Mathematical study of the spread and blocking in inflammatory bowel disease

Ulcerative colitis (UC) is a chronic inflammatory bowel disease (IBD) with mechanisms that are still partially unclear. Unlike other types of IBD, inflammation in UC is limited to the inner lining of the large intestine and rectum, spreading continuously without breaks between affected areas, creating a uniform pattern of inflammation along the colon. In this paper, we develop a mathematical model based on a reaction-diffusion system to describe the inflammation caused by the interaction between a pathogen and immune cells in the context of UC. Our contributions are both theoretical and numerical. We demonstrate the existence of traveling wave solutions, showing how the disease progresses in a homogeneous environment. We then identify the conditions under which the spread of inflammatory waves can be stopped in a heterogeneous environment. Numerical simulations are used to highlight and validate these theoretical results.