Aii , R C ii ,, jj . Formally j transport and the distance to work. This influencing factors simulation the prognoses movements itself are alsobased the investigation of possible to of make which on validated the attribute are real numbers. Furthermore it is A R C are represented by the attributes , and . Formally all influencing factors can be performed. possible to make prognoses which are based on validated i j i , j i j i , j are attribute represented attributes Ai , R j and C i , j . Formally all the are by realthe numbers. influencing factors can be performed. Furthermore it is possible to make prognoses which are based on validated the attribute are real numbers. possible to make prognoses which are based on validated Copyright © 2016 IFAC 221 the attribute are real numbers. Copyright 2016 IFAC 221 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © 2016, IFAC (International Federation of Automatic Control) Peer review©under of International Federation of Automatic Copyright 2016 responsibility IFAC 221Control. Copyright © 2016 IFAC 221 10.1016/j.ifacol.2016.11.054
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In this general form more complexity can be reached by specifying the attributes and involving time dependencies.
After the introduced forms of Gravity Models a lot of different formalisations were used. Ashish Sen and Tony Smith were introducing a general class of gravity models.(see Sen and Smith 1995, p. 4)
3. GRAVITY MODEL The widely used type of Spatial Interaction Models are the Gravity Models. Here the relation which is describing the interaction is based on Newton’s law of gravity. This idea to draw analogies between physics and certain human behaviour has a long history. It came up in the year 1852 by Henry Charles Carey, who described the human migration behaviour as the “tendency to gravitate the fellow man”.(Carey 1852, p. 42) This lead to examination of the relation between migration movement and the attraction of a regions, as well as the distance between regions. Ernest Charles Young was the first one that postulated the formal connection with the law of Gravity.(see Young 1924, p. 88) Investigating the movement of farm population he pointed out this coherence in the following formula.
M =k⋅
F D2
(2)
In the following years this approach was used to de- scribe human shopping behaviour and was refined by John Quincy Steward in the year 1941, who was developing the theory of demographic gravitation. (see Sen and Smith 1995,p. 3)
Pi ⋅ Pj d i2, j
(3)
Here the influence of the fundamental physical law is obvious. The Interaction I i , j between the population centres i and j is direct proportional to the product of the so called population masses Pi and Pj , which are describing the attributes in i and j. Furthermore the interaction in indirect proportional to the squared distance d i , j between the population centres i and j. The constant G is called the demographic gravity constant. A few years later in 1950 Steward developed a formula, which involves the possibility of different impacts of the attributes to the interaction. (see Sen and Smith 1995, p. 3)
I i, j = G ⋅
wi Pi ⋅ w j Pj d i2, j
The general class of Gravity Models has the following form
I i , j = A(i ) ⋅ B ( j ) ⋅ F (d i , j ) ,
(5)
where the interaction I i , j is a real number and is resulting of the product of a weighted function A(i ) of attributes in i, a weighted function of the attributes B ( j ) in j and function of separation attributes F ( d i , j ) between i and j. Here the weighted functions itself are functions from a finite more dimensional vector space over the real numbers and are mapping into the set of real numbers. So it is possible to include a set of different attributes and there heterogeneity. 4. MIGRATION MODEL
Here M is the absolute migration, F the intensity of attraction of a region, D the distance to this region and k a proportional constant. The dependency of attraction and the distance is based on Newton’s law of gravity.
I i, j = G ⋅
analogy to Newton’s law, but it increases the flexibility of the mode la lot.
(4)
Here wi and w j are the population weights. This weights are reflecting the heterogeneity of population masses and are treated as statistical parameters. With this parametric extension the model description steps back from the direct
4.1 Model Equations Based on this ideas the aim is to introduce a migration model, which is fitting to the introduced class of gravity models. So the focus is on the migration behaviour between the regions of origin, the possible regions of destination and the transit regions. The heterogeneity of the attraction or repulsion of this regions influencing the migration behaviour should be included. Furthermore this attribute function should be treated as time dependent to observe structural changes in the migration behaviour over time. The Model description should also have the flexibility to describe migration movement between a list of different included regions in different structure and trough different routes. First of all to find a formal way of describing the regions, these are represented by vertices of a graph, where the neighbouring countries are connected through edges. The graph of migration movement is defined as a directed graph G = (V , E ) , where the finite set of vertices V (G) = {v1 , v 2 ,..., vl } is describing the different regions of
interest and the finite set of edges E (G) = {e1 , e2 ,..., ek }
with ek = vi , , v j
is describing the geographical possibility
to migrate from on region v i to the other region v j . In the adjacency matrix A = (ai ,k ) li , j =1 of the graph the a i , j = 1 if and only if there is the geographical possibility to migrate from i to j. With this formalism it is possible to describe the time dependent interaction I i , j (t ) from the region v i to the region v j with the following equations.
