Diffusion Approximation in a Stochastic Cellular Automaton Model for Epidemics

Although microscopic models are nowadays getting more and more popular among, still the modelling approach lacks of appropriate mathematical theory to confidentally rely on the outputs of the derived models. Especially unexpected chaotic group behaviour and the inability to validate and parametrise the model often leads to unusable simulations. The investigated testcase, a simple cellular automaton (CA) simulating the temporal development of a SIR (Susceptible-InfectedRecovered) type epidemic, shows a field of application for so called complexity theory. In order to explain and analyse the aggregated simulation results of the CA, certain methods usually used in Markov theory for quantum mechanics, basically extensions of so called diffusion approximation [1], are applied. Finally, already suspected, correlations to the solutions of the famous SIR differential equations, formerly derived by Kermack and McKendrick [2], can be proven with analytical methods and extended by convergence results and qualitative error estimations. Introduction Advantages Disadvantages Lower abstraction level compared to reality Difficult to validate Easy to understand for non-experts High computational efforts Very flexible regarding system changes Difficult to document regarding reproducibility More suitable for eyecatching visualisations Sometimes unpredictable and chaotic results Table 1: Some advantages and disadvantages of microscopic models. M Bicher Diffusion Approximation in a Stochastic Cellular Automaton Model for Epidemics 118 SNE 23(3-4) – 12/2013 TN 1 Comparison of two Modelling Approaches Figure 1: Example Result of the Cellular Automaton. Figure 2: Example Result of Classic SIR Differential Equations. M Bicher Diffusion Approximation in a Stochastic Cellular Automaton Model for Epidemics SNE 23(3-4) – 12/2013 119 T N 2 Cellular Automaton – Definition 2.1 Space 2.2 Agents (non-empty cells)