  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.
