The spread of infectious diseases is commonly known to trigger changes in human behaviour, which can subsequently affect epidemic outcomes. Our study introduces an integral epidemic model that links the contagiousness of a disease to the duration of infection, incorporating the impact of human behaviour on the progression of the disease. The model is behavioural in the sense that the force of infection includes the information index that describes the opinion-driven human behavioural changes. The information index contains a memory kernel to mimic how individuals maintain memory of the past values of the infection. We examine the basic properties of the model solution, investigate the existence and stability of equilibria, and apply a non-standard discretization technique to numerically solve the model. This technique, based on finite differences, preserves the key features of the original system, such as positivity, boundedness, and stability of equilibria. Through a comparative analysis with the continuous model, we highlight the efficacy of our numerical approach, particularly in long-time simulations. Our research underlines the crucial role of integrating behavioural dynamics into integral epidemic models, offering new insights into disease transmission and potential strategies for intervention. At the same time, it emphasizes the importance of having numerical methods that are efficient and qualitatively consistent with the model.