In this paper, a novel influenza $ \mathcal{S}\mathcal{I}_N\mathcal{I}_R\mathcal{R} $ model with white noise is investigated. According to the research, white noise has a significant impact on the disease. First, we explain that there is global existence and positivity to the solution. Then we show that the stochastic basic reproduction $ {{\underset{\scriptscriptstyle\centerdot}{\text{R}}}} {_r} $ is a threshold that determines whether the disease is cured or persists. When the noise intensity is high, we get $ {{\underset{\scriptscriptstyle\centerdot}{\text{R}}}}{_r} < 1 $ and the disease goes away; when the white noise intensity is low, we get $ {{\underset{\scriptscriptstyle\centerdot}{\text{R}}}}{_r} > 1 $, and a sufficient condition for the existence of a stationary distribution is obtained, which suggests that the disease is still there. However, the main objective of the study is to produce a stochastic analogue of the deterministic model that we analyze using numerical simulations to get views on the infection dynamics in a stochastic environment that we can relate to the deterministic context.
Keywords: influenza; persistence and extinction of disease; stochastic modeling; white noise.