Epidemic models on lattices
Classic epidemic models of disease transmission are described in Compartmental models in epidemiology. Here we discuss the behavior when such models are simulated on a lattice. Lattice models, which were first explored in the context of cellular automata, act as good first approximations of more complex spatial configurations, although they do not reflect the heterogeneity of space (e.g. population density differences, urban geography and topographical differentiations). Lattice-based epidemic models can also be implemented as fixed agent-based models. Introduction The mathematical modelling of epidemics was originally implemented in terms of differential equations, which effectively assumed that the various states of individuals were uniformly distributed throughout space. To take into account correlations and clustering, lattice-based models have been introduced. Grassberger considered synchronous (cellular automaton) versions of models, and showed how the epidemic growth goes through a critical behavior such that transmission remains local when infection rates are below critical values, and spread throughout the system when they are above a critical value. Cardy and Grassberger argued that this growth is similar to the growth of percolation clusters, which are governed by the "dynamical percolation" universality class (finished clusters are in the same class as static percolation, while growing clusters have additional dynamic