## Agent Based Modelling and Cellular Automata

Cellular Automata (CA) was mainly developed during the 1980s/1990s and Agent based Models (ABM) came later, in the 1990s/2000s. Both can represent and model a diverse range of phenomenon in areas such us medicine, biology, natural systems, urban planning, transport, human migrations and so on.

## 1. Introduction

CA are “models in which contiguous or adjacent cells, such as those that might comprise a rectangular grid, change their states through the repetitive application of simple rules.” (Batty, 1997). CA is a discrete (space, time, and properties have finite, countable states) dynamical system, and the complexity operates from the bottom up. The CA works with a number of cells (which have a state) on a grid interacting with other cells in the neighbourhood accordingly to a set of rules. CA models, simpler than ABM models, “are explicit and simple spatial dynamic models with little or no presumption about the form of the dynamics and rather simple notions about the effect of space. In their strictest form they simulate the spatial diffusion around a point where the diffusion is to immediate neighbours and time and space are treated as one.” (Batty, 2005).

ABM are a “class of models (…) based on representing objects and populations at an elemental or individualistic level which reflects behaviours of those objects through space and time.” (Batty, 2009). Having a bottom up operation as CA, ABM are more complex and better represents “individual heterogeneity, representing explicitly agents’ decision rules.” (Gilbert, 2008). As instance, the agents can perform different rules, have different intentions, attributes, memory or resources.

ABM can reflect the agents’ behaviour, therefore is more appropriated for representing social networks. Nevertheless, the approach of CA is top down and is limited by the neighbourhood and number of states, so it´s less appropriate to simulate social systems.

Overall, ABM presents high level of disaggregation, which makes difficult its handling: “there are many features of the development process that cry out for specification; for example, issues about housing finance and finance for land development, issues about distance from home to work and to other facilities, provision of budgets, life style issues, all crowd into such a model. In a sense, this is why ABMs are so hard to build and test because once this level of detail is broached, it is hard to control the aggregation in such a way as to produce testable propositions.” (Batty, 2005).

The present paper will use ABM to explore 2 different epidemiologic scenarios, where the agents can recover or stay infected - in a tragedy scenario. The environmental parameters (population, initial infected population, immune chance and recovery change) will be manipulated using the software NetLogo in order to compare the population’s states (infected/no infected).

## 2.1 Scenario 1 (Recovery)

Parameters of the first scenario:

The agents got recovered easily in 100 runs (Fig. 1). With an initial population of 200 agents, the minimum number of steps needed for the recovery was 65.8 and the maximum number, 76, with a mean of 22.14. The standard deviation, calculated using R to find out the sample size, is 7.685773 (Fig. 2).

*Fig 1. Scenario 1, 100 runs. Green: Healthy. Red: Infected.*

Fig 2. Histogram with Standard Deviation (7.685773) and mean (22.14)

A margin of error of ±1 will be allowed, with a confidence level of 95%. In order to reach this confidence level and have a statistically significant result, the next formula will be used, finding out the necessary number of total runs:

Once applied this formula, the minimum number of total runs of 110. Repeating the experiment with this number of runs, the results are a mean of 20.5 and a standard deviation of 6.696227 (Fig. 3) and the hypothesis can be accepted, as has a margin of error of ±1.

*Fig 3. Histogram with Standard Deviation (6.696227) and mean (20.5)*

## 2.2 Scenario 2 (Tragedy)

Parameters of the second scenario:

*Fig. 4: Structure and methodology of Amsterdam Smart City initiative.
Source: Amsterdam Smart City website.*

In this scenario a large population of 1000 agents, with a little infected – only 10 –, will not have any chance of being immune or getting recovered, projecting a scenario where no one will be not infected. A total number of 100 run was done in this second scenario.

*Fig 4. Scenario 2, 100 runs. Green: Healthy. Red: Infected.*

*Fig 5. Histogram with standard deviation (8.007996) and mean (12.85153).*

In the fist 100 runs, it was calculated a standard deviation of 8.007996 and a mean of 12.85153, with a margin of error of ±1, and therefore the hypotheses can be accepted. However, the minimal number of ticks has been calculated following the previous methodology: 31 runs should be enough to achieve the steady state.

Fig. 6: Histogram with standard deviation (8.039086) and mean (12.75655).

With a margin of error of ±1 and only 31 runs, we can infer that in a tragedy environment a statistical meaningful results is reached with a small number of runsAlso, the Municipality has impulse and actively backs up the programmes StartupDelta and StartupAmsterdam.

## 3. Bibliography

Batty, M., 2009. Urban Modeling. University College London, London, UK. Elsevier Ltd.

Batty, M., 1997. Cellular automata and urban form: a primer. Journal of the American Planning Association 63, 266–274.

Crooks,A. and Heppenstall, AJ, 2012. Introduction to Agent-Based Modelling. In: Agent-Based Models of Geographical Systems. Springer Science+Business Media B.V.

Dennett, A., 2012. Estimating flows between geographical locations:“get me started in”spatial interaction modelling. Citeseer.

Gilbert, G.N., 2008. Agent-based models. Sage.

Heppenstall, A.J., Crooks, A.T., See, L.M., Batty, M. (Eds.), 2012. Agent-Based Models of Geographical Systems. Springer Netherlands, Dordrecht.

Iltanen, S., 2012. Cellular Automata in Urban Spatial Modelling. In: Agent-Based Models of Geographical Systems. Springer Science+Business Media B.V.

Sharabati,W. ,2004. Basic Properties of Confidence Intervals. [Online]. Available at: <http:// www.stat.purdue.edu/~wsharaba/stat511/Chapter7_print.pdf>. [Acessed 15-05-2016]

Wilson, A., 2010b. The general urban model: Retrospect and prospect: The general urban model: Retrospect and prospect. Papers in Regional Science 89, 27–42. doi:10.1111/j.1435-5957.2010.00282.x

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