Appendicitis: Given the patient's symptoms of abdominal pain that started around the umbilicus and then localized to the right side, along with the recent onset of bloody stool, appendicitis is a strong consideration. The pain pattern and the presence of bloody stool, although not common, can be seen in appendicitis, especially if there's a perforation or an appendiceal abscess.

Inflammatory Bowel Disease (IBD): Conditions like Crohn's disease or ulcerative colitis could explain the abdominal pain, bloody stool, and the duration of symptoms. However, the absence of changes in bowel movements and the lack of other systemic symptoms make this less likely but still a consideration.
Intestinal Obstruction: Although less common, an obstruction could cause abdominal pain and, if involving the distal bowel, could lead to bloody stools. The absence of nausea, vomiting, and significant changes in bowel habits makes this less likely.
Diverticulitis: This condition can cause abdominal pain, usually on the left side, and bloody stools. However, the pain location in this patient is atypical for diverticulitis.

Mesenteric Ischemia: This is a life-threatening condition that requires immediate attention. Although the patient's presentation doesn't strongly suggest mesenteric ischemia, any condition causing abdominal pain and bloody stools warrants consideration of this diagnosis due to its high mortality rate if missed.
Ectopic Pregnancy (if applicable): Although the patient is a male, in female patients, ectopic pregnancy would be a critical diagnosis not to miss, presenting with abdominal pain and potentially bloody stools or vaginal bleeding.
Bowel Infarction: Similar to mesenteric ischemia, any condition that could lead to bowel infarction (e.g., volvulus, intussusception) needs to be considered due to its potential for severe outcomes.

Meckel's Diverticulum: A congenital anomaly that can cause gastrointestinal bleeding, usually in younger patients. It's less likely given the patient's age but could be considered in the differential for bloody stools.
Intussusception: More common in children but can occur in adults, typically with a lead point such as a tumor. It could cause abdominal pain and bloody stools but is less likely given the patient's age and presentation.
Angiodysplasia: Vascular malformations of the gut that can cause bleeding. They are more common in the elderly and might not fully explain the patient's pain pattern but could be a consideration for the bloody stool.