To solve this problem, we first need to define the decision variables, the objective function, and the constraints.

Let's denote:
- $x_b$ as the number of blue gummies purchased.
- $x_r$ as the number of red gummies purchased.

The objective is to minimize the total cost, which can be represented by the function: $2x_b + 3x_r$.

We have three constraints based on the patient's mineral requirements:
1. Calcium requirement: $3x_b + 4x_r \geq 40$
2. Vitamin A requirement: $2x_b + 3x_r \geq 45$
3. Iron requirement: $5x_b + 2x_r \geq 20$

Since the patient cannot purchase a negative number of gummies, we also have:
- $x_b \geq 0$
- $x_r \geq 0$

Given these definitions, we can now formulate the problem in Gurobi.

```python
from gurobipy import *

# Create a new model
m = Model("Gummy_Vitamins")

# Define the decision variables
x_b = m.addVar(vtype=GRB.INTEGER, name="blue_gummies")
x_r = m.addVar(vtype=GRB.INTEGER, name="red_gummies")

# Define the objective function
m.setObjective(2*x_b + 3*x_r, GRB.MINIMIZE)

# Define the constraints
m.addConstr(3*x_b + 4*x_r >= 40, "calcium_requirement")
m.addConstr(2*x_b + 3*x_r >= 45, "vitamin_a_requirement")
m.addConstr(5*x_b + 2*x_r >= 20, "iron_requirement")

# Optimize the model
m.optimize()

# Print the solution
if m.status == GRB.OPTIMAL:
    print(f"Optimal solution: {x_b.varName} = {x_b.x}, {x_r.varName} = {x_r.x}")
    print(f"Total cost: ${2*x_b.x + 3*x_r.x}")
else:
    print("No optimal solution found")
```