  This paper provides a first attempt to incorporate the massive discontinuous
changes in the spatio-temporal dynamics of epidemics. Namely, we propose an
extended class of epidemic models, governed by coupled stochastic semilinear
partial differential equations, driven by pure-jump L\'evy noise. Based on the
considered type of incidence functions, by virtue of semi-group theory, a
truncation technique and Banach fixed point theorem, we prove the existence and
pathwise uniqueness of mild solutions, depending continuously on the initial
datum. Moreover, by means of a regularization technique, based on the resolvent
operator, we acquire that mild solutions can be approximated by a suitable
converging sequence of strong solutions. With this result at hand, for positive
initial states, we derive the almost-sure positiveness of the obtained
solutions. Finally, we present the outcome of several numerical simulations, in
order to exhibit the effect of the considered type of stochastic noise, in
comparison to Gaussian noise, which has been used in the previous literature.
Our established results lay the ground-work for investigating other problems
associated with the new proposed class of epidemic models, such as asymptotic
behavior analyses, optimal control as well as identification problems, which
primarily rely on the existence and uniqueness of biologically feasible
solutions.
