Modelling SIR-type epidemics by ODEs, PDEs, difference equations and cellular automata – A comparative study

The Kermack–McKendrick susceptible-infected-recovered (SIR) model describes the dynamics of epidemics in a cumulative way. This contribution compares different approaches for introducing spatial patterns into these dynamics. The applied techniques cover lattice gas cellular automata (LGCA), stochastic cellular automata (SCA) and partial differential equations (PDE). Even though these methods involve distinct types of spatial interaction, it can be shown, that consistent qualitative and quantitative model behaviour can be obtained by means of parameter adaptions and slight technical modifications. These modifications are motivated by stochastic analysis of distributed interaction (PDE, SCA) and diffusion dynamics (LGCA) as well as prevailing physical analogies. The law of large numbers permits to approximate stochastic contacts by distributed interaction. Diffusion of particles can be approximated through empiric adjustment of a Gaussian diffusion distribution.