2017 IFAC TECIS October 26-28, 2016. Durrës, Albania
n
I i , j (t ) =
Tamara Vobruba et al. / IFAC-PapersOnLine 49-29 (2016) 221–225
m
∑ a k ⋅ (A j (t ) )k ⋅ ∑ rk ⋅ (Ri (t ) )k k =1
k =1
l
∑ c ⋅ (C k =1
k
i, j
(6)
(t ) )k
M i , j (t ) = I i , j (t ) ⋅ M i (t )
(7)
Here the n functions of attractive attributes
(A (t )) : [0, T ] → [0,1] j
n k =1
(8)
in v j and the m functions of repulsive attributes
(Ri (t ) )k =m1 : [0, T ] → [0,1]
(9)
in v i are time dependent weighted function with the parameters n
a k ∈ [0,1], ∑ a k = 1
(10)
k =1
and the parameters m
rk ∈ [0,1], ∑ rk = 1.
j, j
(t ))
: [0, T ] → [1, ∞)
Empirical studies showed that Gravity Models are most successful in describing macro patterns of Spatial Interaction. So it is more reliable in picturing the behaviour of populations rather than individuals. (see Sen and Smith 1995, p. 16) One reason for that is that individual decision are often influenced by a lot of different factors which refer to this specific individual. Therefore the focus on population groups has also the quality of a sufficiently description of the behaviour of interest with just a little required information. It is also important to discuss the treatment of time, where in the introduced migration model the Interaction is only observed at discrete time steps. So it describes static patterns over a period of time. Structural changes, for example of the attribute function from one time step to the next one, are treated as very fast, so as a change of state. This means that the model always treats static patterns. As the model aims to analyse social behaviour, which has a bunch of influencing factors and can only take a finite set of them into consideration, it is by nature a probabilistic approach.
(11) 5.1 Graph of Migration Movement
The l functions of separation attributes
(C
important in terms of understanding the qualities and limits of this approach.
5. SIMULATION AND RESULTS
k =1
l k =1
223
(12)
between the regions v i and v j are also time dependent
With the introduced migration model the movement of migrants from Syria into and within Europe in the summer 2015 is simulated. This situation implies the migration graph shown in figure 1.
weighted functions with the weights l
c k ∈ [0,1], ∑ c k = 1 .
(13)
k =1
Therefore the absolute migration from v i to v j at the time t is represented of the product of the interaction I i , j (t ) ∈ [0,1] and the number of migrants M i (t ) at v i at the tme t. So the migration from one the region to a region is determined by the directed interaction between the regions. This interaction again is a relation of weighted repulsive functions, weighted attractive functions and weighted separation functions. So the interaction is increasing with an increase of the repulsion in the one region and increase of the attraction in the other region and it is decreasing with an increase of the separation. 4.2 Model Characterisation Since social behaviour is modelled it is of great importance to mention that this behaviour is resulting of a complex composition of influences. This is a great challenge in finding a description of the social behaviour. Different approaches are often capable to cover different aspects of this complexity. So the characterization of this migration model is
Fig. 1. The migration graph for the migration movement from Syria into and within Europe in the summer 2015 Since it is not necessary to investigate some of the considered countries in Europe as single countries, some of them are
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considered as regions. A description of the graph in figure 1 is shown in table 1. Table 1. Description of the migration graph vertex v19
region Albania, Bosnia, Montenegro
v18 v17
Italy, Spain, Portugal
v16 v15
v14 v13 v12 v11 v10
UK, Netherlands, Belgium, France
Table 2. Attractive functions Gross domestic product (GPD) Fragile state index (FSI) Maximum attraction of accessible countries Not exceeded capacity Asylum recognition rate Asylum recognition quote in Europe
Norway, Finland, Sweden, Denmark Slovakia, Czech Republic, Romania, Bulgaria, Poland, Lithuania, Estonia, Latvia Germany Austria Slovenia Croatia Hungary
v9
Serbia
v8
Macedonia
v7
Greece
v6
Turkey
v5
Lebanon
v4 v3
Egypt Jordan
v2 v1
Iraq Syria
5.2 Attribute Functions The model focuses on the migration movement and the different migration routes and it is not modelling any reasons for emigration. So this implies a distinction of countries of origin and potential destination countries. So observing the interactions from Syria to the neighbouring countries, the repulsive attributes of the country of origin are not taken into consideration. Here the attractive attributes of the neighbouring countries are determining the value of interaction.
Table 3. Repulsive functions Gross domestic product (GPD) Fragile state index (FSI) Exceeded capacity Asylum recognition rate Asylum recognition quote in Europe In this simulation the separation function is representing the effect of open and closed borders on the interaction. The choice of the attribute functions was taken under the consideration of the outcome of the study of the Australian economic scientist Timothy Hatton. He examined the correlation of specific attributes of developed couturiers and the number of asylum applications. Essentially he observed a big effect of access policy, like the border security, and also of processing policy, like the humanitarian situation. But there was shown just a small positive impact of welfare policy on the number of asylum applications. (see Hatton 2015, p. 21) The access policy of a country is considered by the separation function. If there are closed borders enforced by border security arrangements, the separation function, shown in 12, will reduce the interaction immediately. By integrating the different influences of the attractive and repulsive attributes the weighting parameters where estimated. Figure 2 and figure 3 show that this weighting takes the big effects of processing policy (capacity, FSI, asylum recognition rate and asylum recognition quote) into consideration.
