Abstract
This study investigates the qualitative properties and numerical dynamics of a stochastic epidemiological model incorporating unreported cases with a general contact susceptible function, shedding light on the intricate dynamics of infectious diseases such as COVID-19. In the qualitative analysis, we rigorously examine the mathematical properties of the model, including the existence and positivity of solutions, and identify a critical threshold parameter, pivotal in determining the long-term behavior of the system. Notably, our analysis reveals that the stochastic noise significantly influences the dynamics, leading to distinct outcomes: if exceeds unity, solutions converge exponentially to a unique invariant probability distribution, whereas values below one result in the extinction of infectious diseases at an exponential rate. In the numerical study, we delve into comprehensive simulations to validate our theoretical findings and explore the behavior of the model under various scenarios. Synthetic data simulations provide illustrative examples, showcasing both disease extinction and persistence phenomena. Furthermore, we investigate the impact of the susceptible contact function, g(S), on disease dynamics, and propose a selection method for optimizing this function based on real-world COVID-19 data from the UK. By integrating rigorous mathematical analysis with empirical data-driven insights, our study offers valuable contributions to understanding the complex dynamics of infectious diseases.