  Modeling long-range epidemic spreading in a random environment, we consider a
quenched disordered, $d$-dimensional contact process with infection rates
decaying with the distance as $1/r^{d+\sigma}$. We study the dynamical behavior
of the model at and below the epidemic threshold by a variant of the
strong-disorder renormalization group method and by Monte Carlo simulations in
one and two spatial dimensions. Starting from a single infected site, the
average survival probability is found to decay as $P(t) \sim t^{-d/z}$ up to
multiplicative logarithmic corrections. Below the epidemic threshold, a
Griffiths phase emerges, where the dynamical exponent $z$ varies continuously
with the control parameter and tends to $z_c=d+\sigma$ as the threshold is
approached. At the threshold, the spatial extension of the infected cluster (in
surviving trials) is found to grow as $R(t) \sim t^{1/z_c}$ with a
multiplicative logarithmic correction, and the average number of infected sites
in surviving trials is found to increase as $N_s(t) \sim (\ln t)^{\chi}$ with
$\chi=2$ in one dimension.
