SIR model on one dimensional small world networks

We study the nonequilibrium phase transition for the model of epidemic spreading, Susceptible–Infected–Refractory (SIR), on one dimensional small world networks. This model belongs to the universality class of dynamical percolation class (DyP) and the upper critical dimension corresponding to this class is dc=6. One dimensional case is special case of this class in which the percolation threshold goes to one (boundary value) in thermodynamic limit. This behavior resembles slightly the behavior of equilibrium phase transition in a one dimensional Ising and XY models where the critical thresholds for both models go to zero temperature (boundary value) in thermodynamic limit. By analytical arguments and numerical simulations we demonstrate that, increasing the connectivity (2k) of this model on regular one dimensional lattice does not alter the criticality of the model. However the phase transition study shows that, this model crosses from a one dimensional structure to mean field like for any finite value of the rewiring probability (p). This behavior is similar to what happened in the equilibrium phase transition for the Ising and XY models on small world networks. Thus, this model is a one of nonequilibrium models which behaves similarly to the equilibrium systems on small world networks. Unlike of many nonequilibrium systems on small world networks which have been found to display a mean field like behavior only at finite values of p or even show critical exponents depend on p. Furthermore, we calculate the critical exponents and the full critical phase space of this model on small world network. We also introduce the crossover scaling function of this model from one dimensional behavior to mean field behavior and reveal the similarity between this model and the equilibrium models on the small world networks.