There is also a distinction of potential destination countries and transit countries taken into consideration, because the attractive at- tributes of a transit couturiers have less influence on the interaction. So when the repulsion of a country exceeds a certain amount, it is treated as a transit country, which means that the attractive functions of it are replaced with the maximum of the attractive function of the countries which are accessible from there. In the table 2 and table 3 there is a list of considered attractive and repulsive functions. Fig. 2. The weighting of the repulsive functions
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225
6. CONCLUSIONS AND OUTLOOK A migration model is presented which is fitting into the class of Gravity Models and which is taking the time dependency and the heterogeneity of the attribute functions into consideration. It is applicable to different graph of movements and can also show the impact of border closings. As discussed it is a macroscopic approach which models the migration movement on the basis of whole populations, so the simulation results have to be interpreted in a qualitative way.
Fig. 3. The weighting of the attractive functions 5.3 Simulation Results The described situation was implemented with Data from UNHCR for the time period of 1st of September 2015 till 31. October 2015. The border closing of Hungary to Serbia on the 15th of September 2015 as well as the border closing from Hungary to Croatia on the 16th of October 2015 are implemented as well. The results in figure 4 are showing the migration movements as they were happening in this time period in a qualitative way, compared to UNHCR data. Here one can see the different routes, which are shown by the arrows from one country to another as well as the amount of migrants in the countries, which is shown by the blocks in the countries. The considered countries itself are pictured in different grey shades. The results also show the alternative routes after the border closings.
It was shown that the model is capable of describing the migration movement into and within Europe during September and October 2015 in a qualitative way. But as an outlook it can also be examined if the model can show the qualitative behaviour of migration movements in completely different regions. Furthermore the impact of the different attributes can be studied, which is of great importance for understanding migration behaviour. As a last point it is also possible to model prognostic scenarios. REFERENCES Carey H.C. (1858). Principles of Social Sciences. Philadelphia, Pennsylvania. Dennett A. (2012). Estimating flows between geographical locations: get me started in spatial interaction modelling. Working Paper Series, Paper 181-March 12, UCL, Centre for Advanced Spatial Analysis. Hatton T. and Maloney J. (2015). Applications for Asylum in the Developed World: Modelling Asylum Claims by Origin and Destination. ANU Working Papers in Economics and Econometrics, 625, Australian National University. Sen A. and Smith T. (1995). Gravity Models of Spatial Interaction Behaviour. Springer, Place of publication. Young, E.C. (1924). The movement of farm population. Cornell Agricultural Experiment Station, New York. Appendix A. ACKNOLEDGMENTS We want to thank Irene Hafner and Stefan Emrich (dwh simulation services) as well as Filip Krasinianski (ORF - ¨ Osterreichischer Rundfunk) for the decision support and the performed visualisation.
Fig. 4. Simulation results for the summer 2015
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IFAC-PapersOnLine 49-29 (2016) 221–225 Modelling, Modelling, Analysis Analysis and and Simulation Simulation of of aa Spatial Spatial Interaction Interaction Model Model Modelling, Analysis and Simulation of a Spatial Interaction Model Modelling, Analysis and Simulation of a Spatial Tamara Vobruba*. Andreas Körner**Interaction Model
Tamara Felix Vobruba*. Andreas Körner** Breitenecker*** Tamara Felix Vobruba*. Andreas Körner** Breitenecker*** Tamara Vobruba*. Andreas Körner** Felix Breitenecker*** Felix Breitenecker*** * Institute of Analysis and Scientific Computing Vienna University of Technology, Vienna, Austria * Institute of Analysis and Scientific Computing Vienna University of Technology, Vienna, Austria (e-mail: [email protected]). * Institute of Analysis and Scientific Computing Vienna University of Technology, Vienna, Austria (e-mail: [email protected]). * Institute Computing Vienna ** Instituteof ofAnalysis Analysisand andScientific Scientific Computing ViennaUniversity Universityof ofTechnology, Technology,Vienna, Vienna,Austria Austria (e-mail: [email protected]). ** Institute of Analysis and Scientific Computing Vienna University of Technology, Vienna, Austria (e-mail: [email protected]). (e-mail:[email protected]) ** Institute of Analysis and Scientific Computing Vienna University of Technology, Vienna, Austria (e-mail:[email protected]) ** Institute Computing *** Instituteof ofAnalysis Analysisand andScientific Scientific ComputingVienna ViennaUniversity Universityof ofTechnology, Technology,Vienna, Vienna,Austria Austria (e-mail:[email protected]) *** Institute of Analysis and Scientific Computing Vienna University of Technology, Vienna, Austria (e-mail:[email protected]) (e-mail: [email protected]) *** Institute of Analysis and Scientific Vienna University of Technology, Vienna, Austria (e-mail: Computing [email protected]) *** Institute of Analysis and Scientific Computing Vienna University of Technology, Vienna, Austria (e-mail: [email protected]) (e-mail: [email protected]) Abstract: This paper reviews the theoretical backgrounds and formal structure of Gravity Models for Abstract: This paperBehaviour, reviews the and formal of Gravity Models for Spatial Interaction as theoretical they buildbackgrounds the foundation for thestructure developed Migration Model. Abstract: This paperBehaviour, reviews the theoretical backgrounds and formal structure of Gravity Models for Spatial Interaction as they build the foundation