Joanne Turner 15 Nov 2005 Introduction to Cellular Automata

What is a Cellular Automaton? Type of network model. Nodes (cells) can represent: –individual animals or plants –units such as households or farms Processes are based entirely on local interactions. e.g. Game of Life (Conway, 1970) –If cell is ‘dead’ but has three ‘living’ neighbours, then cell will become ‘alive’. Otherwise cell remains ‘dead’. (e.g. if conditions are right, unoccupied territory becomes occupied by one of neighbour’s offspring). –If cell is ‘alive’ and has 2 or 3 living neighbours, then cell will remain ‘alive’. Otherwise cell will die. only 1 neighbour: die of loneliness > 3 neighbours: die of overcrowding

Rigid spatial structure. Can have any number of dimensions. In 2 dimensions, nodes (cells) form a lattice (grid). Square cells most common, but can also have triangular and hexagonal. Contacts are with ‘nearest neighbours’ (definition can vary). e.g. neighbours of yellow cell are shown in green Model consists of a program that records the states of cells and then updates the information according to a set of rules that involve the definition of the neighbourhood. All states are updated simultaneously (i.e. population evolves in discrete time steps). Model Definition Von NeumannMoore

Lattice can be wrapped on itself. Rules can involve probabilities (Probabilistic Cellular Automata). Outcome can depend on initial state of the population. Simple (local) rules can generate very complex global behaviour. Applications include –development of computing –image processing –cryptography –traffic flow –earthquakes –tumour growth –apparent competition in a predator-prey system –spread of an invasive plant species –waves of invading predators (Sherratt) Further Details

Disease Transmission: FCA vs. FCN CA model in conjunction with Roger Bowers and Mike Begon. Study was inspired by work on bank voles and wood mice at Manor Wood and Rake Hey. Q: If transmission is density-dependent at the local level, does it appear to be density-dependent at the global level? Square lattice consisting of 50 x 50 cells. Cells could be either occupied by a single individual or unoccupied. 2 contact mechanisms: –fixed contact area (FCA) → local density-dependent transmission –fixed contact number (FCN) → local frequency-dependent transmission Only difference was in the definition of the ‘nearest neighbours’.

Fixed contact area (FCA): ‘nearest neighbours’ were those with whom the individual shared a boundary (green). 4 on a fully occupied lattice. Fixed contact number (FCN): ‘nearest neighbours’ were 4 ‘nearest individuals’ (e.g. blue). Same as FCA ‘nearest neighbours’ when lattice is fully occupied. Nearest Neighbours: FCA vs. FCN Fixed contact area (local density- dependent transmission) Fixed contact number (local frequency- dependent transmission) FCA = FCN when lattice is fully occupied.

Possible outcomes: Pathogen extinction (short infectious period) Susceptible Infected Immune Unoccupied FCA: Local density-dependent transmission prevalence time prevalence time Host-pathogen coexistence (long infectious period)

Possible outcomes: Pathogen extinction (short infectious period) Susceptible Infected Immune Unoccupied FCN: Local frequency-dependent transmission prevalence time prevalence time Host-pathogen coexistence (long infectious period)

Other possible outcomes are ‘low host’ pathogen extinction and, in the case of FCN, pathogen-induced host extinction. Parameter Space

Q: If transmission is density-dependent at the local level, does it appear to be density- dependent at the global level? Global frequency-dependent term is βSI/N (short dashes). Global density-dependent term is βSI (long dashes). Used GLM to compare the number of new infections predicted by these two terms with the observed number (solid line). A: Global frequency-dependent term was the better descriptor of both local frequency- dependent (FCN) and local density-dependent (FCA) transmission. Modelling Transmission Globally local den-dep CA local freq-dep CA

Where probability of contact depends on ‘distance apart’ (as on a lattice), elements of the adjacency matrix might look something like this However, an adjacency matrix can incorporate a more complicated contact pattern. For example, here m ij  m ji. Cellular automata are good for modelling systems with a regular spatial structure. However, network models that describe contacts using an adjacency matrix are more versatile. Each element m ij of the adjacency matrix can be thought of as the probability of a suitable contact between nodes i and j. Adjacency Matrix Approach