Mathematical modeling and dynamics of gonorrhoea epidemic within stochastic fractional delay differential equations

Stochastic fractional delayed modeling (stochastic fractional delay differential equations (SFDDEs) with delay parameters) is a significant non-pharmaceutical intervention to control transmission dynamics of infectious diseases and memory effects to the reality of nature. According to the report of the World Health Organization (WHO), 374 million people were infected in adults 15 to 49 in 2020 and approximately 8 million people reported positive tests for the new infection of gonorrhoea in 2022. Gonorrhoea and its related group of diseases are still a threat in developing and developed countries and appear in different variants globally. The present study extends the deterministic gonorrhoea model into a stochastic fractional delayed model. The main aim is to capture the complex interplay between random environmental fluctuations and the memory effect inherent in the spread of the disease by incorporating both stochastic perturbations and fractional-order derivatives. The incorporation of stochasticity with delay was studied in non-integer change concerning time for each compartment of the population like the S(t) susceptible, E(t) exposed, I(t) infected, and R(t) recovered classes. The essential properties like positivity, boundedness, existence, uniqueness, equilibria (Gonorrhoea-free equilibrium (GFE) and Gonorrhoea-present equilibrium (GPE)), reproduction number, sensitivity analysis, and stability result in the sense of local, and first-order stability result in the sense of global are studied rigorously. Also, some well-known theorems verified the positivity, boundedness, extinction, and persistence of disease for the support of the non-integer stochastic version of the model. Due to the high complexity of nonlinear stochastic fractional delay differential equations, the Grunwald-Letnikov approximation is implemented in the nonstandard finite difference method (NSFD) to visualize the results with data from the model. The proposed method is efficient and restores all dynamic properties of the model with a free choice of time steps. For convenience, a computational code of the non-standard discrete stochastic fractional delayed model may be provided to the readers at their request.