for the developed Migration Model. Abstract: This reviews the theoretical backgrounds and offormal of examined. Gravity Models Furthermore the paper qualities and possibilities as well as the limits these structure models are Based for on Spatial Interaction Behaviour, as they build theas foundation for themodels developed MigrationBased Model. Furthermore the qualities possibilities as is well the limitstoofsimulate these are examined. on Spatial Interaction Behaviour, as they build the foundation for the Migration developed Migration into Model. this a Migration Model isand introduced, which implemented Movement and Furthermore the qualities and possibilities as is well as the limitstoofsimulate these models are examined. Based on this a Migration Model is introduced, which implemented Migration Movement into and Furthermore theThis qualities andused possibilities asmigration well as the limitstoofsimulate models examined. Based on within Europe. canisbe to analyse behaviour inthese general andare in terms of influencing this a Migration Model which is implemented Migration Movement into and within Europe. This canisbeintroduced, used to analyse behaviour in general and in terms of influencing this a Migration Model introduced, whichmigration is implemented to simulate Migration Movement into and parameters. within Europe. This can be used to analyse migration behaviour in general and in terms of influencing parameters. within Europe. This can be used to analyse migration behaviour in general and in terms of influencing parameters. © 2016, IFACQualitative (International FederationMathematical of Automatic Control) by Elsevier Ltd. AllMacro, rights reserved. Keywords: Simulation, Models,Hosting Probabilistic Models, Interaction parameters. Keywords: Qualitative Simulation, Mathematical Models, Probabilistic Models, Macro, Interaction Mechanism, Modelling, Simulation Keywords: Qualitative Simulation, Mathematical Models, Probabilistic Models, Macro, Interaction Mechanism, Modelling, Simulation Keywords: Qualitative Simulation, Mathematical Models, Probabilistic Models, Macro, Interaction Mechanism, Modelling, Simulation Mechanism, Modelling, Simulation models and this can be the used for decision support. 1. INTRODUCTION models and this can be the used for decision support. 1. INTRODUCTION Furthermore, this model is usedsupport. to simulate the models and this canmigration be the used for decision Furthermore, this model is used toSyria simulate the 1. INTRODUCTION andmovement this canmigration beinthe used for decision support. The Spatial Interaction Model is a macroscopic approach for models migration the summer 2015 from into and 1. INTRODUCTION The Spatial Interaction Model is a macroscopic approach for Furthermore, this migration model is used to simulate the migration movement in the summer 2015 from Syria into and describing any kind of spatial interaction behaviour between Furthermore, this migration modelrefugee is used toSyria simulate the Europe during called crisis, to into test this The Spatialany Interaction Model interaction is a macroscopic approach for within describing kind of spatial behaviour movement inthe theso summer 2015 from and within Europe during the so called refugee crisis, to test this The Spatial Interaction Model is a macroscopic approach for migration populations or regions. This Modelling approach hasbetween a wide migration movement in the summer 2015 from Syria into and kind of models in terms of usability in a specific scenario. describing any kind of spatial interactionapproach behaviour between populations or regions. ThistheModelling a wide called refugee crisis, to test this kind ofEurope models during in termsthe ofso usability in a specific scenario. describing any kind of spatial interaction behaviour between range of applications, simulationapproach of traffichas flows, the within within Europe during the so called refugee crisis, to test this populations or regions.from ThistheModelling has a wide range of applications, from simulation of traffic flows, the kind of models in terms of usability in a specific scenario. populations regions. This Modelling hasgoods a wide movement oforcommuters orthe migrants, theapproach trade with or kind of models in terms of usability in a specific scenario. 2. SPATIAL INTERACTION MODEL range of applications, from simulation of traffic flows, the movement of commuters orthe migrants, with goods 2. SPATIAL INTERACTION MODEL range of applications, from simulation ofSmith traffic flows, the transmission of messages. (see Senthe andtrade 2012, p. the 1)or movement of commuters or migrants, the trade with goods or the transmission of messages. (see Sen and Smith 2012, p. 2. SPATIAL INTERACTION MODELover space, movement of commuters or migrants, the trade with goods1)or Spatial interaction is a movement or transmission 2. SPATIAL INTERACTION MODELover space, the transmission of messages. (seetype Sen of andSpatial SmithInter2012,action p. 1) Spatial interaction is a movement or transmission The best known and widely used the transmission of messages. (seetype Sen of andSpatial SmithInter2012,action p. 1) which is resulting of a decision process involving different The best known and widely used Spatial interaction is a movement or transmission over space, which is resulting of a decision process involving different Models is the Gravity Model. The basic idea is the Spatial interaction is a movement transmission over space, influences. Interaction as physicalormovement is for example The best known and widelyModel. used type of Spatial Inter- isaction Models is the Gravity The basic idea the which is resulting of a decision process involving different influences. Interaction as physical movement is for example The best known widely used type of Spatial Interaction which description of theand Interaction between populations or regions is resulting of a decision process involving different migration movement, where nonphysical movement could be Models is ofthe Gravity