A study of the influence of the mobility on the phase transitions of the   synchronous SIR model
  By using an appropriate version of the synchronous SIR model, we studied the effects of dilution and mobility on the critical immunization rate. We showed that, by applying time-dependent Monte Carlo (MC) simulations at criticality, and taking into account the optimization of the power law for the density of infected individuals, the critical immunization necessary to block the epidemic in two-dimensional lattices decreases as dilution increases with a logarithmic dependence. On the other hand, the mobility minimizes such effects and the critical immunizations is greater when the probability of movement of the individuals increases. 
Roberto da Silva, Henrique A. Fernandes
physics.soc-ph cond-mat.stat-mech