Mathematical analysis of a stochastic delay model for respiratory syncytial virus dynamics.

Respiratory syncytial virus (RSV) is a single-stranded RNA virus responsible for a wide range of respiratory tract infections, including those affecting the lungs, airways, and middle ear. Understanding its transmission dynamics remains essential for effective disease control. A bio-inspired stochastic delay model for RSV transmission is proposed and analyzed. The model's qualitative properties including positivity, boundedness, equilibrium states, and the basic reproduction number are rigorously established through well-posedness theorems. Parameter sensitivity is also examined. To investigate the system's stochastic behavior, numerical schemes such as Stochastic Euler, Runge-Kutta, and Euler-Maruyama methods are applied. However, these traditional approaches fail to fully preserve the dynamic characteristics of the model. To address these limitations, a stochastic nonstandard finite difference (NSFD) scheme with delay is developed. This approach ensures non-negativity, boundedness, consistency, and unconditional convergence, overcoming issues of instability and divergence often observed in standard stochastic numerical methods. Comparative simulations demonstrate that the NSFD method reliably reproduces the true dynamic states of the model. The proposed stochastic delayed modeling framework enhances our understanding of RSV dynamics and provides a stable computational tool for analyzing complex biological systems. The findings open new avenues for exploring nonlinear stochastic processes in epidemiological and neurobiological modeling.