Model. Thepopulations basic idea is the influences. description thereferring Interaction or regions Interactionwhere as physical movement is for could example migration movement, nonphysical movement be Models is the Gravity Model. The law basicofidea is the with a relation to between Newton’s gravity. In influences. Interaction as physical movement is for example the transmission of messages or the exchange of knowledge. description of thereferring Interaction between populations or regions with a relation to Gravity Newton’s law are of gravity. movement, where nonphysical movement could be the transmission of messages or the exchange of knowledge. description of the Interaction between populations or regions geography and demography Models used for Ina migration movement, where nonphysical movement could be with a relation referring to Gravity Newton’s law are of gravity. Ina migration geography demography Models used for of messages or theis exchange of knowledge. with a relation referring to ofNewton’s law of gravity. In the Thetransmission Spatial Interaction Model describing such spatial long time toand analyse the flow people, goods or capital. But the transmission of messages or theis exchange of knowledge. geography and demography Gravity Models are used for a The Spatial Interaction Model describing such spatial long time to analyse the flow of people, goods or capital. But geography and demography Gravity Modelsand are used for a The interaction with a relation, which is depending on different often there is a lack of mathematical theoretical Spatialwith Interaction Model isisdescribing such spatial long time to analyse the flow of people, goodsand or capital. But interaction a relation, which depending on different often there is a lack of mathematical theoretical Spatial Modelform, is describing spatial long time to to analyse the flowthe of people, goods or capital. But The attributes. In Interaction the most general this modelsuch is given by foundation understand backgrounds. (see Dennett often there is a lack of mathematical and theoretical interaction with a relation, which is depending on different In the general form, this model on is given by foundation backgrounds. (seetheoretical Dennett attributes. interaction with amost relation, which is depending different often there is understand athislack ofthe mathematical the following equation. 2012, p. 2) toSo work aims to review and the In the most general form, this model is given by foundation toSounderstand the backgrounds. (seetheoretical Dennett attributes. the following equation. 2012, p. 2) this work aims to review the foundation the backgrounds. (see a Dennett backgroundstoofunderstand Gravity Models and to develop formal attributes. In the most general form, this model is given by following equation. (1) 2012, p. 2) So this work aims and to review the theoretical backgrounds of Gravity Models to develop a formal the I i , j = f ( Ai , R j , C i , j ) the following equation. 2012, p. 2) So this work aims to review the theoretical model description. (1) I i , j = f ( Ai , R j , C i , j ) backgrounds of Gravity Models and to develop a formal model description. (1) backgrounds of Gravity Models and to develop a formal I ii ,, jj = f ( Aii , R jj , C ii ,, jj ) (1) model description. I i , j = f ( Ai , R j , C i , j ) Based on this formalism a Migration Model is developed, model description. Based onaims this formalism a Migration Modelmovements is developed, I i , j is a real number and is representing the interaction which to describe international of Here Based onaims this formalism a Migration Modelmovements is developed, which tostructure describe international of Here I i , j is a real number and is representing the interaction Based onaims this a Migration Modelmovements developed, migration. The formalism of such a model, asiswell as the Here I i , jthe is apopulations real number is representing interaction orand regions i and j. It isthe described by which to describe international of between Here I ii ,, jjthe is apopulations real number is representing interaction migration. The ofare such a model, as well as the orand regions i attributes and j. It isthe described by which aims tostructure describe international movements of between of this the opportunities and the limits examined. As part in i, in j and a function of attributes i migration. The structure ofare such a model, as well as the the between j in j and the of populations orA regions i attributes and j. It is R described by the limits examined. As part of this opportunities and in i, a function attributes A R migration. The structure of such aand model, as well as the between the populations or regions i model is methodologically analysed characterised. i and j. It is described by j opportunities and the limits are examined. As part of this the in i, attributes in j and a function of attributes between i and j. separation attributes model is methodologically and characterised. C i , j Aii in i, attributes R jj in j and opportunities and the limitsanalysed are examined. As part of this the separation a functionattributes of attributes between i and j. A R C i, j i j model is methodologically analysedcan andhave characterised. International migration movements a wide range of separation attributes C i , j between i and j. model is methodologically analysedcan andhave characterised. International migration movements a wide range of i , j between i and j. separation attributes C The interaction is depending on a decision process which is impacts and may influence political decisions and from i, j International migration movements can have aand wide range of The interaction depending on aconditions. decision process whichthe is impacts may influence political decisions from happening on theisbasis of certain For example International migration movements can haverelations. a wide range national and integration politics to international In of The interaction isbasis depending on aconditions. decision process