Universality class of epidemic percolation transitions driven by random walks

Inspired by the recent viral epidemic outbreak and its consequent worldwide pandemic, we devise a model to capture the dynamics and the universality of the spread of such infectious diseases. The transition from a pre-critical to the post-critical phase is modeled by a percolation problem driven by random walks on a two-dimensional lattice with an extra average number $\rho$ of nonlocal links per site. Using the finite-size scaling analysis, we find that the effective exponents of the percolation transitions as well as the corresponding time thresholds, extrapolated to the infinite system size, are $\rho$-dependent. We argue that the $\rho$-dependence of our estimated exponents represents a crossover-type behavior caused by the finite-size effects between the two limiting regimes of the system. We also find that the universal scaling functions governing the critical behavior in every single realization of the model can be well described by the theory of extreme values for the maximum jumps in the order parameter and by the central limit theorem for the transition threshold.