In this article, we successfully generalize the SEIR (Susceptible, Exposed, Infected, and Recovered) model to account for random and intermittent instances of disease transmission. This model explains the increased probability of disease transmission during specific periods, particularly for seasonal illnesses. Furthermore, we investigate the conditions necessary to ensure at least one solution for this model by employing the measure of noncompactness. Following this, we introduce a new approach that examines convergence, after which we apply the proposed numerical method to study practical examples.