whichthe is impacts and may influence political decisions and from happening on the of certain For example national integration politics to international relations. In The interaction isbasis depending aconditions. decision whichthe is impacts may influence political decisions and from decision ofona the commuter aoncertain trafficprocess route could be terms of and analysing and understanding this movements happening of for certain For example integration politics to international relations. In national decision of a commuter for a certain traffic route could be terms of analysing and understanding this movements happening on the basis of certain conditions. For example the national integration politics to international relations. In the influenced by costs this route, the availability of public modelling and simulation is of great importance. Beside decision ofbya the commuter aroute, certain traffic route of could be terms of analysing and understanding this movements influenced costs offor this the availability public modelling and simulation is ofitself greatalso importance. Beside the decision a the commuter for certain traffic route could be terms of analysing and understanding this transport of and distance to awork. This influencing factors simulation of the movements themovements investigation of influenced by the costs of this route, the availability of public modelling and simulation is of great importance. Beside the transport and the distance to work. This influencing factors simulation of the movements the investigation of influenced by the of this route, the availability of public modelling and simulation is ofitself greatalso importance. Beside the are represented bycosts the attributes , and . Formally all influencing factors can be performed. Furthermore it is A R C j and simulation of the movements itself alsoFurthermore the investigation transport and the to work. This influencing factors are represented by distance the attributes all influencing factors can be performed. it is of Aii , R C ii ,, jj . Formally j transport and the distance to work. This influencing factors simulation the prognoses movements itself are alsobased the investigation of possible to of make which on validated the attribute are real numbers. Furthermore it is A R C are represented by the attributes , and . Formally all influencing factors can be performed. possible to make prognoses which are based on validated i j i , j i j i , j are attribute represented attributes Ai , R j and C i , j . Formally all the are by realthe numbers. influencing factors can be performed. Furthermore it is possible to make prognoses which are based on validated the attribute are real numbers. possible to make prognoses which are based on validated Copyright © 2016 IFAC 221 the attribute are real numbers. Copyright 2016 IFAC 221 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © 2016, IFAC (International Federation of Automatic Control) Peer review©under of International Federation of Automatic Copyright 2016 responsibility IFAC 221Control. Copyright © 2016 IFAC 221 10.1016/j.ifacol.2016.11.054
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In this general form more complexity can be reached by specifying the attributes and involving time dependencies.
After the introduced forms of Gravity Models a lot of different formalisations were used. Ashish Sen and Tony Smith were introducing a general class of gravity models.(see Sen and Smith 1995, p. 4)
3. GRAVITY MODEL The widely used type of Spatial Interaction Models are the Gravity Models. Here the relation which is describing the interaction is based on Newton’s law of gravity. This idea to draw analogies between physics and certain human behaviour has a long history. It came up in the year 1852 by Henry Charles Carey, who described the human migration behaviour as the “tendency to gravitate the fellow man”.(Carey 1852, p. 42) This lead to examination of the relation between migration movement and the attraction of a regions, as well as the distance between regions. Ernest Charles Young was the first one that postulated the formal connection with the law of Gravity.(see Young 1924, p. 88) Investigating the movement of farm population he pointed out this coherence in the following formula.
M =k⋅
F D2
(2)
In the following years this approach was used to de- scribe human shopping behaviour and was refined by John Quincy Steward in the year 1941, who was developing the theory of demographic gravitation. (see Sen and Smith 1995,p. 3)
Pi ⋅ Pj d i2, j
(3)
Here the influence of the fundamental physical law is obvious. The Interaction I i , j between the population centres i and j is direct proportional to the product of the so called population masses Pi and Pj , which are describing the attributes in i and j. Furthermore the interaction in indirect proportional to the squared distance d i , j between the population centres i and j. The constant G is called the demographic gravity constant. A few years later in 1950 Steward developed a formula, which involves the possibility of different impacts of the attributes to the interaction. (see Sen and Smith 1995, p. 3)
I i, j = G ⋅
wi Pi ⋅ w j Pj d i2, j
The general class of Gravity Models has the following form
I i , j = A(i ) ⋅ B ( j ) ⋅ F (d i , j ) ,
(5)
where the interaction I i , j is a real number and is resulting of the product of a weighted function A(i ) of attributes in i, a weighted function of the attributes B ( j ) in j and function of separation attributes F ( d i , j ) between i and j. Here the weighted functions itself are functions from a finite more dimensional vector space over the real numbers and are mapping into the set of real numbers. So it is possible to include a set of different attributes and there heterogeneity. 4. MIGRATION MODEL
Here M is the absolute migration, F the intensity of attraction of a region, D the distance to this region and k a proportional constant. The dependency of attraction and the distance is based on Newton’s law of gravity.
