Stochastic Delay Modelling of Streptococcal Transmission: Stability Analysis and Dynamically Consistent Numerical...

This study investigates the transmission dynamics of streptococcal disease using a stochastic delay differential equation (SDDE) framework. The population is divided into four compartments: susceptible individuals $S$ , infected individuals $I$ , carriers of streptococcus $C$ and individuals who progress to the post-infectious complication of acute rheumatic fever (ARF), representing severe disease outcomes rather than an infectious transmission class, $A$ . The incorporated discrete time delay represents the effective latency associated with bacterial colonisation, disease incubation and the early host immune response prior to disease progression. The deterministic delayed model is first analysed to establish fundamental dynamical properties, including positivity, boundedness and the existence and uniqueness of solutions. The equilibrium states are derived and their local and global stability are examined in relation to the basic reproduction number, $R_0$ , which serves as the epidemiological threshold determining whether the disease dies out or persists within the population. A stochastic delayed version of the model is then formulated, and conditions ensuring the existence of a unique global positive solution are obtained. Furthermore, a dynamically consistent stochastic nonstandard finite difference (NSFD) scheme is constructed by combining the nonstandard finite difference methodology with an Euler-Maruyama discretisation of the stochastic perturbation terms to preserve the qualitative dynamics of the continuous SDDE model. The proposed scheme preserves essential dynamical properties of the system and is shown to converge to the corresponding equilibrium states without restrictions on the step size. Numerical simulations are carried out to compare the proposed scheme with several standard stochastic numerical methods. The results demonstrate that the nonstandard finite difference scheme maintains positivity and boundedness of the solutions and closely follows the mean trajectory of the deterministic model, whereas the standard schemes may violate these properties. The proposed framework therefore provides a reliable and dynamically consistent computational tool for investigating the influence of stochastic perturbations and delayed disease progression on streptococcal transmission dynamics. Finally, the influence of stochasticity and time delay on the dynamics of susceptible and infected populations is illustrated through numerical experiments.