Generally, stochastic functional differential equations (SFDEs) pose a challenge as they often lack explicit exact solutions. Consequently, it becomes necessary to seek certain favorable conditions under which numerical solutions can converge towards the exact solutions. This article aims to delve into the convergence analysis of solutions for stochastic functional differential equations by employing the framework of G-Brownian motion. To establish the goal, we find a set of useful monotone type conditions and work within the space Cr((−∞,0];Rn). The investigation conducted in this article confirms the mean square boundedness of solutions. Furthermore, this study enables us to compute both LG2 and exponential estimates.

In this paper, we propose a new mathematical model based on the association between susceptible and recovered individual, where the association between susceptible and recovered individual is disturbed by white noise. This model is based on demographic changes and is used for long term behavior. We study the stability of equilibria of the deterministic model and prove the conditions for the extinction of diseases. Then, we investigate and obtain the critical condition of the stochastic epidemic model for the extinction and the permanence in mean of the disease with the white noise. To verify our results, we present some numerical simulations for real data related to disease.