I i, j = G ⋅
analogy to Newton’s law, but it increases the flexibility of the mode la lot.
(4)
Here wi and w j are the population weights. This weights are reflecting the heterogeneity of population masses and are treated as statistical parameters. With this parametric extension the model description steps back from the direct
4.1 Model Equations Based on this ideas the aim is to introduce a migration model, which is fitting to the introduced class of gravity models. So the focus is on the migration behaviour between the regions of origin, the possible regions of destination and the transit regions. The heterogeneity of the attraction or repulsion of this regions influencing the migration behaviour should be included. Furthermore this attribute function should be treated as time dependent to observe structural changes in the migration behaviour over time. The Model description should also have the flexibility to describe migration movement between a list of different included regions in different structure and trough different routes. First of all to find a formal way of describing the regions, these are represented by vertices of a graph, where the neighbouring countries are connected through edges. The graph of migration movement is defined as a directed graph G = (V , E ) , where the finite set of vertices V (G) = {v1 , v 2 ,..., vl } is describing the different regions of
interest and the finite set of edges E (G) = {e1 , e2 ,..., ek }
with ek = vi , , v j
is describing the geographical possibility
to migrate from on region v i to the other region v j . In the adjacency matrix A = (ai ,k ) li , j =1 of the graph the a i , j = 1 if and only if there is the geographical possibility to migrate from i to j. With this formalism it is possible to describe the time dependent interaction I i , j (t ) from the region v i to the region v j with the following equations.
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n
I i , j (t ) =
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m
∑ a k ⋅ (A j (t ) )k ⋅ ∑ rk ⋅ (Ri (t ) )k k =1
k =1
l
∑ c ⋅ (C k =1
k
i, j
(6)
(t ) )k
M i , j (t ) = I i , j (t ) ⋅ M i (t )
(7)
Here the n functions of attractive attributes
(A (t )) : [0, T ] → [0,1] j
n k =1
(8)
in v j and the m functions of repulsive attributes
(Ri (t ) )k =m1 : [0, T ] → [0,1]
(9)
in v i are time dependent weighted function with the parameters n
a k ∈ [0,1], ∑ a k = 1
(10)
k =1
and the parameters m
rk ∈ [0,1], ∑ rk = 1.
j, j
(t ))
: [0, T ] → [1, ∞)
Empirical studies showed that Gravity Models are most successful in describing macro patterns of Spatial Interaction. So it is more reliable in picturing the behaviour of populations rather than individuals. (see Sen and Smith 1995, p. 16) One reason for that is that individual decision are often influenced by a lot of different factors which refer to this specific individual. Therefore the focus on population groups has also the quality of a sufficiently description of the behaviour of interest with just a little required information. It is also important to discuss the treatment of time, where in the introduced migration model the Interaction is only observed at discrete time steps. So it describes static patterns over a period of time. Structural changes, for example of the attribute function from one time step to the next one, are treated as very fast, so as a change of state. This means that the model always treats static patterns. As the model aims to analyse social behaviour, which has a bunch of influencing factors and can only take a finite set of them into consideration, it is by nature a probabilistic approach.
(11) 5.1 Graph of Migration Movement
The l functions of separation attributes
(C
important in terms of understanding the qualities and limits of this approach.
5. SIMULATION AND RESULTS
k =1
l k =1
223
(12)
between the regions v i and v j are also time dependent
With the introduced migration model the movement of migrants from Syria into and within Europe in the summer 2015 is simulated. This situation implies the migration graph shown in figure 1.
weighted functions with the weights l
c k ∈ [0,1], ∑ c k = 1 .
