Inflammatory Bowel Diseases (IBDs), characterized by the autoimmune response of the immune system in the gastrointestinal tract, have Crohn’s Disease and Ulcerative Retocolitis as their main manifestations and although they have no cure, they can be treated based on the patient’s diagnosis. Studies throughout the 19th and 20th centuries revealed the genetic, immunological and environmental influence on these diseases, defining them as we know them today. Due to the complexity of IBDs, this research aims to understand the behavior of the disease by studying a mathematical model using Ordinary Differential Equations (ODEs) that allows us to mold and simulate the phenomenon of the disease. The work analyzes the model proposed by Park and Jung (2016), essentially using Differential and Integral Calculus to analyze the model and Optimal Control Theory to add control to it, divided into sections on IBD, preliminary concepts, analysis of the model with and without control, and numerical simulation to validate the model’s solutions.
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