This study investigates the transmission dynamics of conjunctivitis using stochastic delay differential equations (SDDEs). A delayed stochastic model is formulated by dividing the population into five distinct compartments: susceptible, exposed, infected, environmental irritants, and recovered individuals. The model undergoes thorough analytical examination, addressing key dynamical properties including positivity, boundedness, existence, and uniqueness of solutions. Local and global stability around the equilibrium points is studied with respect to the basic reproduction number. The existence of a unique global positive solution for the stochastic delayed model is established. In addition, a stochastic nonstandard finite difference scheme is developed, which is shown to be dynamically consistent and convergent toward the equilibrium states. The scheme preserves the essential qualitative features of the model and demonstrates improved performance when compared to existing numerical methods. Finally, the impact of time delays and stochastic fluctuations on the susceptible and infected populations is analyzed.

Based on the World Health Organization (WHO), Meningitis is a severe infection of the meninges, the membranes covering the brain and spinal cord. It is a devastating disease and remains a significant public health challenge. This study investigates a bacterial meningitis model through deterministic and stochastic versions. Four-compartment population dynamics explain the concept, particularly the susceptible population, carrier, infected, and recovered. The model predicts the nonnegative equilibrium points and reproduction number, i.e., the Meningitis-Free Equilibrium (MFE), and Meningitis-Existing Equilibrium (MEE). For the stochastic version of the existing deterministic model, the two methodologies studied are transition probabilities and non-parametric perturbations. Also, positivity, boundedness, extinction, and disease persistence are studied rigorously with the help of well-known theorems. Standard and nonstandard techniques such as Euler Maruyama, stochastic Euler, stochastic Runge Kutta, and stochastic nonstandard finite difference in the sense of delay have been presented for computational analysis of the stochastic model. Unfortunately, standard methods fail to restore the biological properties of the model, so the stochastic nonstandard finite difference approximation is offered as an efficient, low-cost, and independent of time step size. In addition, the convergence, local, and global stability around the equilibria of the nonstandard computational method is studied by assuming the perturbation effect is zero. The simulations and comparison of the methods are presented to support the theoretical results and for the best visualization of results.