The disease characterized by a high blood glucose due to a deficiency of insulin production or failure of cells to respond to insulin is mellitus.

The disease characterized by high blood glucose levels due to a deficiency in insulin production or the failure of cells to respond to insulin is called Diabetes Mellitus. This condition can be categorized into two main types: Type 1 and Type 2 diabetes.

Type 1 diabetes is an autoimmune condition where the body's immune system attacks and destroys the insulin-producing cells in the pancreas. This results in the body not being able to produce insulin, leading to high blood glucose levels.

Type 2 diabetes, on the other hand, is a condition where the body either doesn't produce enough insulin or the cells in the body don't respond properly to the insulin that is produced (insulin resistance). This also leads to high blood glucose levels.

Both types of diabetes require medical intervention and management to prevent complications.