  This paper is devoted to the study of a stochastic epidemiological model
which is a variant of the SIR model to which we add an extra factor in the
transition rate from susceptible to infected accounting for the inflow of
infection due to immigration or environmental sources of infection. This factor
yields the formation of new clusters of infections, without having to specify a
priori and explicitly their date and place of appearance.We establish an {exact
deterministic description} for such stochastic processes on 1D lattices (finite
lines, semi-infinite lines, infinite lines, circles) by showing that the
probability of infection at a given point in space and time can be obtained as
the solution of a deterministic ODE system on the lattice. Our results allow
stochastic initial conditions and arbitrary spatio-temporal heterogeneities on
the parameters.We then apply our results to some concrete situations and obtain
useful qualitative results and explicit formulae on the macroscopic dynamics
and also the local temporal behavior of each individual. In particular, we
provide a fine analysis of some aspects of cluster formation through the study
of {patient-zero problems} and the effects of {time-varying point
sources}.Finally, we show that the space-discrete model gives rise to new
space-continuous models, which are either ODEs or PDEs, depending on the
rescaling regime assumed on the parameters.
