On a new method for the stochastic perturbation of the disease transmission coefficient in SIS Models

In this study, first a necessary background is provided to readers. Then a novel approach to stochastically perturb the disease transmission coefficient was investigated, which is a key parameter in susceptible-infected-susceptible (SIS) models. Motivated by the papers [5] and [2], the disease transmission coefficient was perturbed with a Gaussian white noise, formally modelled as the time derivative of a mean reverting Ornstein-Uhlenbeck process. It has been remarked 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. It has been proven 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. Then the class of perturbing noises for which this property holds were identified and propose simple sufficient conditions for that. All the theoretical discoveries are illustrated and discussed with the help of several numerical simulations.