  Two versions of the susceptible-infected-susceptible epidemic model, which
have different transmission rules, are analysed. Both models are considered on
a weighted network to simulate a mitigation in the connection between the
individuals. The analysis is performed through a heterogeneous mean-field
approach on a scale-free network. For a suitable choice of the parameters, both
models exhibit a positive infection threshold, when they share the same
critical exponents associated with the behaviour of the prevalence against the
infection rate. Nevertheless, when the infection threshold vanishes, the
prevalence of these models display different algebraic decays to zero for low
values of the infection rate.
