11.1 The Model

Spatial interaction modelling is usually associated with trips or other kinds of flows. In this Chapter, we aim to show how it can be used to represent threat. We illustrate this with a model of maritime piracy: the pirates threaten shipping, naval vessels can offer some defence. We are aware, however, that this idea can be applied much more widely.

Our study area is shown in Figure 11.1. This shows the broader geographical context and then, specifically, we focus on Somalia and the Gulf of Aden. This work complements Marchione et al. (2014).

The specific area is shown in Figure 11.2 – divided into a kilometre-square grid. In this figure, we show the likely density of vessels passing through the Gulf to and from the Suez Canal. We show below how this is constructed as part of the model.

To formulate the model, we define the following variables: , measure of propensity to attack from ; , volume field of type ship in cell ; , volume field of naval units in cell ; , level of threat from origin to cell for ship ; total threat at cell of type k ship; , distance from origin to cell .

The model can then be constructed as follows. In Equation (11.1), is the level of threat delivered from a pirate base at zone to vessels in cell – measured as . is a measure of the volume of shipping in cell and is a measure – in commensurable units (see below, through the parameter in Equation (11.3)) – of the defence that can be offered by naval vessels.

11.1

A spatial deterrence function with parameter is added. and are calculated from Equations (11.2) and (11.3), respectively.

11.2

11.3

These also include spatial deterrence functions with parameters and . determines the range of the pirates from bases, the extent to which ships are likely to be found at some distance from what is regarded as the optimal route, and the effectiveness of naval deterrence in a cell some distance away from the actual vessel.

The total threat at cell can then be calculated by summing over all pirate sources, , and this is done in Equation (11.4).

11.4

The next step is to test the model for an idealised but plausible situation.

11.2 The Test Case

We construct a test case by assigning shipping to an optimum route and then using Equation (11.2) to allocate ships away from this route (as happens in practice). The set of lines that determine the optimum route and the means of generating the ‘volume of shipping’ field are shown in Appendix. These volumes by cell can be interpreted as forming a ‘volume of shipping’ field – indeed as a field of probabilities. This field has already been shown in Figure 11.2. We have now been able to explain how it was constructed. In the case of naval vessels, we assign three to particular points for this test and the areas of influence are described by Equation (11.3). These generate a ‘naval defence’ field via Equation (11.3) (see Figure 11.3).

These terms can be fed into Equation (11.1) to generate the level of threat from a particular pirate source. For the test case, we assume three port bases in Somali, the three mother ships, which act as bases at the sea. The summation in Equation (11.4) then generates a threat field–, see Figure 11.4. We use the following parameters to produce the fields:

11.3 Uses of the Model

It is interesting to compare the threat field with the observed attacks in 2010 in Figure 11.5.

There is a reasonable degree of fit – by eye, since this is not a formal process. However, it does indicate that it should be possible to calibrate a model of this kind against data. More importantly, it should then be possible to optimise naval strategies on a real-time basis –, given appropriate intelligence on the likely location of pirate sources. Such optimisation might be along the lines: let the naval units available be 3, then the by varying where and keeping and pirates' port and mother ships' locations constant would identify the best naval units location to deter the threat coming from pirates.

This may be a particularly interesting problem because it would have to be run in real time as possible pirate positions shift or new intelligence is collected.

Reference

1. Marchione, E., Johnson, S., and Wilson, A. (2014) Modelling maritime piracy: a spatial approach. Journal of Artificial Societies and Social Simulation, 17 (2), 9.

Appendix

A.1 Volume Field of Type Ship

Figure A.2 shows the volume field of type ship. It is obtained assuming that are all those cells that touched the polyline Lin1 + Line2 or Line1 + Line3 (see Figure A.6):

• Line1
  1. 1. point: Longitude = 42.75476, Latitude = 13.57411
  2. 2. point: Longitude = 44.04895, Latitude = 11.76767
  3. 3. point: Longitude = 47.93152, Latitude = 12.49025
  4. 4. point: Longitude = 50.51990, Latitude = 13.21282
• Line2
  1. 1. point: Longitude = 50.51990, Latitude = 13.21282
  2. 2. point: Longitude = 55.26526, Latitude = 13.93539
• Line3
  1. 1. point: Longitude = 50.51990, Latitude = 13.212820
  2. 2. point: Longitude = 53.10828, Latitude = 11.767672
  3. 3. point: Longitude = 53.53967, Latitude = 4.180644

A.2 Volume Field of Naval Units

Three naval force units at Longitude/Latitude

1. 1. 44.87579 / 12.12896
2. 2. 47.03278 / 12.49025
3. 3. 49.18976 / 13.21282

A.3 Pirates Ports and Mother Ships

Three pirates ports at Longitude / Latitude

1. 1. 47.857500 / 4.654444 (Harardhere)
2. 2. 49.815066 / 7.980800 (Eyl)
3. 3. 48.524975 / 5.351592 (Hobyo)

and three mother ships at Longitude / Latitude

1. 1. 43.725586 / 11.931852
2. 2. 45.791016 / 12.039321
3. 3. 51.789551 / 13.859414

Figure A.7 shows their locations and Figure A.8 shows their distances.