A note about the invariance of the basic reproduction number for stochastically perturbed SIS models

We try to justify rigorously, using a Wong-Zakai approximation argument, the susceptible-infected-susceptible (SIS) stochastic differential equation proposed in [2]. We discover that according to this approach the “right” stochastic model to be considered should be the Stratonovich version of the It^o equation analyzed in [2]. Surprisingly, this alternative model presents the following feature: the threshold value characterizing the two different asymptotic regimes of the solution coincides with the one describing the classical SIS deterministic equation.