Benevento, Spring 2011

1. Benevento, Spring 2011 Cellular Automata Models : Intro and Paradigms Costas Siettos School of Applied Mathematics and Physical Sciences, NTUA, Athens, Greece

2. Motivation Different time and space scales Macro scales much much bigger than the bigger Microscopic scale Macroscale Meso-scale Fokker Planck Moments Spherical Harmonics ODE’s PDE’s Wavelets The Analysis and Control is usually sought at this level A Big! number of available Microscopic/ Stochastic/ Models Simulating the Time Evolution of Real World- Complex Systems (Fluid Mechanics, Material Science, Bio, Ecology, Process Engineering,…) Micro-scale Microscopic/ Stochastic models - Fokker Fokker Planck Planck Brownian Dynamics Monte Carlo Molecular Dynamics Cellular Automata εξίσωση εξίσωση

3. Real Complex Problems: Fire-Spreading Greece: Summer of 2007: -2x105 hectares of Forest Burned -74 Died Island of Spetses, 1990: Burned the 1/3 of the Island (around 8 km2)

4. Real Complex Problems: Fire-Spreading Modeling Atomistic/Stochastic Models Like Cellular Automata Can! Predict Large & Multiscale Complex Problems Evolution! Agents: Tree =f (Type, Density, Height, Ground Slope)

5. Cellular Automata: A simplistic model A cell can take each time one of the three states: 1:Black,empty/burned 2:Green,trees. 3:Red: Fire The update rule has as follows: The fire on a site will spread to the trees at its nearest-neighbor sites at the next time step with probability p. All trees on fire will burn down and return to empty sites at the next time step. At time t+1 fire fire At time t With probability p At time t+1 fire Empty sites At time t

6. Cellular Automata: Phase transitions Probability of Spreading: 42% Probability of Spreading: 46%

7. Coarse-Grained Computations and Control for Cellular Automata Models of Randomly Connected Individuals • Two CA Models • Network of Neurons • B. Infection Spreading among Individuals

8. Each neuron is described by 2 states Every neuron has 4 links (5 with itself) which influence the fate of its state. The topology depends on the number of remote & local connections Remote connections Local connections Neurons Connections-Topology of the network a(t)=1: activated a(t)=0: inactivated

9. The CA model: The Evolution of the network : Majority rule Determined by two functions: • Arousal function s(x): the probability that an inactivated neuron becomes activated . • the depression function r(x): the probability that an activated • Neuron becomes inactivated where |Λ(x)| =the number of connections of each neuron (including itself) (here 5)

10. The rules: an example s(x)=ε (if ε<<1: small probability to become activated) r(x)=ε (small probability to become inactivated) r(x)=1-ε (bigger probability to become inactivated) s(x)=1-ε (bigger probability to become activated)

11. Temporal Simulations ε=0.215, 2 remote neighbors Why two states?  Bifurcation Analysis Transition rates? Rare Events Analysis

12. Coarse-Grained Computations for Infection Spreading Among Individuals

13. N number of individuals Each individual can be in one of the 3 following states 1:Susceptible : not yet infected; probabilistic potential to be infected 2:Infected 3:Recovered; recovers from the infection; immunized from infection He/ She interacts with 4 other The CA model • Rules of Evolution: • A susceptible gets infected with probability pS->I if one of his links is infected • An infected recovers with probability pI->R • A recovered becomes susceptible again with probability pR->S • otherwise has immunity

14. Temporal Simulations S=95%, I=5% pS->I = 0.9 S=90%, I=10%