(13)
k =1
Therefore the absolute migration from v i to v j at the time t is represented of the product of the interaction I i , j (t ) ∈ [0,1] and the number of migrants M i (t ) at v i at the tme t. So the migration from one the region to a region is determined by the directed interaction between the regions. This interaction again is a relation of weighted repulsive functions, weighted attractive functions and weighted separation functions. So the interaction is increasing with an increase of the repulsion in the one region and increase of the attraction in the other region and it is decreasing with an increase of the separation. 4.2 Model Characterisation Since social behaviour is modelled it is of great importance to mention that this behaviour is resulting of a complex composition of influences. This is a great challenge in finding a description of the social behaviour. Different approaches are often capable to cover different aspects of this complexity. So the characterization of this migration model is
Fig. 1. The migration graph for the migration movement from Syria into and within Europe in the summer 2015 Since it is not necessary to investigate some of the considered countries in Europe as single countries, some of them are
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considered as regions. A description of the graph in figure 1 is shown in table 1. Table 1. Description of the migration graph vertex v19
region Albania, Bosnia, Montenegro
v18 v17
Italy, Spain, Portugal
v16 v15
v14 v13 v12 v11 v10
UK, Netherlands, Belgium, France
Table 2. Attractive functions Gross domestic product (GPD) Fragile state index (FSI) Maximum attraction of accessible countries Not exceeded capacity Asylum recognition rate Asylum recognition quote in Europe
Norway, Finland, Sweden, Denmark Slovakia, Czech Republic, Romania, Bulgaria, Poland, Lithuania, Estonia, Latvia Germany Austria Slovenia Croatia Hungary
v9
Serbia
v8
Macedonia
v7
Greece
v6
Turkey
v5
Lebanon
v4 v3
Egypt Jordan
v2 v1
Iraq Syria
5.2 Attribute Functions The model focuses on the migration movement and the different migration routes and it is not modelling any reasons for emigration. So this implies a distinction of countries of origin and potential destination countries. So observing the interactions from Syria to the neighbouring countries, the repulsive attributes of the country of origin are not taken into consideration. Here the attractive attributes of the neighbouring countries are determining the value of interaction.
Table 3. Repulsive functions Gross domestic product (GPD) Fragile state index (FSI) Exceeded capacity Asylum recognition rate Asylum recognition quote in Europe In this simulation the separation function is representing the effect of open and closed borders on the interaction. The choice of the attribute functions was taken under the consideration of the outcome of the study of the Australian economic scientist Timothy Hatton. He examined the correlation of specific attributes of developed couturiers and the number of asylum applications. Essentially he observed a big effect of access policy, like the border security, and also of processing policy, like the humanitarian situation. But there was shown just a small positive impact of welfare policy on the number of asylum applications. (see Hatton 2015, p. 21) The access policy of a country is considered by the separation function. If there are closed borders enforced by border security arrangements, the separation function, shown in 12, will reduce the interaction immediately. By integrating the different influences of the attractive and repulsive attributes the weighting parameters where estimated. Figure 2 and figure 3 show that this weighting takes the big effects of processing policy (capacity, FSI, asylum recognition rate and asylum recognition quote) into consideration.
There is also a distinction of potential destination countries and transit countries taken into consideration, because the attractive at- tributes of a transit couturiers have less influence on the interaction. So when the repulsion of a country exceeds a certain amount, it is treated as a transit country, which means that the attractive functions of it are replaced with the maximum of the attractive function of the countries which are accessible from there. In the table 2 and table 3 there is a list of considered attractive and repulsive functions. Fig. 2. The weighting of the repulsive functions
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225
6. CONCLUSIONS AND OUTLOOK A migration model is presented which is fitting into the class of Gravity Models and which is taking the time dependency and the heterogeneity of the attribute functions into consideration. It is applicable to different graph of movements and can also show the impact of border closings. As discussed it is a macroscopic approach which models the migration movement on the basis of whole populations, so the simulation results have to be interpreted in a qualitative way.
Fig. 3. The weighting of the attractive functions 5.3 Simulation Results The described situation was implemented with Data from UNHCR for the time period of 1st of September 2015 till 31. October 2015. The border closing of Hungary to Serbia on the 15th of September 2015 as well as the border closing from Hungary to Croatia on the 16th of October 2015 are implemented as well. The results in figure 4 are showing the migration movements as they were happening in this time period in a qualitative way, compared to UNHCR data. Here one can see the different routes, which are shown by the arrows from one country to another as well as the amount of migrants in the countries, which is shown by the blocks in the countries. The considered countries itself are pictured in different grey shades. The results also show the alternative routes after the border closings.
It was shown that the model is capable of describing the migration movement into and within Europe during September and October 2015 in a qualitative way. But as an outlook it can also be examined if the model can show the qualitative behaviour of migration movements in completely different regions. Furthermore the impact of the different attributes can be studied, which is of great importance for understanding migration behaviour. As a last point it is also possible to model prognostic scenarios. REFERENCES Carey H.C. (1858). Principles of Social Sciences. Philadelphia, Pennsylvania. Dennett A. (2012). Estimating flows between geographical locations: get me started in spatial interaction modelling. Working Paper Series, Paper 181-March 12, UCL, Centre for Advanced Spatial Analysis. Hatton T. and Maloney J. (2015). Applications for Asylum in the Developed World: Modelling Asylum Claims by Origin and Destination. ANU Working Papers in Economics and Econometrics, 625, Australian National University. Sen A. and Smith T. (1995). Gravity Models of Spatial Interaction Behaviour. Springer, Place of publication. Young, E.C. (1924). The movement of farm population. Cornell Agricultural Experiment Station, New York. Appendix A. ACKNOLEDGMENTS We want to thank Irene Hafner and Stefan Emrich (dwh simulation services) as well as Filip Krasinianski (ORF - ¨ Osterreichischer Rundfunk) for the decision support and the performed visualisation.
Fig. 4. Simulation results for the summer 2015