  In this study we investigate a novel approach to stochastically perturb the
disease transmission coefficient, which is a key parameter in
susceptible-infected-susceptible (SIS) models. Motivated by the papers [2] and
[5], we perturb the disease transmission coefficient with a Gaussian white
noise, formally modelled as the time derivative of a mean reverting
Ornstein-Uhlenbeck process. We remark that, thanks to a suitable representation
of the solution to the deterministic SIS model, this perturbation is rigorous
and supported by a Wong-Zakai approximation argument that consists in smoothing
the singular Gaussian white noise and then taking limit of the solution from
the approximated model. We prove that the stochastic version of the classic SIS
model obtained this way preserves a crucial feature of the deterministic
equation: the reproduction number dictating the two possible asymptotic regimes
for the infection, i.e. extinction and persistence, remains unchanged. We then
identify the class of perturbing noises for which this property holds and
propose simple sufficient conditions for that. All the theoretical discoveries
are illustrated and discussed with the help of several numerical simulations.
