  Our study is devoted to a four-compartment epidemic model of a constant
population of independent random walkers. Each walker is in one of four
compartments (S-susceptible, C-infected but not infectious (period of
incubation), I-infected and infectious, R-recovered and immune) characterizing
the states of health. The walkers navigate independently on a periodic 2D
lattice. Infections occur by collisions of susceptible and infectious walkers.
Once infected, a walker undergoes the delayed cyclic transition pathway S $\to$
C $\to$ I $\to$ R $\to$ S. The random delay times between the transitions
(sojourn times in the compartments) are drawn from independent probability
density functions (PDFs). We analyze the existence of the endemic equilibrium
and stability of the globally healthy state and derive a condition for the
spread of the epidemics which we connect with the basic reproduction number
$R_0>1$. We give quantitative numerical evidence that a simple approach based
on random walkers offers an appropriate microscopic picture of the dynamics for
this class of epidemics.
