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

TL;DR

This paper introduces a rigorous stochastic perturbation method for the disease transmission coefficient in SIS models using Gaussian white noise and Ornstein-Uhlenbeck processes, preserving key epidemiological thresholds.

Contribution

It presents a novel, mathematically rigorous approach to perturb the transmission coefficient in SIS models, maintaining the basic reproduction number and providing conditions for noise compatibility.

Findings

The stochastic model preserves the basic reproduction number.

A Wong-Zakai approximation supports the noise modeling.

Numerical simulations illustrate theoretical results.

Abstract

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.