Mean-field-type equations for spread of epidemics: the ‘small world’ model

In the paper we study a cellular automata (CA) model of epidemic dynamics. The effects of local spatial correlations on a temporal (aggregated) spread of single epidemics are studied, as a function of increasing proportion of global contacts (‘small world’ model). We conjecture that even in the presence of high local correlations, the aggregated (mean-field-type) models can be quite successful, if the contact rate is treated as a free parameter. The dependence of the (estimated) contact rate on the mixing parameter can be understood in terms of a simple probabilistic model. The contact rate reflects not only a microscopic and epidemiological situation, but also a complicated social pattern, including short- and long-range contacts as well as a possibly hierarchical structure of human society.