This chapter significantly contributes to improving human health and well-being, aligned with the Sustainable Development Goals (SDGs). A focus is placed on combating infectious diseases and epidemics, reducing health risks, and promoting sustainable solutions, particularly within the framework of the Decade of Action to achieve the SDGs. In this context, a stochastic SEIS model that incorporates logistic birth rates and saturated incidence is presented. This model allows for the dynamics of certain infectious diseases, which pose a serious threat to human health and sustainable development, to be modeled and studied. These diseases impact various aspects of daily life and hinder socio-economic progress on a global scale. The mathematical study begins with demonstrating the existence and uniqueness of a global positive solution for the proposed model. Conditions under which the infection may either persist or become extinct are then discussed. Additionally, sufficient conditions for the existence and uniqueness of ergodic stationary distributions among the model’s solution are described by developing an appropriate stochastic Lyapunov function. The theoretical results are supported by numerical simulations, providing empirical validation of the mathematical conclusions and reinforcing the model’s relevance in studying infectious diseases and their impact on public health and sustainable development.