  In this paper, we introduce and study one-dimensional models for the behavior
of pedestrians in a narrow street or corridor. We begin at the microscopic
level by formulating a stochastic cellular automata model with explicit rules
for pedestrians moving in two opposite directions. Coarse-grained mesoscopic
and macroscopic analogs are derived leading to the coupled system of PDEs for
the density of the pedestrian traffic. The obtained PDE system is of a mixed
hyperbolic-elliptic type and therefore, we rigorously derive higher-order
nonlinear diffusive corrections for the macroscopic PDE model. We perform
numerical experiments, which compare and contrast the behavior of the
microscopic stochastic model and the resulting coarse-grained PDEs for various
parameter settings and initial conditions. We also demonstrate that the
nonlinear diffusion is essential for reproducing the behavior of the stochastic
system in the nonhyperbolic regime.
